Chapter1.tex
Contents
1 Introduction 3
1.1 Machine Perception . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Related fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 The Sub-problems of Pattern Classification . . . . . . . . . . . . . . . 11
1.3.1 Feature Extraction . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.3 Overfitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.4 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.5 Prior Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.6 Missing Features . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.7 Mereology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.8 Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.9 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.10 Invariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.11 Evidence Pooling . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.12 Costs and Risks . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.13 Computational Complexity . . . . . . . . . . . . . . . . . . . . 16
1.4 Learning and Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 Supervised Learning . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.2 Unsupervised Learning . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.3 Reinforcement Learning . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Summary by Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Bibliographical and Historical Remarks . . . . . . . . . . . . . . . . . . . . 19
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1
2 CONTENTS
Chapter 1
Introduction
The ease with which we recognize a face, understand spoken words, read handwrit-ten characters, identify our car keys in our pocket by feel, and decide whether
an apple is ripe by its smell belies the astoundingly complex processes that underlie
these acts of pattern recognition. Pattern recognition — the act of taking in raw
data and taking an action based on the “category” of the pattern — has been crucial
for our survival, and over the past tens of millions of years we have evolved highly
sophisticated neural and cognitive systems for such tasks.
1.1 Machine Perception
It is natural that we should seek to design and build machines that can recognize
patterns. From automated speech recognition, fingerprint identification, optical char-
acter recognition, DNA sequence identification and much more, it is clear that reli-
able, accurate pattern recognition by machine would be immensely useful. Moreover,
in solving the myriad problems required to build such systems, we gain deeper un-
derstanding and appreciation for pattern recognition systems in the natural world —
most particularly in humans. For some applications, such as speech and visual recog-
nition, our design efforts may in fact be influenced by knowledge of how these are
solved in nature, both in the algorithms we employ and the design of special purpose
hardware.
1.2 An Example
To illustrate the complexity of some of the types of problems involved, let us consider
the following imaginary and somewhat fanciful example. Suppose that a fish packing
plant wants to automate the process of sorting incoming fish on a conveyor belt
according to species. As a pilot project it is decided to try to separate sea bass from
salmon using optical sensing. We set up a camera, take some sample images and begin
to note some physical differences between the two types of fish — length, lightness,
width, number and shape of fins, position of the mouth, and so on — and these suggest
features to explore for use in our classifier. We also notice noise or variations in the
3
4 CHAPTER 1. INTRODUCTION
images — variations in lighting, position of the fish on the conveyor, even “static”
due to the electronics of the camera itself.
Given that there truly are differences between the population of sea bass and that
of salmon, we view them as having different models — different descriptions, whichmodel
are typically mathematical in form. The overarching goal and approach in pattern
classification is to hypothesize the class of these models, process the sensed data
to eliminate noise (not due to the models), and for any sensed pattern choose the
model that corresponds best. Any techniques that further this aim should be in the
conceptual toolbox of the designer of pattern recognition systems.
Our prototype system to perform this very specific task might well have the form
shown in Fig. 1.1. First the camera captures an image of the fish. Next, the camera’s
signals are preprocessed to simplify subsequent operations without loosing relevantpre-
processing information. In particular, we might use a segmentation operation in which the images
segmentation
of different fish are somehow isolated from one another and from the background. The
information from a single fish is then sent to a feature extractor, whose purpose is to
feature
extraction
reduce the data by measuring certain “features” or “properties.” These features
(or, more precisely, the values of these features) are then passed to a classifier that
evaluates the evidence presented and makes a final decision as to the species.
The preprocessor might automatically adjust for average light level, or threshold
the image to remove the background of the conveyor belt, and so forth. For the
moment let us pass over how the images of the fish might be segmented and consider
how the feature extractor and classifier might be designed. Suppose somebody at the
fish plant tells us that a sea bass is generally longer than a salmon. These, then,
give us our tentative models for the fish: sea bass have some typical length, and this
is greater than that for salmon. Then length becomes an obvious feature, and we
might attempt to classify the fish merely by seeing whether or not the length l of
a fish exceeds some critical value l∗. To choose l∗ we could obtain some design or
training samples of the different types of fish, (somehow) make length measurements,training
samples and inspect the results.
Suppose that we do this, and obtain the histograms shown in Fig. 1.2. These
disappointing histograms bear out the statement that sea bass are somewhat longer
than salmon, on average, but it is clear that this single criterion is quite poor; no
matter how we choose l∗, we cannot reliably separate sea bass from salmon by length
alone.
Discouraged, but undeterred by these unpromising results, we try another feature
— the average lightness of the fish scales. Now we are very careful to eliminate
variations in illumination, since they can only obscure the models and corrupt our
new classifier. The resulting histograms, shown in Fig. 1.3, are much more satisfactory
— the classes are much better separated.
So far we have tacitly assumed that the consequences of our actions are equally
costly: deciding the fish was a sea bass when in fact it was a salmon was just as
undesirable as the converse. Such a symmetry in the cost is often, but not invariablycost
the case. For instance, as a fish packing company we may know that our customers
easily accept occasional pieces of tasty salmon in their cans labeled “sea bass,” but
they object vigorously if a piece of sea bass appears in their cans labeled “salmon.”
If we want to stay in business, we should adjust our decision boundary to avoid
antagonizing our customers, even if it means that more salmon makes its way into
the cans of sea bass. In this case, then, we should move our decision boundary x∗ to
smaller values of lightness, thereby reducing the number of sea bass that are classified
as salmon (Fig. 1.3). The more our customers object to getting sea bass with their
1.2. AN EXAMPLE 5
Figure 1.1: The objects to be classified are first sensed by a transducer (camera),
whose signals are preprocessed, then the features extracted and finally the classifi-
cation emitted (here either “salmon” or “sea bass”). Although the information flow
is often chosen to be from the source to the classifier (“bottom-up”), some systems
employ “top-down” flow as well, in which earlier levels of processing can be altered
based on the tentative or preliminary response in later levels (gray arrows). Yet others
combine two or more stages into a unified step, such as simultaneous segmentation
and feature extraction.
salmon — i.e., the more costly this type of error — the lower we should set the decision
threshold x∗ in Fig. 1.3.
Such considerations suggest that there is an overall single cost associated with our
decision, and our true task is to make a decision rule (i.e., set a decision boundary)
so as to minimize such a cost. This is the central task of decision theory of which decision
theorypattern classification is perhaps the most important subfield.
Even if we know the costs associated with our decisions and choose the optimal
decision boundary x∗, we may be dissatisfied with the resulting performance. Our
first impulse might be to seek yet a different feature on which to separate the fish.
Let us assume, though, that no other single visual feature yields better performance
than that based on lightness. To improve recognition, then, we must resort to the use
6 CHAPTER 1. INTRODUCTION
salmon sea bass
Length
Count
l*
0
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8
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5 10 2015 25
Figure 1.2: Histograms for the length feature for the two categories. No single thresh-
old value l∗ (decision boundary) will serve to unambiguously discriminate between
the two categories; using length alone, we will have some errors. The value l∗ marked
will lead to the smallest number of errors, on average.
2 4 6 8 10
0
2
4
6
8
10
12
14
Lightness
Count
x*
salmon sea bass
Figure 1.3: Histograms for the lightness feature for the two categories. No single
threshold value x∗ (decision boundary) will serve to unambiguously discriminate be-
tween the two categories; using lightness alone, we will have some errors. The value
x∗ marked will lead to the smallest number of errors, on average.
1.2. AN EXAMPLE 7
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Width
Lightness
salmon sea bass
Figure 1.4: The two features of lightness and width for sea bass and salmon. The
dark line might serve as a decision boundary of our classifier. Overall classification
error on the data shown is lower than if we use only one feature as in Fig. 1.3, but
there will still be some errors.
of more than one feature at a time.
In our search for other features, we might try to capitalize on the observation that
sea bass are typically wider than salmon. Now we have two features for classifying
fish — the lightness x1 and the width x2. If we ignore how these features might be
measured in practice, we realize that the feature extractor has thus reduced the image
of each fish to a point or feature vector x in a two-dimensional feature space, where
x =
[
x1
x2
]
.
Our problem now is to partition the feature space into two regions, where for all
patterns in one region we will call the fish a sea bass, and all points in the other we
call it a salmon. Suppose that we measure the feature vectors for our samples and
obtain the scattering of points shown in Fig. 1.4. This plot suggests the following rule
for separating the fish: Classify the fish as sea bass if its feature vector falls above the
decision boundary shown, and as salmon otherwise. decision
boundaryThis rule appears to do a good job of separating our samples and suggests that
perhaps incorporating yet more features would be desirable. Besides the lightness
and width of the fish, we might include some shape parameter, such as the vertex
angle of the dorsal fin, or the placement of the eyes (as expressed as a proportion of
the mouth-to-tail distance), and so on. How do we know beforehand which of these
features will work best? Some features might be redundant: for instance if the eye
color of all fish correlated perfectly with width, then classification performance need
not be improved if we also include eye color as a feature. Even if the difficulty or
computational cost in attaining more features is of no concern, might we ever have
too many features?
Suppose that other features are too expensive or expensive to measure, or provide
little improvement (or possibly even degrade the performance) in the approach de-
scribed above, and that we are forced to make our decision based on the two features
in Fig. 1.4. If our models were extremely complicated, our classifier would have a
decision boundary more complex than the simple straight line. In that case all the
8 CHAPTER 1. INTRODUCTION
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Width
Lightness
?
salmon sea bass
Figure 1.5: Overly complex models for the fish will lead to decision boundaries that are
complicated. While such a decision may lead to perfect classification of our training
samples, it would lead to poor performance on future patterns. The novel test point
marked ? is evidently most likely a salmon, whereas the complex decision boundary
shown leads it to be misclassified as a sea bass.
training patterns would be separated perfectly, as shown in Fig. 1.5. With such a
“solution,” though, our satisfaction would be premature because the central aim of
designing a classifier is to suggest actions when presented with novel patterns, i.e.,
fish not yet seen. This is the issue of generalization. It is unlikely that the complexgeneral-
ization decision boundary in Fig. 1.5 would provide good generalization, since it seems to be
“tuned” to the particular training samples, rather than some underlying characteris-
tics or true model of all the sea bass and salmon that will have to be separated.
Naturally, one approach would be to get more training samples for obtaining a
better estimate of the true underlying characteristics, for instance the probability
distributions of the categories. In most pattern recognition problems, however, the
amount of such data we can obtain easily is often quite limited. Even with a vast
amount of training data in a continuous feature space though, if we followed the
approach in Fig. 1.5 our classifier would give a horrendously complicated decision
boundary — one that would be unlikely to do well on novel patterns.
Rather, then, we might seek to “simplify” the recognizer, motivated by a belief
that the underlying models will not require a decision boundary that is as complex as
that in Fig. 1.5. Indeed, we might be satisfied with the slightly poorer performance
on the training samples if it means that our classifier will have better performance
on novel patterns.∗ But if designing a very complex recognizer is unlikely to give
good generalization, precisely how should we quantify and favor simpler classifiers?
How would our system automatically determine that the simple curve in Fig. 1.6
is preferable to the manifestly simpler straight line in Fig. 1.4 or the complicated
boundary in Fig. 1.5? Assuming that we somehow manage to optimize this tradeoff,
can we then predict how well our system will generalize to new patterns? These are
some of the central problems in statistical pattern recognition.
For the same incoming patterns, we might need to use a drastically different cost
∗ The philosophical underpinnings of this approach derive from William of Occam (1284-1347?), who
advocated favoring simpler explanations over those that are needlessly complicated — Entia non
sunt multiplicanda praeter necessitatem (“Entities are not to be multiplied without necessity”).
Decisions based on overly complex models often lead to lower accuracy of the classifier.
1.2. AN EXAMPLE 9
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22
Width
Lightness
salmon sea bass
Figure 1.6: The decision boundary shown might represent the optimal tradeoff be-
tween performance on the training set and simplicity of classifier.
function, and this will lead to different actions altogether. We might, for instance,
wish instead to separate the fish based on their sex — all females (of either species)
from all males if we wish to sell roe. Alternatively, we might wish to cull the damaged
fish (to prepare separately for cat food), and so on. Different decision tasks may
require features and yield boundaries quite different from those useful for our original
categorization problem.
This makes it quite clear that our decisions are fundamentally task or cost specific,
and that creating a single general purpose artificial pattern recognition device — i.e.,
one capable of acting accurately based on a wide variety of tasks — is a profoundly
difficult challenge. This, too, should give us added appreciation of the ability of
humans to switch rapidly and fluidly between pattern recognition tasks.
Since classification is, at base, the task of recovering the model that generated the
patterns, different classification techniques are useful depending on the type of candi-
date models themselves. In statistical pattern recognition we focus on the statistical
properties of the patterns (generally expressed in probability densities), and this will
command most of our attention in this book. Here the model for a pattern may be a
single specific set of features, though the actual pattern sensed has been corrupted by
some form of random noise. Occasionally it is claimed that neural pattern recognition
(or neural network pattern classification) should be considered its own discipline, but
despite its somewhat different intellectual pedigree, we will consider it a close descen-
dant of statistical pattern recognition, for reasons that will become clear. If instead
the model consists of some set of crisp logical rules, then we employ the methods of
syntactic pattern recognition, where rules or grammars describe our decision. For ex-
ample we might wish to classify an English sentence as grammatical or not, and here
statistical descriptions (word frequencies, word correlations, etc.) are inapapropriate.
It was necessary in our fish example to choose our features carefully, and hence
achieve a representation (as in Fig. 1.6) that enabled reasonably successful pattern
classification. A central aspect in virtually every pattern recognition problem is that
of achieving such a “good” representation, one in which the structural relationships
among the components is simply and naturally revealed, and one in which the true
(unknown) model of the patterns can be expressed. In some cases patterns should be
represented as vectors of real-valued numbers, in others ordered lists of attributes, in
yet others descriptions of parts and their relations, and so forth. We seek a represen-
10 CHAPTER 1. INTRODUCTION
tation in which the patterns that lead to the same action are somehow “close” to one
another, yet “far” from those that demand a different action. The extent to which we
create or learn a proper representation and how we quantify near and far apart will
determine the success of our pattern classifier. A number of additional characteris-
tics are desirable for the representation. We might wish to favor a small number of
features, which might lead to simpler decision regions, and a classifier easier to train.
We might also wish to have features that are robust, i.e., relatively insensitive to noise
or other errors. In practical applications we may need the classifier to act quickly, or
use few electronic components, memory or processing steps.
A central technique, when we have insufficient training data, is to incorporate
knowledge of the problem domain. Indeed the less the training data the more impor-
tant is such knowledge, for instance how the patterns themselves were produced. One
method that takes this notion to its logical extreme is that of analysis by synthesis,analysis
by
synthesis
where in the ideal case one has a model of how each pattern is generated. Con-
sider speech recognition. Amidst the manifest acoustic variability among the possible
“dee”s that might be uttered by different people, one thing they have in common is
that they were all produced by lowering the jaw slightly, opening the mouth, placing
the tongue tip against the roof of the mouth after a certain delay, and so on. We
might assume that “all” the acoustic variation is due to the happenstance of whether
the talker is male or female, old or young, with different overall pitches, and so forth.
At some deep level, such a “physiological” model (or so-called “motor” model) for
production of the utterances is appropriate, and different (say) from that for “doo”
and indeed all other utterances. If this underlying model of production can be deter-
mined from the sound (and that is a very big if ), then we can classify the utterance by
how it was produced. That is to say, the production representation may be the “best”
representation for classification. Our pattern recognition systems should then analyze
(and hence classify) the input pattern based on how one would have to synthesize
that pattern. The trick is, of course, to recover the generating parameters from the
sensed pattern.
Consider the difficulty in making a recognizer of all types of chairs — standard
office chair, contemporary living room chair, beanbag chair, and so forth — based on
an image. Given the astounding variety in the number of legs, material, shape, and
so on, we might despair of ever finding a representation that reveals the unity within
the class of chair. Perhaps the only such unifying aspect of chairs is functional: a
chair is a stable artifact that supports a human sitter, including back support. Thus
we might try to deduce such functional properties from the image, and the property
“can support a human sitter” is very indirectly related to the orientation of the larger
surfaces, and would need to be answered in the affirmative even for a beanbag chair.
Of course, this requires some reasoning about the properties and naturally touches
upon computer vision rather than pattern recognition proper.
Without going to such extremes, many real world pattern recognition systems seek
to incorporate at least some knowledge about the method of production of the pat-
terns or their functional use in order to insure a good representation, though of course
the goal of the representation is classification, not reproduction. For instance, in op-
tical character recognition (OCR) one might confidently assume that handwritten
characters are written as a sequence of strokes, and first try to recover a stroke rep-
resentation from the sensed image, and then deduce the character from the identified
strokes.
1.3. THE SUB-PROBLEMS OF PATTERN CLASSIFICATION 11
1.2.1 Related fields
Pattern classification differs from classical statistical hypothesis testing, wherein the
sensed data are used to decide whether or not to reject a null hypothesis in favor of
some alternative hypothesis. Roughly speaking, if the probability of obtaining the
data given some null hypothesis falls below a “significance” threshold, we reject the
null hypothesis in favor of the alternative. For typical values of this criterion, there is
a strong bias or predilection in favor of the null hypothesis; even though the alternate
hypothesis may be more probable, we might not be able to reject the null hypothesis.
Hypothesis testing is often used to determine whether a drug is effective, where the
null hypothesis is that it has no effect. Hypothesis testing might be used to determine
whether the fish on the conveyor belt belong to a single class (the null hypothesis) or
from two classes (the alternative). In contrast, given some data, pattern classification
seeks to find the most probable hypothesis from a set of hypotheses — “this fish is
probably a salmon.”
Pattern classification differs, too, from image processing. In image processing, the image
processinginput is an image and the output is an image. Image processing steps often include
rotation, contrast enhancement, and other transformations which preserve all the
original information. Feature extraction, such as finding the peaks and valleys of the
intensity, lose information (but hopefully preserve everything relevant to the task at
hand.)
As just described, feature extraction takes in a pattern and produces feature values.
The number of features is virtually always chosen to be fewer than the total necessary
to describe the complete target of interest, and this leads to a loss in information. In
acts of associative memory, the system takes in a pattern and emits another pattern associative
memorywhich is representative of a general group of patterns. It thus reduces the information
somewhat, but rarely to the extent that pattern classification does. In short, because
of the crucial role of a decision in pattern recognition information, it is fundamentally
an information reduction process. The classification step represents an even more
radical loss of information, reducing the original several thousand bits representing
all the color of each of several thousand pixels down to just a few bits representing
the chosen category (a single bit in our fish example.)
1.3 The Sub-problems of Pattern Classification
We have alluded to some of the issues in pattern classification and we now turn to a
more explicit list of them. In practice, these typically require the bulk of the research
and development effort. Many are domain or problem specific, and their solution will
depend upon the knowledge and insights of the designer. Nevertheless, a few are of
sufficient generality, difficulty, and interest that they warrant explicit consideration.
1.3.1 Feature Extraction
The conceptual boundary between feature extraction and classification proper is some-
what arbitrary: an ideal feature extractor would yield a representation that makes
the job of the classifier trivial; conversely, an omnipotent classifier would not need the
help of a sophisticated feature extractor. The distinction is forced upon us for practi-
cal, rather than theoretical reasons. Generally speaking, the task of feature extraction
is much more problem and domain dependent than is classification proper, and thus
requires knowledge of the domain. A good feature extractor for sorting fish would
12 CHAPTER 1. INTRODUCTION
surely be of little use for identifying fingerprints, or classifying photomicrographs of
blood cells. How do we know which features are most promising? Are there ways to
automatically learn which features are best for the classifier? How many shall we use?
1.3.2 Noise
The lighting of the fish may vary, there could be shadows cast by neighboring equip-
ment, the conveyor belt might shake — all reducing the reliability of the feature values
actually measured. We define noise very general terms: any property of the sensed
pattern due not to the true underlying model but instead to randomness in the world
or the sensors. All non-trivial decision and pattern recognition problems involve noise
in some form. In some cases it is due to the transduction in the signal and we may
consign to our preprocessor the role of cleaning up the signal, as for instance visual
noise in our video camera viewing the fish. An important problem is knowing some-
how whether the variation in some signal is noise or instead to complex underlying
models of the fish. How then can we use this information to improve our classifier?
1.3.3 Overfitting
In going from Fig 1.4 to Fig. 1.5 in our fish classification problem, we were, implicitly,
using a more complex model of sea bass and of salmon. That is, we were adjusting
the complexity of our classifier. While an overly complex model may allow perfect
classification of the training samples, it is unlikely to give good classification of novel
patterns — a situation known as overfitting. One of the most important areas of re-
search in statistical pattern classification is determining how to adjust the complexity
of the model — not so simple that it cannot explain the differences between the cat-
egories, yet not so complex as to give poor classification on novel patterns. Are there
principled methods for finding the best (intermediate) complexity for a classifier?
1.3.4 Model Selection
We might have been unsatisfied with the performance of our fish classifier in Figs. 1.4
& 1.5, and thus jumped to an entirely different class of model, for instance one based
on some function of the number and position of the fins, the color of the eyes, the
weight, shape of the mouth, and so on. How do we know when a hypothesized model
differs significantly from the true model underlying our patterns, and thus a new
model is needed? In short, how are we to know to reject a class of models and try
another one? Are we as designers reduced to random and tedious trial and error in
model selection, never really knowing whether we can expect improved performance?
Or might there be principled methods for knowing when to jettison one class of models
and invoke another? Can we automate the process?
1.3.5 Prior Knowledge
In one limited sense, we have already seen how prior knowledge — about the lightness
of the different fish categories helped in the design of a classifier by suggesting a
promising feature. Incorporating prior knowledge can be far more subtle and difficult.
In some applications the knowledge ultimately derives from information about the
production of the patterns, as we saw in analysis-by-synthesis. In others the knowledge
may be about the form of the underlying categories, or specific attributes of the
patterns, such as the fact that a face has two eyes, one nose, and so on.
1.3. THE SUB-PROBLEMS OF PATTERN CLASSIFICATION 13
1.3.6 Missing Features
Suppose that during classification, the value of one of the features cannot be deter-
mined, for example the width of the fish because of occlusion by another fish (i.e., occlusion
the other fish is in the way). How should the categorizer compensate? Since our
two-feature recognizer never had a single-variable threshold value x∗ determined in
anticipation of the possible absence of a feature (cf., Fig. 1.3), how shall it make the
best decision using only the feature present? The naive method, of merely assuming
that the value of the missing feature is zero or the average of the values for the train-
ing patterns, is provably non-optimal. Likewise we occasionally have missing features
during the creation or learning in our recognizer. How should we train a classifier or
use one when some features are missing?
1.3.7 Mereology
We effortlessly read a simple word such as BEATS. But consider this: Why didn’t
we read instead other words that are perfectly good subsets of the full pattern, such
as BE, BEAT, EAT, AT, and EATS? Why don’t they enter our minds, unless
explicitly brought to our attention? Or when we saw the B why didn’t we read a P
or an I, which are “there” within the B? Conversely, how is it that we can read the
two unsegmented words in POLOPONY — without placing the entire input into a
single word category?
This is the problem of subsets and supersets — formally part of mereology, the
study of part/whole relationships. It is closely related to that of prior knowledge and
segmentation. In short, how do we recognize or group together the “proper” number
of elements — neither too few nor too many? It appears as though the best classifiers
try to incorporate as much of the input into the categorization as “makes sense,” but
not too much. How can this be done?
1.3.8 Segmentation
In our fish example, we have tacitly assumed that the fish were isolated, separate
on the conveyor belt. In practice, they would often be abutting or overlapping, and
our system would have to determine where one fish ends and the next begins — the
individual patterns have to be segmented. If we have already recognized the fish then
it would be easier to segment them. But how can we segment the images before they
have been categorized or categorize them before they have been segmented? It seems
we need a way to know when we have switched from one model to another, or to know
when we just have background or “no category.” How can this be done?
Segmentation is one of the deepest problems in automated speech recognition.
We might seek to recognize the individual sounds (e.g., phonemes, such as “ss,” “k,”
…), and then put them together to determine the word. But consider two nonsense
words, “sklee” and “skloo.” Speak them aloud and notice that for “skloo” you push
your lips forward (so-called “rounding” in anticipation of the upcoming “oo”) before
you utter the “ss.” Such rounding influences the sound of the “ss,” lowering the
frequency spectrum compared to the “ss” sound in “sklee” — a phenomenon known
as anticipatory coarticulation. Thus, the “oo” phoneme reveals its presence in the “ss”
earlier than the “k” and “l” which nominally occur before the “oo” itself! How do we
segment the “oo” phoneme from the others when they are so manifestly intermingled?
Or should we even try? Perhaps we are focusing on groupings of the wrong size, and
that the most useful unit for recognition is somewhat larger, as we saw in subsets and
14 CHAPTER 1. INTRODUCTION
supersets (Sect. 1.3.7). A related problem occurs in connected cursive handwritten
character recognition: How do we know where one character “ends” and the next one
“begins”?
1.3.9 Context
We might be able to use context — input-dependent information other than from the
target pattern itself — to improve our recognizer. For instance, it might be known
for our fish packing plant that if we are getting a sequence of salmon, that it is highly
likely that the next fish will be a salmon (since it probably comes from a boat that just
returned from a fishing area rich in salmon). Thus, if after a long series of salmon our
recognizer detects an ambiguous pattern (i.e., one very close to the nominal decision
boundary), it may nevertheless be best to categorize it too as a salmon. We shall see
how such a simple correlation among patterns — the most elementary form of context
— might be used to improve recognition. But how, precisely, should we incorporate
such information?
Context can be highly complex and abstract. The utterance “jeetyet?” may seem
nonsensical, unless you hear it spoken by a friend in the context of the cafeteria at
lunchtime — “did you eat yet?” How can such a visual and temporal context influence
your speech recognition?
1.3.10 Invariances
In seeking to achieve an optimal representation for a particular pattern classification
task, we confront the problem of invariances. In our fish example, the absolute
position on the conveyor belt is irrelevant to the category and thus our representation
should also be insensitive to absolute position of the fish. Here we seek a representation
that is invariant to the transformation of translation (in either horizontal or vertical
directions). Likewise, in a speech recognition problem, it might be required only that
we be able to distinguish between utterances regardless of the particular moment they
were uttered; here the “translation” invariance we must ensure is in time.
The “model parameters” describing the orientation of our fish on the conveyor
belt are horrendously complicated — due as they are to the sloshing of water, the
bumping of neighboring fish, the shape of the fish net, etc. — and thus we give up hope
of ever trying to use them. These parameters are irrelevant to the model parameters
that interest us anyway, i.e., the ones associated with the differences between the fish
categories. Thus here we try to build a classifier that is invariant to transformations
such as rotation.
The orientation of the fish on the conveyor belt is irrelevant to its category. Hereorientation
the transformation of concern is a two-dimensional rotation about the camera’s line
of sight. A more general invariance would be for rotations about an arbitrary line in
three dimensions. The image of even such a “simple” object as a coffee cup undergoes
radical variation as the cup is rotated to an arbitrary angle — the handle may become
hidden, the bottom of the inside volume come into view, the circular lip appear oval or
a straight line or even obscured, and so forth. How might we insure that our pattern
recognizer is invariant to such complex changes?
The overall size of an image may be irrelevant for categorization. Such differencessize
might be due to variation in the range to the object; alternatively we may be genuinely
unconcerned with differences between sizes — a young, small salmon is still a salmon.
1.3. THE SUB-PROBLEMS OF PATTERN CLASSIFICATION 15
For patterns that have inherent temporal variation, we may want our recognizer
to be insensitive to the rate at which the pattern evolves. Thus a slow hand wave and rate
a fast hand wave may be considered as equivalent. Rate variation is a deep problem
in speech recognition, of course; not only do different individuals talk at different
rates, but even a single talker may vary in rate, causing the speech signal to change
in complex ways. Likewise, cursive handwriting varies in complex ways as the writer
speeds up — the placement of dots on the i’s, and cross bars on the t’s and f’s, are
the first casualties of rate increase, while the appearance of l’s and e’s are relatively
inviolate. How can we make a recognizer that changes its representations for some
categories differently from that for others under such rate variation?
A large number of highly complex transformations arise in pattern recognition,
and many are domain specific. We might wish to make our handwritten optical
character recognizer insensitive to the overall thickness of the pen line, for instance.
Far more severe are transformations such as non-rigid deformations that arise in three- deformation
dimensional object recognition, such as the radical variation in the image of your hand
as you grasp an object or snap your fingers. Similarly, variations in illumination or
the complex effects of cast shadows may need to be taken into account.
The symmetries just described are continuous — the pattern can be translated,
rotated, sped up, or deformed by an arbitrary amount. In some pattern recognition
applications other — discrete — symmetries are relevant, such as flips left-to-right, discrete
symmetryor top-to-bottom.
In all of these invariances the problem arises: How do we determine whether an
invariance is present? How do we efficiently incorporate such knowledge into our
recognizer?
1.3.11 Evidence Pooling
In our fish example we saw how using multiple features could lead to improved recog-
nition. We might imagine that we could do better if we had several component
classifiers. If these categorizers agree on a particular pattern, there is no difficulty.
But suppose they disagree. How should a “super” classifier pool the evidence from the
component recognizers to achieve the best decision?
Imagine calling in ten experts for determining if a particular fish is diseased or
not. While nine agree that the fish is healthy, one expert does not. Who is right?
It may be that the lone dissenter is the only one familiar with the particular very
rare symptoms in the fish, and is in fact correct. How would the “super” categorizer
know when to base a decision on a minority opinion, even from an expert in one small
domain who is not well qualified to judge throughout a broad range of problems?
1.3.12 Costs and Risks
We should realize that a classifier rarely exists in a vacuum. Instead, it is generally
to be used to recommend actions (put this fish in this bucket, put that fish in that
bucket), each action having an associated cost or risk. Conceptually, the simplest
such risk is the classification error: what percentage of new patterns are called the
wrong category. However the notion of risk is far more general, as we shall see. We
often design our classifier to recommend actions that minimize some total expected
cost or risk. Thus, in some sense, the notion of category itself derives from the cost
or task. How do we incorporate knowledge about such risks and how will they affect
our classification decision?
16 CHAPTER 1. INTRODUCTION
Finally, can we estimate the total risk and thus tell whether our classifier is ac-
ceptable even before we field it? Can we estimate the lowest possible risk of any
classifier, to see how close ours meets this ideal, or whether the problem is simply too
hard overall?
1.3.13 Computational Complexity
Some pattern recognition problems can be solved using algorithms that are highly
impractical. For instance, we might try to hand label all possible 20 × 20 binary pixel
images with a category label for optical character recognition, and use table lookup
to classify incoming patterns. Although we might achieve error-free recognition, the
labeling time and storage requirements would be quite prohibitive since it would
require a labeling each of 220×20 ≈ 10120 patterns. Thus the computational complexity
of different algorithms is of importance, especially for practical applications.
In more general terms, we may ask how an algorithm scales as a function of the
number of feature dimensions, or the number of patterns or the number of categories.
What is the tradeoff between computational ease and performance? In some prob-
lems we know we can design an excellent recognizer, but not within the engineering
constraints. How can we optimize within such constraints? We are typically less
concerned with the complexity of learning, which is done in the laboratory, than the
complexity of making a decision, which is done with the fielded application. While
computational complexity generally correlates with the complexity of the hypothe-
sized model of the patterns, these two notions are conceptually different.
This section has catalogued some of the central problems in classification. It has
been found that the most effective methods for developing classifiers involve learning
from examples, i.e., from a set of patterns whose category is known. Throughout this
book, we shall see again and again how methods of learning relate to these central
problems, and are essential in the building of classifiers.
1.4 Learning and Adaptation
In the broadest sense, any method that incorporates information from training sam-
ples in the design of a classifier employs learning. Because nearly all practical or
interesting pattern recognition problems are so hard that we cannot guess classifi-
cation decision ahead of time, we shall spend the great majority of our time here
considering learning. Creating classifiers then involves posit some general form of
model, or form of the classifier, and using training patterns to learn or estimate the
unknown parameters of the model. Learning refers to some form of algorithm for
reducing the error on a set of training data. A range of gradient descent algorithms
that alter a classifier’s parameters in order to reduce an error measure now permeate
the field of statistical pattern recognition, and these will demand a great deal of our
attention. Learning comes in several general forms.
1.4.1 Supervised Learning
In supervised learning, a teacher provides a category label or cost for each pattern
in a training set, and we seek to reduce the sum of the costs for these patterns.
How can we be sure that a particular learning algorithm is powerful enough to learn
the solution to a given problem and that it will be stable to parameter variations?
1.5. CONCLUSION 17
How can we determine if it will converge in finite time, or scale reasonably with the
number of training patterns, the number of input features or with the perplexity of
the problem? How can we insure that the learning algorithm appropriately favors
“simple” solutions (as in Fig. 1.6) rather than complicated ones (as in Fig. 1.5)?
1.4.2 Unsupervised Learning
In unsupervised learning or clustering there is no explicit teacher, and the system forms
clusters or “natural groupings” of the input patterns. “Natural” is always defined
explicitly or implicitly in the clustering system itself, and given a particular set of
patterns or cost function, different clustering algorithms lead to different clusters.
Often the user will set the hypothesized number of different clusters ahead of time,
but how should this be done? How do we avoid inappropriate representations?
1.4.3 Reinforcement Learning
The most typical way to train a classifier is to present an input, compute its tentative
category label, and use the known target category label to improve the classifier. For
instance, in optical character recognition, the input might be an image of a character,
the actual output of the classifier the category label “R,” and the desired output a “B.”
In reinforcement learning or learning with a critic, no desired category signal is given; critic
instead, the only teaching feedback is that the tentative category is right or wrong.
This is analogous to a critic who merely states that something is right or wrong, but
does not say specifically how it is wrong. (Thus only binary feedback is given to the
classifier; reinforcement learning also describes the case where a single scalar signal,
say some number between 0 and 1, is given by the teacher.) In pattern classification,
it is most common that such reinforcement is binary — either the tentative decision
is correct or it is not. (Of course, if our problem involves just two categories and
equal costs for errors, then learning with a critic is equivalent to standard supervised
learning.) How can the system learn which are important from such non-specific
feedback?
1.5 Conclusion
At this point the reader may be overwhelmed by the number, complexity and mag-
nitude of these sub-problems. Further, these sub-problems are rarely addressed in
isolation and they are invariably interrelated. Thus for instance in seeking to reduce
the complexity of our classifier, we might affect its ability to deal with invariance. We
point out, though, that the good news is at least three-fold: 1) there is an “existence
proof” that many of these problems can indeed be solved — as demonstrated by hu-
mans and other biological systems, 2) mathematical theories solving some of these
problems have in fact been discovered, and finally 3) there remain many fascinating
unsolved problems providing opportunities for progress.
Summary by Chapters
The overall organization of this book is to address first those cases where a great deal
of information about the models is known (such as the probability densities, category
labels, …) and to move, chapter by chapter, toward problems where the form of the
18 CHAPTER 1. INTRODUCTION
distributions are unknown and even the category membership of training patterns is
unknown. We begin in Chap. ?? (Bayes decision theory) by considering the ideal case
in which the probability structure underlying the categories is known perfectly. While
this sort of situation rarely occurs in practice, it permits us to determine the optimal
(Bayes) classifier against which we can compare all other methods. Moreover in some
problems it enables us to predict the error we will get when we generalize to novel
patterns. In Chap. ?? (Maximum Likelihood and Bayesian Parameter Estimation)
we address the case when the full probability structure underlying the categories
is not known, but the general forms of their distributions are — i.e., the models.
Thus the uncertainty about a probability distribution is represented by the values of
some unknown parameters, and we seek to determine these parameters to attain the
best categorization. In Chap. ?? (Nonparametric techniques) we move yet further
from the Bayesian ideal, and assume that we have no prior parameterized knowledge
about the underlying probability structure; in essence our classification will be based
on information provided by training samples alone. Classic techniques such as the
nearest-neighbor algorithm and potential functions play an important role here.
We then in Chap. ?? (Linear Discriminant Functions) return somewhat toward
the general approach of parameter estimation. We shall assume that the so-called
“discriminant functions” are of a very particular form — viz., linear — in order to de-
rive a class of incremental training rules. Next, in Chap. ?? (Nonlinear Discriminants
and Neural Networks) we see how some of the ideas from such linear discriminants
can be extended to a class of very powerful algorithms such as backpropagation and
others for multilayer neural networks; these neural techniques have a range of use-
ful properties that have made them a mainstay in contemporary pattern recognition
research. In Chap. ?? (Stochastic Methods) we discuss simulated annealing by the
Boltzmann learning algorithm and other stochastic methods. We explore the behavior
of such algorithms with regard to the matter of local minima that can plague other
neural methods. Chapter ?? (Non-metric Methods) moves beyond models that are
statistical in nature to ones that can be best described by (logical) rules. Here we
discuss tree-based algorithms such as CART (which can also be applied to statistical
data) and syntactic based methods, such as grammar based, which are based on crisp
rules.
Chapter ?? (Theory of Learning) is both the most important chapter and the
most difficult one in the book. Some of the results described there, such as the
notion of capacity, degrees of freedom, the relationship between expected error and
training set size, and computational complexity are subtle but nevertheless crucial
both theoretically and practically. In some sense, the other chapters can only be
fully understood (or used) in light of the results presented here; you cannot expect to
solve important pattern classification problems without using the material from this
chapter.
We conclude in Chap. ?? (Unsupervised Learning and Clustering), by addressing
the case when input training patterns are not labeled, and that our recognizer must
determine the cluster structure. We also treat a related problem, that of learning
with a critic, in which the teacher provides only a single bit of information during
the presentation of a training pattern — “yes,” that the classification provided by the
recognizer is correct, or “no,” it isn’t. Here algorithms for reinforcement learning will
be presented.
1.5. BIBLIOGRAPHICAL AND HISTORICAL REMARKS 19
Bibliographical and Historical Remarks
Classification is among the first crucial steps in making sense of the blooming buzzing
confusion of sensory data that intelligent systems confront. In the western world,
the foundations of pattern recognition can be traced to Plato [2], later extended by
Aristotle [1], who distinguished between an “essential property” (which would be
shared by all members in a class or “natural kind” as he put it) from an “accidental
property” (which could differ among members in the class). Pattern recognition can
be cast as the problem of finding such essential properties of a category. It has been a
central theme in the discipline of philosophical epistemology, the study of the nature
of knowledge. A more modern treatment of some philosophical problems of pattern
recognition, relating to the technical matter in the current book can be found in
[22, 4, 18]. In the eastern world, the first Zen patriarch, Bodhidharma, would point
at things and demand students to answer “What is that?” as a way of confronting the
deepest issues in mind, the identity of objects, and the nature of classification and
decision. A delightful and particularly insightful book on the foundations of artificial
intelligence, including pattern recognition, is [9].
Early technical treatments by Minsky [14] and Rosenfeld [16] are still valuable, as
are a number of overviews and reference books [5]. The modern literature on decision
theory and pattern recognition is now overwhelming, and comprises dozens of journals,
thousands of books and conference proceedings and innumerable articles; it continues
to grow rapidly. While some disciplines such as statistics [7], machine learning [17]
and neural networks [8], expand the foundations of pattern recognition, others, such
as computer vision [6, 19] and speech recognition [15] rely on it heavily. Perceptual
Psychology, Cognitive Science [12], Psychobiology [21] and Neuroscience [10] analyze
how pattern recognition is achieved in humans and other animals. The extreme view
that everything in human cognition — including rule-following and logic — can be
reduced to pattern recognition is presented in [13]. Pattern recognition techniques
have been applied in virtually every scientific and technical discipline.
20 CHAPTER 1. INTRODUCTION
Bibliography
[1] Aristotle, Robin Waterfield, and David Bostock. Physics. Oxford University
Press, Oxford, UK, 1996.
[2] Allan Bloom. The Republic of Plato. Basic Books, New York, NY, 2nd edition,
1991.
[3] Bodhidharma. The Zen Teachings of Bodhidharma. North Point Press, San
Francisco, CA, 1989.
[4] Mikhail M. Bongard. Pattern Recognition. Spartan Books, Washington, D.C.,
1970.
[5] Chi-hau Chen, Louis François Pau, and Patrick S. P. Wang, editors. Handbook
of Pattern Recognition & Computer Vision. World Scientific, Singapore, 2nd
edition, 1993.
[6] Marty Fischler and Oscar Firschein. Readings in Computer Vision: Issues, Prob-
lems, Principles and Paradigms. Morgan Kaufmann, San Mateo, CA, 1987.
[7] Keinosuke Fukunaga. Introduction to Statistical Pattern Recognition. Academic
Press, New York, NY, 2nd edition, 1990.
[8] John Hertz, Anders Krogh, and Richard G. Palmer. Introduction to the Theory
of Neural Computation. Addison-Wesley Publishing Company, Redwood City,
CA, 1991.
[9] Douglas Hofstadter. Gödel, Escher, Bach: an Eternal Golden Braid. Basic
Books, Inc., New York, NY, 1979.
[10] Eric R. Kandel and James H. Schwartz. Principles of Neural Science. Elsevier,
New York, NY, 2nd edition, 1985.
[11] Immanuel Kant. Critique of Pure Reason. Prometheus Books, New York, NY,
1990.
[12] George F. Luger. Cognitive Science: The Science of Intelligent Systems. Aca-
demic Press, New York, NY, 1994.
[13] Howard Margolis. Patterns, Thinking, and Cognition: A Theory of Judgement.
University of Chicago Press, Chicago, IL, 1987.
[14] Marvin Minsky. Steps toward artificial intelligence. Proceedings of the IEEE,
49:8–30, 1961.
21
22 BIBLIOGRAPHY
[15] Lawrence Rabiner and Biing-Hwang Juang. Fundamentals of Speech Recognition.
Prentice Hall, Englewood Cliffs, NJ, 1993.
[16] Azriel Rosenfeld. Picture Processing by Computer. Academic Press, New York,
1969.
[17] Jude W. Shavlik and Thomas G. Dietterich, editors. Readings in Machine Learn-
ing. Morgan Kaufmann, San Mateo, CA, 1990.
[18] Brian Cantwell Smith. On the Origin of Objects. MIT Press, Cambridge, MA,
1996.
[19] Louise Stark and Kevin Bower. Generic Object Recognition using Form & Func-
tion. World Scientific, River Edge, NJ, 1996.
[20] Donald R. Tveter. The Pattern Recognition basis of Artificial Intelligence. IEEE
Press, New York, NY, 1998.
[21] William R. Uttal. The psychobiology of sensory coding. HarperCollins, New York,
NY, 1973.
[22] Satoshi Watanabe. Knowing and Guessing: A quantitative study of inference and
information. John Wiley, New York, NY, 1969.
Index
analysis by synthesis, 10
anticipatory coarticulation, see coartic-
ulation, anticipatory
beanbag chair
example, 10
BEATS example, see subset/superset
camera
for pattern recognition, 4
classification, see pattern recognition
cost, 4, 15
model, 4
risk, see classification, cost
clustering, see learning, unsupervised,
17
coarticulation
anticipatory, 13
complexity
computational, see computational
complexity
computational complexity, 16
and feature dimensions, 16
and number of categories, 16
and number of patterns, 16
context, 14
decision
boundary, 7, 8
complex, 8
simple, 10
decision theory, 5
deformations
non-rigid, 15
distribution, see probability, distribu-
tion
DNA sequence identification, 3
evidence pooling, 15
feature
extraction, 4, 11
missing, 13
robust, 10
space, 7
continuous, 8
vector, 7
fingerprint identification, 3
fish
categorization example, 3–9
generalization, 8
grammar, 9
hardware, 3
hypothesis
null, see null hypothesis
hypothesis testing, 11
image
processing, 11
threshold, 4
information
loss, 11
invariance, 14–15
illumination, 15
line thickness, 15
jeetyet example, 14
knowledge
incorporating, 10
prior, 12
learning
and adaptation, 16
reinforcement, 17
supervised, 16
unsupervised, 17
machine perception, see perception, ma-
chine
memory
associative, 11
23
24 INDEX
mereology, 13
missing feature, see feature, missing
model, 4
selection, 12
noise, 12
null hypothesis, 11
Occam
William of, 8
OCR, see optical character recognition
optical character recognition, 3
exhaustive training, 16
handwritten, 10
rate variation, 15
segmentation, 14
orientation, 14
overfitting, 8, 12
pattern classification, see pattern recog-
nition
pattern recognition, 3
general purpose, 9
information reduction, 11
neural, 9
statistical, 8
syntactic, 9
perception
machine, 3
phoneme, 13
POLOPONY example, see subset/superset
preprocessing, 4
prior knowledge, see knowledge, prior
probability
density, 9
distribution, 8
rate variation, 15
recognition
chair example, 10
reinforcement learning, see learning, re-
inforcement
representation, 9
scatter plot, 7
segmentation, 4, 13
speech, 13
shadow, 15
significance threshold, 11
size, 14
sklee
coarticulation in, 13
skloo
coarticulation in, 13
speech recognition
rate variation, 15
rounding, 13
subset/superset, 13
supervised learning, see learning, su-
pervised
symmetry
discrete, 15
unsupervised learning, see learning, un-
supervised
William of Occam, see Occam, William
of
Contents
2 Bayesian decision theory 3
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Bayesian Decision Theory – Continuous Features . . . . . . . . . . . . 7
2.2.1 Two-Category Classification . . . . . . . . . . . . . . . . . . . . 8
2.3 Minimum-Error-Rate Classification . . . . . . . . . . . . . . . . . . . . 9
2.3.1 *Minimax Criterion . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 *Neyman-Pearson Criterion . . . . . . . . . . . . . . . . . . . . 12
2.4 Classifiers, Discriminants and Decision Surfaces . . . . . . . . . . . . . 13
2.4.1 The Multi-Category Case . . . . . . . . . . . . . . . . . . . . . 13
2.4.2 The Two-Category Case . . . . . . . . . . . . . . . . . . . . . . 14
2.5 The Normal Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5.1 Univariate Density . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.2 Multivariate Density . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Discriminant Functions for the Normal Density . . . . . . . . . . . . . 19
2.6.1 Case 1: Σi = σ2I . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6.2 Case 2: Σi = Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6.3 Case 3: Σi = arbitrary . . . . . . . . . . . . . . . . . . . . . . . 25
Example 1: Decisions for Gaussian data . . . . . . . . . . . . . . . . . 29
2.7 *Error Probabilities and Integrals . . . . . . . . . . . . . . . . . . . . . 30
2.8 *Error Bounds for Normal Densities . . . . . . . . . . . . . . . . . . . 31
2.8.1 Chernoff Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.8.2 Bhattacharyya Bound . . . . . . . . . . . . . . . . . . . . . . . 32
Example 2: Error bounds . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.8.3 Signal Detection Theory and Operating Characteristics . . . . 33
2.9 Bayes Decision Theory — Discrete Features . . . . . . . . . . . . . . . 36
2.9.1 Independent Binary Features . . . . . . . . . . . . . . . . . . . 36
Example 3: Bayes decisions for binary data . . . . . . . . . . . . . . . 38
2.10 *Missing and Noisy Features . . . . . . . . . . . . . . . . . . . . . . . 39
2.10.1 Missing Features . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.10.2 Noisy Features . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.11 *Compound Bayes Decision Theory and Context . . . . . . . . . . . . 41
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Bibliographical and Historical Remarks . . . . . . . . . . . . . . . . . . . . 43
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Computer exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1
2 CONTENTS
Chapter 2
Bayesian decision theory
2.1 Introduction
Bayesian decision theory is a fundamental statistical approach to the problem ofpattern classification. This approach is based on quantifying the tradeoffs be-
tween various classification decisions using probability and the costs that accompany
such decisions. It makes the assumption that the decision problem is posed in proba-
bilistic terms, and that all of the relevant probability values are known. In this chapter
we develop the fundamentals of this theory, and show how it can be viewed as being
simply a formalization of common-sense procedures; in subsequent chapters we will
consider the problems that arise when the probabilistic structure is not completely
known.
While we will give a quite general, abstract development of Bayesian decision
theory in Sect. ??, we begin our discussion with a specific example. Let us reconsider
the hypothetical problem posed in Chap. ?? of designing a classifier to separate two
kinds of fish: sea bass and salmon. Suppose that an observer watching fish arrive
along the conveyor belt finds it hard to predict what type will emerge next and that
the sequence of types of fish appears to be random. In decision-theoretic terminology
we would say that as each fish emerges nature is in one or the other of the two possible
states: either the fish is a sea bass or the fish is a salmon. We let ω denote the state state of
natureof nature, with ω = ω1 for sea bass and ω = ω2 for salmon. Because the state of
nature is so unpredictable, we consider ω to be a variable that must be described
probabilistically.
If the catch produced as much sea bass as salmon, we would say that the next fish
is equally likely to be sea bass or salmon. More generally, we assume that there is
some a priori probability (or simply prior) P (ω1) that the next fish is sea bass, and prior
some prior probability P (ω2) that it is salmon. If we assume there are no other types
of fish relevant here, then P (ω1) and P (ω2) sum to one. These prior probabilities
reflect our prior knowledge of how likely we are to get a sea bass or salmon before
the fish actually appears. It might, for instance, depend upon the time of year or the
choice of fishing area.
Suppose for a moment that we were forced to make a decision about the type of
fish that will appear next without being allowed to see it. For the moment, we shall
3
4 CHAPTER 2. BAYESIAN DECISION THEORY
assume that any incorrect classification entails the same cost or consequence, and that
the only information we are allowed to use is the value of the prior probabilities. If a
decision must be made with so little information, it seems logical to use the following
decision rule: Decide ω1 if P (ω1) > P (ω2); otherwise decide ω2.decision
rule This rule makes sense if we are to judge just one fish, but if we are to judge many
fish, using this rule repeatedly may seem a bit strange. After all, we would always
make the same decision even though we know that both types of fish will appear.
How well it works depends upon the values of the prior probabilities. If P (ω1) is very
much greater than P (ω2), our decision in favor of ω1 will be right most of the time.
If P (ω1) = P (ω2), we have only a fifty-fifty chance of being right. In general, the
probability of error is the smaller of P (ω1) and P (ω2), and we shall see later that
under these conditions no other decision rule can yield a larger probability of being
right.
In most circumstances we are not asked to make decisions with so little informa-
tion. In our example, we might for instance use a lightness measurement x to improve
our classifier. Different fish will yield different lightness readings and we express this
variability in probabilistic terms; we consider x to be a continuous random variable
whose distribution depends on the state of nature, and is expressed as p(x|ω1).∗ This
is the class-conditional probability density function. Strictly speaking, the probabil-
ity density function p(x|ω1) should be written as pX(x|ω1) to indicate that we are
speaking about a particular density function for the random variable X. This more
elaborate subscripted notation makes it clear that pX(·) and pY (·) denote two differ-
ent functions, a fact that is obscured when writing p(x) and p(y). Since this potential
confusion rarely arises in practice, we have elected to adopt the simpler notation.
Readers who are unsure of our notation or who would like to review probability the-
ory should see Appendix ??). This is the probability density function for x given that
the state of nature is ω1. (It is also sometimes called state-conditional probability
density.) Then the difference between p(x|ω1) and p(x|ω2) describes the difference in
lightness between populations of sea bass and salmon (Fig. 2.1).
Suppose that we know both the prior probabilities P (ωj) and the conditional
densities p(x|ωj). Suppose further that we measure the lightness of a fish and discover
that its value is x. How does this measurement influence our attitude concerning the
true state of nature — that is, the category of the fish? We note first that the (joint)
probability density of finding a pattern that is in category ωj and has feature value x
can be written two ways: p(ωj , x) = P (ωj |x)p(x) = p(x|ωj)P (ωj). Rearranging these
leads us to the answer to our question, which is called Bayes’ formula:
P (ωj |x) =
p(x|ωj)P (ωj)
p(x)
, (1)
where in this case of two categories
p(x) =
2∑
j=1
p(x|ωj)P (ωj). (2)
Bayes’ formula can be expressed informally in English by saying that
posterior =
likelihood× prior
evidence
. (3)
∗ We generally use an upper-case P (·) to denote a probability mass function and a lower-case p(·)
to denote a probability density function.
2.1. INTRODUCTION 5
Bayes’ formula shows that by observing the value of x we can convert the prior
probability P (ωj) to the a posteriori probability (or posterior) probability P (ωj |x) posterior
— the probability of the state of nature being ωj given that feature value x has
been measured. We call p(x|ωj) the likelihood of ωj with respect to x (a term likelihood
chosen to indicate that, other things being equal, the category ωj for which p(x|ωj)
is large is more “likely” to be the true category). Notice that it is the product of the
likelihood and the prior probability that is most important in determining the psterior
probability; the evidence factor, p(x), can be viewed as merely a scale factor that
guarantees that the posterior probabilities sum to one, as all good probabilities must.
The variation of P (ωj |x) with x is illustrated in Fig. 2.2 for the case P (ω1) = 2/3 and
P (ω2) = 1/3.
9 10 11 12 13 14 15
0.1
0.2
0.3
0.4
p(x|ωi)
x
ω1
ω2
Figure 2.1: Hypothetical class-conditional probability density functions show the
probability density of measuring a particular feature value x given the pattern is
in category ωi. If x represents the length of a fish, the two curves might describe
the difference in length of populations of two types of fish. Density functions are
normalized, and thus the area under each curve is 1.0.
If we have an observation x for which P (ω1|x) is greater than P (ω2|x), we would
naturally be inclined to decide that the true state of nature is ω1. Similarly, if P (ω2|x)
is greater than P (ω1|x), we would be inclined to choose ω2. To justify this decision
procedure, let us calculate the probability of error whenever we make a decision.
Whenever we observe a particular x,
P (error|x) =
{
P (ω1|x) if we decide ω2
P (ω2|x) if we decide ω1.
(4)
Clearly, for a given x we can minimize the probability of error by deciding ω1 if
P (ω1|x) > P (ω2|x) and ω2 otherwise. Of course, we may never observe exactly the
same value of x twice. Will this rule minimize the average probability of error? Yes,
because the average probability of error is given by
6 CHAPTER 2. BAYESIAN DECISION THEORY
9 10 11 12 13 14 15
0.2
0.4
0.6
0.8
1
x
P(ωi|x)
ω2
ω1
Figure 2.2: Posterior probabilities for the particular priors P(ω1) = 2/3 and P(ω2) =
1/3 for the class-conditional probability densities shown in Fig. 2.1. Thus in this case,
given that a pattern is measured to have feature value x = 14, the probability it is
in category ω2 is roughly 0.08, and that it is in ω1 is 0.92. At every x, the posteriors
sum to 1.0.
P (error) =
∞∫
−∞
P (error, x) dx =
∞∫
−∞
P (error|x)p(x) dx (5)
and if for every x we insure that P (error|x) is as small as possible, then the integral
must be as small as possible. Thus we have justified the following Bayes’ decision
rule for minimizing the probability of error:Bayes’
decision
rule Decide ω1 if P (ω1|x) > P (ω2|x); otherwise decide ω2, (6)
and under this rule Eq. 4 becomes
P (error|x) = min [P (ω1|x), P (ω2|x)]. (7)
This form of the decision rule emphasizes the role of the posterior probabilities.
By using Eq. 1, we can instead express the rule in terms of the conditional and prior
probabilities. First note that the evidence, p(x), in Eq. 1 is unimportant as far asevidence
making a decision is concerned. It is basically just a scale factor that states how
frequently we will actually measure a pattern with feature value x; its presence in
Eq. 1 assures us that P (ω1|x) + P (ω2|x) = 1. By eliminating this scale factor, we
obtain the following completely equivalent decision rule:
Decide ω1 if p(x|ω1)P (ω1) > p(x|ω2)P (ω2); otherwise decide ω2. (8)
Some additional insight can be obtained by considering a few special cases. If
for some x we have p(x|ω1) = p(x|ω2), then that particular observation gives us no
2.2. BAYESIAN DECISION THEORY – CONTINUOUS FEATURES 7
information about the state of nature; in this case, the decision hinges entirely on the
prior probabilities. On the other hand, if P (ω1) = P (ω2), then the states of nature are
equally probable; in this case the decision is based entirely on the likelihoods p(x|ωj).
In general, both of these factors are important in making a decision, and the Bayes
decision rule combines them to achieve the minimum probability of error.
2.2 Bayesian Decision Theory – Continuous Fea-
tures
We shall now formalize the ideas just considered, and generalize them in four ways:
• by allowing the use of more than one feature
• by allowing more than two states of nature
• by allowing actions other than merely deciding the state of nature
• by introducing a loss function more general than the probability of error.
These generalizations and their attendant notational complexities should not ob-
scure the central points illustrated in our simple example. Allowing the use of more
than one feature merely requires replacing the scalar x by the feature vector x, where
x is in a d-dimensional Euclidean space Rd, called the feature space. Allowing more feature
spacethan two states of nature provides us with a useful generalization for a small notational
expense. Allowing actions other than classification primarily allows the possibility of
rejection, i.e., of refusing to make a decision in close cases; this is a useful option if
being indecisive is not too costly. Formally, the loss function states exactly how costly loss
functioneach action is, and is used to convert a probability determination into a decision. Cost
functions let us treat situations in which some kinds of classification mistakes are more
costly than others, although we often discuss the simplest case, where all errors are
equally costly. With this as a preamble, let us begin the more formal treatment.
Let ω1, …, ωc be the finite set of c states of nature (“categories”) and α1, …, αa
be the finite set of a possible actions. The loss function λ(αi|ωj) describes the loss
incurred for taking action αi when the state of nature is ωj . Let the feature vector
x be a d-component vector-valued random variable, and let p(x|ωj) be the state-
conditional probability density function for x — the probability density function for
x conditioned on ωj being the true state of nature. As before, P (ωj) describes the
prior probability that nature is in state ωj . Then the posterior probability P (ωj |x)
can be computed from p(x|ωj) by Bayes’ formula:
P (ωj |x) =
p(x|ωj)P (ωj)
p(x)
, (9)
where the evidence is now
p(x) =
c∑
j=1
p(x|ωj)P (ωj). (10)
Suppose that we observe a particular x and that we contemplate taking action
αi. If the true state of nature is ωj , by definition we will incur the loss λ(αi|ωj).
Since P (ωj |x) is the probability that the true state of nature is ωj , the expected loss
associated with taking action αi is merely
8 CHAPTER 2. BAYESIAN DECISION THEORY
R(αi|x) =
c∑
j=1
λ(αi|ωj)P (ωj |x). (11)
In decision-theoretic terminology, an expected loss is called a risk, and R(αi|x) isrisk
called the conditional risk. Whenever we encounter a particular observation x, we can
minimize our expected loss by selecting the action that minimizes the conditional risk.
We shall now show that this Bayes decision procedure actually provides the optimal
performance on an overall risk.
Stated formally, our problem is to find a decision rule against P (ωj) that mini-
mizes the overall risk. A general decision rule is a function α(x) that tells us whichdecision
rule action to take for every possible observation. To be more specific, for every x the
decision function α(x) assumes one of the a values α1, …, αa. The overall risk R is the
expected loss associated with a given decision rule. Since R(αi|x) is the conditional
risk associated with action αi, and since the decision rule specifies the action, the
overall risk is given by
R =
∫
R(α(x)|x)p(x) dx, (12)
where dx is our notation for a d-space volume element, and where the integral extends
over the entire feature space. Clearly, if α(x) is chosen so that R(αi(x)) is as small
as possible for every x, then the overall risk will be minimized. This justifies the
following statement of the Bayes decision rule: To minimize the overall risk, compute
the conditional risk
R(αi|x) =
c∑
j=1
λ(αi|ωj)P (ωj |x) (13)
for i = 1,…,a and select the action αi for which R(αi|x) is minimum.∗ The resulting
minimum overall risk is called the Bayes risk, denoted R∗, and is the best performanceBayes risk
that can be achieved.
2.2.1 Two-Category Classification
Let us consider these results when applied to the special case of two-category classifi-
cation problems. Here action α1 corresponds to deciding that the true state of nature
is ω1, and action α2 corresponds to deciding that it is ω2. For notational simplicity,
let λij = λ(αi|ωj) be the loss incurred for deciding ωi when the true state of nature
is ωj . If we write out the conditional risk given by Eq. 13, we obtain
R(α1|x) = λ11P (ω1|x) + λ12P (ω2|x) and
R(α2|x) = λ21P (ω1|x) + λ22P (ω2|x). (14)
There are a variety of ways of expressing the minimum-risk decision rule, each
having its own minor advantages. The fundamental rule is to decide ω1 if R(α1|x) <
R(α2|x). In terms of the posterior probabilities, we decide ω1 if
(λ21 − λ11)P (ω1|x) > (λ12 − λ22)P (ω2|x). (15)
∗ Note that if more than one action minimizes R(α|x), it does not matter which of these actions is
taken, and any convenient tie-breaking rule can be used.
2.3. MINIMUM-ERROR-RATE CLASSIFICATION 9
Ordinarily, the loss incurred for making an error is greater than the loss incurred for
being correct, and both of the factors λ21 − λ11 and λ12 − λ22 are positive. Thus in
practice, our decision is generally determined by the more likely state of nature, al-
though we must scale the posterior probabilities by the loss differences. By employing
Bayes’ formula, we can replace the posterior probabilities by the prior probabilities
and the conditional densities. This results in the equivalent rule, to decide ω1 if
(λ21 − λ11)p(x|ω1)P (ω1) > (λ12 − λ22)p(x|ω2)P (ω2), (16)
and ω2 otherwise.
Another alternative, which follows at once under the reasonable assumption that
λ21 > λ11, is to decide ω1 if
p(x|ω1)
p(x|ω2)
>
λ12 − λ22
λ21 − λ11
P (ω2)
P (ω1)
. (17)
This form of the decision rule focuses on the x-dependence of the probability densities.
We can consider p(x|ωj) a function of ωj (i.e., the likelihood function), and then form likelihood
ratiothe likelihood ratio p(x|ω1)/p(x|ω2). Thus the Bayes decision rule can be interpreted
as calling for deciding ω1 if the likelihood ratio exceeds a threshold value that is
independent of the observation x.
2.3 Minimum-Error-Rate Classification
In classification problems, each state of nature is usually associated with a different
one of the c classes, and the action αi is usually interpreted as the decision that the
true state of nature is ωi. If action αi is taken and the true state of nature is ωj , then
the decision is correct if i = j, and in error if i �= j. If errors are to be avoided, it is
natural to seek a decision rule that minimizes the probability of error, i.e., the error
rate.
The loss function of interest for this case is hence the so-called symmetrical or
zero-one loss function, zero-one
loss
λ(αi|ωj) =
{
0 i = j
1 i �= j i, j = 1, …, c. (18)
This loss function assigns no loss to a correct decision, and assigns a unit loss to any
error; thus, all errors are equally costly.∗ The risk corresponding to this loss function
is precisely the average probability of error, since the conditional risk is
R(αi|x) =
c∑
j=1
λ(αi|ωj)P (ωj |x)
=
∑
j �=i
P (ωj |x)
= 1− P (ωi|x) (19)
∗ We note that other loss functions, such as quadratic and linear-difference, find greater use in
regression tasks, where there is a natural ordering on the predictions and we can meaningfully
penalize predictions that are “more wrong” than others.
10 CHAPTER 2. BAYESIAN DECISION THEORY
and P (ωi|x) is the conditional probability that action αi is correct. The Bayes decision
rule to minimize risk calls for selecting the action that minimizes the conditional
risk. Thus, to minimize the average probability of error, we should select the i that
maximizes the posterior probability P (ωi|x). In other words, for minimum error rate:
Decide ωi if P (ωi|x) > P (ωj |x) for all j �= i. (20)
This is the same rule as in Eq. 6.
We saw in Fig. 2.2 some class-conditional probability densities and the posterior
probabilities; Fig. 2.3 shows the likelihood ratio p(x|ω1)/p(x|ω2) for the same case. In
general, this ratio can range between zero and infinity. The threshold value θa marked
is from the same prior probabilities but with zero-one loss function. Notice that this
leads to the same decision boundaries as in Fig. 2.2, as it must. If we penalize mistakes
in classifying ω1 patterns as ω2 more than the converse (i.e., λ21 > λ12), then Eq. 17
leads to the threshold θb marked. Note that the range of x values for which we classify
a pattern as ω1 gets smaller, as it should.
x
θa
p(x|ω1)
p(x|ω2)
R1 R1R2R2
θb
Figure 2.3: The likelihood ratio p(x|ω1)/p(x|ω2) for the distributions shown in Fig. 2.1.
If we employ a zero-one or classification loss, our decision boundaries are determined
by the threshold θa. If our loss function penalizes miscategorizing ω2 as ω1 patterns
more than the converse, (i.e., λ12 > λ21), we get the larger threshold θb, and hence
R1 becomes smaller.
2.3.1 *Minimax Criterion
Sometimes we must design our classifier to perform well over a range of prior proba-
bilities. For instance, in our fish categorization problem we can imagine that whereas
the physical properties of lightness and width of each type of fish remain constant, the
prior probabilities might vary widely and in an unpredictable way, or alternatively
we want to use the classifier in a different plant where we do not know the prior
probabilities. A reasonable approach is then to design our classifier so that the worst
overall risk for any value of the priors is as small as possible — that is, minimize the
maximum possible overall risk.
2.3. MINIMUM-ERROR-RATE CLASSIFICATION 11
In order to understand this, we let R1 denote that (as yet unknown) region in
feature space where the classifier decides ω1 and likewise for R2 and ω2, and then
write our overall risk Eq. 12 in terms of conditional risks:
R =
∫
R1
[λ11P (ω1) p(x|ω1) + λ12P (ω2) p(x|ω2)] dx
+
∫
R2
[λ21P (ω1) p(x|ω1) + λ22P (ω2) p(x|ω2)] dx. (21)
We use the fact that P (ω2) = 1− P (ω1) and that
∫
R1
p(x|ω1) dx = 1−
∫
R2
p(x|ω1) dx
to rewrite the risk as:
R(P (ω1)) =
= Rmm, minimax risk︷ ︸︸ ︷
λ22 + (λ12 − λ22)
∫
R1
p(x|ω2) dx (22)
+ P (ω1)
(λ11 − λ22)− (λ21 − λ11)
∫
R2
p(x|ω1) dx− (λ12 − λ22)
∫
R1
p(x|ω2) dx
︸ ︷︷ ︸
= 0 for minimax solution
.
This equation shows that once the decision boundary is set (i.e., R1 and R2
determined), the overall risk is linear in P (ω1). If we can find a boundary such that
the constant of proportionality is 0, then the risk is independent of priors. This is the
minimax solution, and the minimax risk, Rmm, can be read from Eq. 22: minimax
risk
Rmm = λ22 + (λ12 − λ22)
∫
R1
p(x|ω2) dx
= λ11 + (λ21 − λ11)
∫
R2
p(x|ω1) dx. (23)
Figure 2.4 illustrates the approach. Briefly stated, we search for the prior for which
the Bayes risk is maximum, the corresponding decision boundary gives the minimax
solution. The value of the minimax risk, Rmm, is hence equal to the worst Bayes risk.
In practice, finding the decision boundary for minimax risk may be difficult, partic-
ularly when distributions are complicated. Nevertheless, in some cases the boundary
can be determined analytically (Problem 3).
The minimax criterion finds greater use in game theory then it does in traditional
pattern recognition. In game theory, you have a hostile opponent who can be expected
to take an action maximally detrimental to you. Thus it makes great sense for you to
take an action (e.g., make a classification) where your costs — due to your opponent’s
subsequent actions — are minimized.
12 CHAPTER 2. BAYESIAN DECISION THEORY
0 0.2 0.4 0.6 0.8 1
P(ω1)
P(error)
.1
.2
.3
.4
.1
.2
.3
.4
Figure 2.4: The curve at the bottom shows the minimum (Bayes) error as a function of
prior probability P (ω1) in a two-category classification problem of fixed distributions.
For each value of the priors (e.g., P (ω1) = 0.25) there is a corresponding optimal
decision boundary and associated Bayes error rate. For any (fixed) such boundary, if
the priors are then changed, the probability of error will change as a linear function of
P (ω1) (shown by the dashed line). The maximum such error will occur at an extreme
value of the prior, here at P (ω1) = 1. To minimize the maximum of such error, we
should design our decision boundary for the maximum Bayes error (here P (ω1) = 0.6),
and thus the error will not change as a function of prior, as shown by the solid red
horizontal line.
2.3.2 *Neyman-Pearson Criterion
In some problems, we may wish to minimize the overall risk subject to a constraint;
for instance, we might wish to minimize the total risk subject to the constraint∫
R(αi|x) dx < constant for some particular i. Such a constraint might arise when
there is a fixed resource that accompanies one particular action αi, or when we must
not misclassify pattern from a particular state of nature ωi at more than some limited
frequency. For instance, in our fish example, there might be some government regu-
lation that we must not misclassify more than 1% of salmon as sea bass. We might
then seek a decision that minimizes the chance of classifying a sea bass as a salmon
subject to this condition.
We generally satisfy such a Neyman-Pearson criterion by adjusting decision bound-
aries numerically. However, for Gaussian and some other distributions, Neyman-
Pearson solutions can be found analytically (Problems 5 & 6). We shall have cause
to mention Neyman-Pearson criteria again in Sect. 2.8.3 on operating characteristics.
2.4. CLASSIFIERS, DISCRIMINANTS AND DECISION SURFACES 13
2.4 Classifiers, Discriminant Functions and Deci-
sion Surfaces
2.4.1 The Multi-Category Case
There are many different ways to represent pattern classifiers. One of the most useful
is in terms of a set of discriminant functions gi(x), i = 1, ..., c. The classifier is said
to assign a feature vector x to class ωi if
gi(x) > gj(x) for all j �= i. (24)
Thus, the classifier is viewed as a network or machine that computes c discriminant
functions and selects the category corresponding to the largest discriminant. A net-
work representation of a classifier is illustrated in Fig. 2.5.
Discriminant
functions
Input
g1(x) g2(x) gc(x). . .
x1 x2 xd. . .x3
Costs
Action
(e.g., classification)
Figure 2.5: The functional structure of a general statistical pattern classifier which
includes d inputs and c discriminant functions gi(x). A subsequent step determines
which of the discriminant values is the maximum, and categorizes the input pat-
tern accordingly. The arrows show the direction of the flow of information, though
frequently the arrows are omitted when the direction of flow is self-evident.
A Bayes classifier is easily and naturally represented in this way. For the gen-
eral case with risks, we can let gi(x) = −R(αi|x), since the maximum discriminant
function will then correspond to the minimum conditional risk. For the minimum-
error-rate case, we can simplify things further by taking gi(x) = P (ωi|x), so that the
maximum discriminant function corresponds to the maximum posterior probability.
Clearly, the choice of discriminant functions is not unique. We can always multiply
all the discriminant functions by the same positive constant or shift them by the same
additive constant without influencing the decision. More generally, if we replace every
gi(x) by f(gi(x)), where f(·) is a monotonically increasing function, the resulting
classification is unchanged. This observation can lead to significant analytical and
computational simplifications. In particular, for minimum-error-rate classification,
any of the following choices gives identical classification results, but some can be
much simpler to understand or to compute than others:
14 CHAPTER 2. BAYESIAN DECISION THEORY
gi(x) = P (ωi|x) =
p(x|ωi)P (ωi)
c∑
j=1
p(x|ωj)P (ωj)
(25)
gi(x) = p(x|ωi)P (ωi) (26)
gi(x) = ln p(x|ωi) + ln P (ωi), (27)
where ln denotes natural logarithm.
Even though the discriminant functions can be written in a variety of forms, the
decision rules are equivalent. The effect of any decision rule is to divide the feature
space into c decision regions, R1,…,Rc. If gi(x) > gj(x) for all j �= i, then x is indecision
region Ri, and the decision rule calls for us to assign x to ωi. The regions are separated
by decision boundaries, surfaces in feature space where ties occur among the largest
discriminant functions (Fig. 2.6).
0
5
0
5
0
0.1
0.2
0.3
0
5
0
0.1
0.2
0.3
R2
Decision
Boundary
R1
R2
p(x|ω1)P(ω1)
p(x|ω2)P(ω2)
Figure 2.6: In this two-dimensional two-category classifier, the probability densities
are Gaussian (with 1/e ellipses shown), the decision boundary consists of two hyper-
bolas, and thus the decision region R2 is not simply connected.
2.4.2 The Two-Category Case
While the two-category case is just a special instance of the multicategory case, it has
traditionally received separate treatment. Indeed, a classifier that places a pattern in
2.5. THE NORMAL DENSITY 15
one of only two categories has a special name — a dichotomizer.∗ Instead of using two dichotomizer
discriminant functions g1 and g2 and assigning x to ω1 if g1 > g2, it is more common
to define a single discriminant function
g(x) ≡ g1(x)− g2(x), (28)
and to use the following decision rule: Decide ω1 if g(x) > 0; otherwise decide ω2.
Thus, a dichotomizer can be viewed as a machine that computes a single discriminant
function g(x), and classifies x according to the algebraic sign of the result. Of the
various forms in which the minimum-error-rate discriminant function can be written,
the following two (derived from Eqs. 25 & 27) are particularly convenient:
g(x) = P (ω1|x)− P (ω2|x) (29)
g(x) = ln
p(x|ω1)
p(x|ω2)
+ ln
P (ω1)
P (ω2)
. (30)
2.5 The Normal Density
The structure of a Bayes classifier is determined by the conditional densities p(x|ωi)
as well as by the prior probabilities. Of the various density functions that have
been investigated, none has received more attention than the multivariate normal or
Gaussian density. To a large extent this attention is due to its analytical tractability.
However the multivariate normal density is also an appropriate model for an important
situation, viz., the case where the feature vectors x for a given class ωi are continuous
valued, randomly corrupted versions of a single typical or prototype vector µi. In this
section we provide a brief exposition of the multivariate normal density, focusing on
the properties of greatest interest for classification problems.
First, recall the definition of the expected value of a scalar function f(x), defined expectation
for some density p(x):
E [f(x)] ≡
∞∫
−∞
f(x)p(x)dx. (31)
If we have samples in a set D from a discrete distribution, we must sum over all
samples as
E [f(x)] =
∑
x∈D
f(x)P (x), (32)
where P (x) is the probability mass at x. We shall often have call to calculate expected
values — by these and analogous equations defined in higher dimensions (see Appendix
Secs. ??, ?? & ??).∗
∗ A classifier for more than two categories is called a polychotomizer.
∗ We will often use somewhat loose engineering terminology and refer to a single point as a “sample.”
Statisticians, though, always refer to a sample as a collection of points, and discuss “a sample of
size n.” When taken in context, there are rarely ambiguities in such usage.
16 CHAPTER 2. BAYESIAN DECISION THEORY
2.5.1 Univariate Density
We begin with the continuous univariate normal or Gaussian density,
p(x) =
1√
2πσ
exp
[
−1
2
(
x− µ
σ
)2]
, (33)
for which the expected value of x (an average, here taken over the feature space) is
µ ≡ E [x] =
∞∫
−∞
xp(x) dx, (34)
and where the expected squared deviation or variance isvariance
σ2 ≡ E [(x− µ)2] =
∞∫
−∞
(x− µ)2p(x) dx. (35)
The univariate normal density is completely specified by two parameters: its mean
µ and variance σ2. For simplicity, we often abbreviate Eq. 33 by writing p(x) ∼mean
N(µ, σ2) to say that x is distributed normally with mean µ and variance σ2. Samples
from normal distributions tend to cluster about the mean, with a spread related to
the standard deviation σ (Fig. 2.7).
x
2.5% 2.5%
σ
p(x)
µ + σ µ + 2σµ − σµ − 2σ µ
Figure 2.7: A univariate normal distribution has roughly 95% of its area in the range
|x− µ| ≤ 2σ, as shown. The peak of the distribution has value p(µ) = 1/
√
2πσ.
There is a deep relationship between the normal distribution and entropy. Weentropy
shall consider entropy in greater detail in Chap. ??, but for now we merely state that
the entropy of a distribution is given by
H(p(x)) = −
∫
p(x) ln p(x) dx, (36)
and measured in nats. If a log2 is used instead, the unit is the bit. The entropy is a non-nat
bit
negative quantity that describes the fundamental uncertainty in the values of points
2.5. THE NORMAL DENSITY 17
selected randomly from a distribution. It can be shown that the normal distribution
has the maximum entropy of all distributions having a given mean and variance
(Problem 20). Moreover, as stated by the Central Limit Theorem, the aggregate Central
Limit
Theorem
effect of a large number of small, independent random disturbances will lead to a
Gaussian distribution (Computer exercise ??). Because many patterns — from fish
to handwritten characters to some speech sounds — can be viewed as some ideal or
prototype pattern corrupted by a large number of random processes, the Gaussian is
often a good model for the actual probability distribution.
2.5.2 Multivariate Density
The general multivariate normal density in d dimensions is written as
p(x) =
1
(2π)d/2|Σ|1/2
exp
[
−1
2
(x− µ)tΣ−1(x− µ)
]
, (37)
where x is a d-component column vector, µ is the d-component mean vector, Σ is the
d-by-d covariance matrix, |Σ| and Σ−1 are its determinant and inverse, respectively, covariance
matrixand (x− µ)t is the transpose of x− µ.∗ Our notation for the inner product is
inner
productatb =
d∑
i=1
aibi, (38)
and often called a dot product.
For simplicity, we often abbreviate Eq. 37 as p(x) ∼ N(µ,Σ). Formally, we have
µ ≡ E [x] =
∫
xp(x) dx (39)
and
Σ ≡ E [(x− µ)(x− µ)t] =
∫
(x− µ)(x− µ)tp(x) dx, (40)
where the expected value of a vector or a matrix is found by taking the expected
values of its components. In other words, if xi is the ith component of x, µi the ith
component of µ, and σij the ijth component of Σ, then
µi = E [xi] (41)
and
σij = E [(xi − µi)(xj − µj)]. (42)
The covariance matrix Σ is always symmetric and positive semidefinite. We shall
restrict our attention to the case in which Σ is positive definite, so that the deter-
minant of Σ is strictly positive.† The diagonal elements σii are the variances of the
respective xi (i.e., σ2i ), and the off-diagonal elements σij are the covariances of xi and covariance
xj . We would expect a positive covariance for the length and weight features of a
population of fish, for instance. If xi and xj are statistically independent, σij = 0. If statistical
indepen-
dence
∗ The mathematical expressions for the multivariate normal density are greatly simplified by em-
ploying the concepts and notation of linear algebra. Readers who are unsure of our notation or
who would like to review linear algebra should see Appendix ??.
† If sample vectors are drawn from a linear subspace, |Σ| = 0 and p(x) is degenerate. This occurs,
for example, when one component of x has zero variance, or when two components are identical
or multiples of one another.
18 CHAPTER 2. BAYESIAN DECISION THEORY
all the off-diagonal elements are zero, p(x) reduces to the product of the univariate
normal densities for the components of x.
Linear combinations of jointly normally distributed random variables, independent
or not, are normally distributed. In particular, if A is a d-by-k matrix and y = Atx
is a k-component vector, then p(y) ∼ N(Atµ,AtΣA), as illustrated in Fig. 2.8. In
the special case where k = 1 and A is a unit-length vector a, y = atx is a scalar that
represents the projection of x onto a line in the direction of a; in that case atΣa is the
variance of the projection of x onto a. In general then, knowledge of the covariance
matrix allows us to calculate the dispersion of the data in any direction, or in any
subspace.
It is sometimes convenient to perform a coordinate transformation that converts
an arbitrary multivariate normal distribution into a spherical one, i.e., one having a
covariance matrix proportional to the identity matrix I. If we define Φ to be the ma-
trix whose columns are the orthonormal eigenvectors of Σ, and Λ the diagonal matrix
of the corresponding eigenvalues, then the transformation Aw = ΦΛ
−1/2 applied to
the coordinates insures that the transformed distribution has covariance matrix equal
to the identity matrix. In signal processing, the transform Aw is called a whiten-
ing transformation, since it makes the spectrum of eigenvectors of the transformedwhitening
transform distribution uniform.
The multivariate normal density is completely specified by d + d(d + 1)/2 pa-
rameters — the elements of the mean vector µ and the independent elements of the
covariance matrix Σ. Samples drawn from a normal population tend to fall in a single
cloud or cluster (Fig. 2.9); the center of the cluster is determined by the mean vector,
and the shape of the cluster is determined by the covariance matrix. If follows from
Eq. 37 that the loci of points of constant density are hyperellipsoids for which the
quadratic form (x−µ)tΣ−1(x−µ) is constant. The principal axes of these hyperellip-
soids are given by the eigenvectors of Σ (described by Φ); the eigenvalues (described
by Λ) determine the lengths of these axes. The quantity
r2 = (x− µ)tΣ−1(x− µ) (43)
is sometimes called the squared Mahalanobis distance from x to µ. Thus, the contoursMahalanobis
distance of constant density are hyperellipsoids of constant Mahalanobis distance to µ and the
volume of these hyperellipsoids measures the scatter of the samples about the mean. It
can be shown (Problems 15 & 16) that the volume of the hyperellipsoid corresponding
to a Mahalanobis distance r is given by
V = Vd|Σ|1/2rd, (44)
where Vd is the volume of a d-dimensional unit hypersphere:
Vd =
πd/2/(d/2)! d even
2dπ(d−1)/2(d−1
2
)!/(d)! d odd.
(45)
Thus, for a given dimensionality, the scatter of the samples varies directly with |Σ|1/2
(Problem 17).
2.6. DISCRIMINANT FUNCTIONS FOR THE NORMAL DENSITY 19
2 4 6 8 10
0
2
4
6
8
10
µ
Ptµ
Atµ
N(µ, ΣΣ)
P
N(Atµ, AtΣ A)
A
a
N(Aw
tµ, I)
Aw
Aw
tµ
σ
x1
x2
Figure 2.8: The action of a linear transformation on the feature space will convert an
arbitrary normal distribution into another normal distribution. One transformation,
A, takes the source distribution into distribution N(Atµ,AtΣA). Another linear
transformation — a projection P onto line a — leads to N(µ, σ2) measured along a.
While the transforms yield distributions in a different space, we show them super-
imposed on the original x1 − x2 space. A whitening transform leads to a circularly
symmetric Gaussian, here shown displaced.
2.6 Discriminant Functions for the Normal Density
In Sect. 2.4.1 we saw that the minimum-error-rate classification can be achieved by
use of the discriminant functions
gi(x) = ln p(x|ωi) + ln P (ωi). (46)
This expression can be readily evaluated if the densities p(x|ωi) are multivariate nor-
mal, i.e., if p(x|ωi) ∼ N(µi,Σi). In this case, then, from Eq. 37 we have
gi(x) = −
1
2
(x− µi)tΣ−1i (x− µi)−
d
2
ln 2π − 1
2
ln |Σi|+ ln P (ωi). (47)
Let us examine the discriminant function and resulting classification for a number of
special cases.
2.6.1 Case 1: Σi = σ
2I
The simplest case occurs when the features are statistically independent, and when
each feature has the same variance, σ2. In this case the covariance matrix is diagonal,
20 CHAPTER 2. BAYESIAN DECISION THEORY
x2
x1
µ
Figure 2.9: Samples drawn from a two-dimensional Gaussian lie in a cloud centered on
the mean µ. The red ellipses show lines of equal probability density of the Gaussian.
being merely σ2 times the identity matrix I. Geometrically, this corresponds to the
situation in which the samples fall in equal-size hyperspherical clusters, the cluster
for the ith class being centered about the mean vector µi. The computation of the
determinant and the inverse of Σi is particularly easy: |Σi| = σ2d and Σ−1i = (1/σ2)I.
Since both |Σi| and the (d/2) ln 2π term in Eq. 47 are independent of i, they are
unimportant additive constants that can be ignored. Thus we obtain the simple
discriminant functions
gi(x) = −
‖x− µi‖2
2σ2
+ ln P (ωi), (48)
where ‖ · ‖ is the Euclidean norm, that is,Euclidean
norm
‖x− µi‖2 = (x− µi)t(x− µi). (49)
If the prior probabilities are not equal, then Eq. 48 shows that the squared distance
‖x−µ‖2 must be normalized by the variance σ2 and offset by adding ln P (ωi); thus,
if x is equally near two different mean vectors, the optimal decision will favor the a
priori more likely category.
Regardless of whether the prior probabilities are equal or not, it is not actually
necessary to compute distances. Expansion of the quadratic form (x − µi)t(x − µi)
yields
gi(x) = −
1
2σ2
[xtx− 2µtix + µtiµi] + ln P (ωi), (50)
which appears to be a quadratic function of x. However, the quadratic term xtx is
the same for all i, making it an ignorable additive constant. Thus, we obtain the
equivalent linear discriminant functionslinear
discriminant
gi(x) = w
t
ix + wi0, (51)
where
2.6. DISCRIMINANT FUNCTIONS FOR THE NORMAL DENSITY 21
-2 2 4
0.1
0.2
0.3
0.4
P(ω1)=.5 P(ω2)=.5
p(x|ωi)
x
ω1 ω2
R1 R2
-2
0
2
4
-2
0
2
4
0
0.05
0.1
0.15
-2
0
2
4
-2
0
2
4
R1
R2
P(ω2)=.5
P(ω1)=.5
-2
-1
0
1
2
0
1
2
-2
-1
0
1
2
-2
-1
0
1
2
0
1
2
P(ω1)=.5
P(ω2)=.5
R2
R1
Figure 2.10: If the covariances of two distributions are equal and proportional to the
identity matrix, then the distributions are spherical in d dimensions, and the boundary
is a generalized hyperplane of d− 1 dimensions, perpendicular to the line separating
the means. In these 1-, 2-, and 3-dimensional examples, we indicate p(x|ωi) and the
boundaries for the case P (ω1) = P (ω2). In the 3-dimensional case, the grid plane
separates R1 from R2.
wi =
1
σ2
µi (52)
and
wi0 =
−1
2σ2
µtiµi + ln P (ωi). (53)
We call wi0 the threshold or bias in the ith direction. threshold
bias
A classifier that uses linear discriminant functions is called a linear machine. This
linear
machine
kind of classifier has many interesting theoretical properties, some of which will be
discussed in detail in Chap. ??. At this point we merely note that the decision
surfaces for a linear machine are pieces of hyperplanes defined by the linear equations
gi(x) = gj(x) for the two categories with the highest posterior probabilities. For our
particular case, this equation can be written as
wt(x− x0) = 0, (54)
where
w = µi − µj (55)
and
x0 =
1
2
(µi + µj)−
σ2
‖µi − µj‖2
ln
P (ωi)
P (ωj)
(µi − µj). (56)
This equation defines a hyperplane through the point x0 and orthogonal to the
vector w. Since w = µi − µj , the hyperplane separating Ri and Rj is orthogonal to
the line linking the means. If P (ωi) = P (ωj), the second term on the right of Eq. 56
vanishes, and thus the point x0 is halfway between the means, and the hyperplane is
the perpendicular bisector of the line between the means (Fig. 2.11). If P (ωi) �= P (ωj),
the point x0 shifts away from the more likely mean. Note, however, that if the variance
22 CHAPTER 2. BAYESIAN DECISION THEORY
-2 2 4
0.1
0.2
0.3
0.4
P(ω1) = .7 P(ω2) = .3
ω2ω1
p(x|ωi)
R1 R2
P(ω2) = .1P(ω1) = .9
-2 2 4
0.1
0.2
0.3
0.4
ω2ω1
p(x|ωi)
R1 R2
-2
0
2
4
-2
0
2
4
0
0.05
0.1
0.15
-2
0
2
4
-2
0
2
4
P(ω2)=.2
P(ω1)=.8 R1
R2
-2
0
2
4
-2
0
2
4
0
0.05
0.1
0.15
-2
0
2
4
-2
0
2
4
P(ω1)=.99
P(ω2)=.01
R1
R2
-2
-1
0
1
2
0
1
2
3
-2
-1
0
1
2
-2
-1
0
1
2
0
1
2
3
P(ω1)=.8
P(ω2)=.2
R1
R2
-2
-1
0
1
2
0
2
4
-2
-1
0
1
2
-2
-1
0
1
2
0
2
4
P(ω1)=.99
P(ω2)=.01
R1
R2
Figure 2.11: As the priors are changed, the decision boundary shifts; for sufficiently
disparate priors the boundary will not lie between the means of these 1-, 2- and
3-dimensional spherical Gaussian distributions.
σ2 is small relative to the squared distance ‖µi − µj‖, then the position of the decision
boundary is relatively insensitive to the exact values of the prior probabilities.
If the prior probabilities P (ωi) are the same for all c classes, then the ln P (ωi)
term becomes another unimportant additive constant that can be ignored. When this
happens, the optimum decision rule can be stated very simply: to classify a feature
vector x, measure the Euclidean distance ‖x − µi‖ from each x to each of the c
mean vectors, and assign x to the category of the nearest mean. Such a classifier is
called a minimum distance classifier. If each mean vector is thought of as being anminimum
distance
classifier
ideal prototype or template for patterns in its class, then this is essentially a template-
matching procedure (Fig. 2.10), a technique we will consider again in Chap. ?? Sect. ??
template-
matching
on the nearest-neighbor algorithm.
2.6. DISCRIMINANT FUNCTIONS FOR THE NORMAL DENSITY 23
2.6.2 Case 2: Σi = Σ
Another simple case arises when the covariance matrices for all of the classes are
identical but otherwise arbitrary. Geometrically, this corresponds to the situation in
which the samples fall in hyperellipsoidal clusters of equal size and shape, the cluster
for the ith class being centered about the mean vector µi. Since both |Σi| and the
(d/2) ln 2π term in Eq. 47 are independent of i, they can be ignored as superfluous
additive constants. This simplification leads to the discriminant functions
gi(x) = −
1
2
(x− µi)tΣ−1(x− µi) + ln P (ωi). (57)
If the prior probabilities P (ωi) are the same for all c classes, then the ln P (ωi)
term can be ignored. In this case, the optimal decision rule can once again be stated
very simply: to classify a feature vector x, measure the squared Mahalanobis distance
(x − µi)tΣ−1(x − µi) from x to each of the c mean vectors, and assign x to the
category of the nearest mean. As before, unequal prior probabilities bias the decision
in favor of the a priori more likely category.
Expansion of the quadratic form (x−µi)tΣ−1(x−µi) results in a sum involving
a quadratic term xtΣ−1x which here is independent of i. After this term is dropped
from Eq. 57, the resulting discriminant functions are again linear:
gi(x) = w
t
ix + wi0, (58)
where
wi = Σ
−1µi (59)
and
wi0 = −
1
2
µtiΣ
−1µi + ln P (ωi). (60)
Since the discriminants are linear, the resulting decision boundaries are again
hyperplanes (Fig. 2.10). If Ri and Rj are contiguous, the boundary between them
has the equation
wt(x− x0) = 0, (61)
where
w = Σ−1(µi − µj) (62)
and
x0 =
1
2
(µi + µj)−
ln [P (ωi)/P (ωj)]
(µi − µj)tΣ−1(µi − µj)
(µi − µj). (63)
Since w = Σ−1(µi−µj) is generally not in the direction of µi−µj , the hyperplane
separating Ri and Rj is generally not orthogonal to the line between the means.
However, it does intersect that line at the point x0 which is halfway between the
means if the prior probabilities are equal. If the prior probabilities are not equal, the
optimal boundary hyperplane is shifted away from the more likely mean (Fig. 2.12).
As before, with sufficient bias the decision plane need not lie between the two mean
vectors.
24 CHAPTER 2. BAYESIAN DECISION THEORY
p
P(ω1)=.5
P(ω2)=.5
-5
0
5
-5
0
5
0
0.1
0.2
p
-5
0
5
P(ω1)=.9
P(ω2)=.1
-2
0
2
-2
0
2
4
-2.5
0
2.5
5
7.5
-2
0
2
-2.5
0
2.5
5
7.5
P(ω2)=.5
P(ω1)=.5
-2
0
2
-2
0
2
4
0
2.5
5
7.5
10
-2
0
2
0
2.5
5
7.5
10
P(ω1)=.9
P(ω2)=.1
Figure 2.12: Probability densities (indicated by the surfaces in two dimensions and
ellipsoidal surfaces in three dimensions) and decision regions for equal but asymmetric
Gaussian distributions. The decision hyperplanes need not be perpendicular to the
line connecting the means.
2.6. DISCRIMINANT FUNCTIONS FOR THE NORMAL DENSITY 25
2.6.3 Case 3: Σi = arbitrary
In the general multivariate normal case, the covariance matrices are different for each
category. The only term that can be dropped from Eq. 47 is the (d/2) ln 2π term,
and the resulting discriminant functions are inherently quadratic:
gi(x) = x
tWix + w
t
ix + wi0, (64)
where
Wi = −
1
2
Σ−1i , (65)
wi = Σ
−1
i µi (66)
and
wi0 = −
1
2
µtiΣ
−1
i µi −
1
2
ln |Σi|+ ln P (ωi). (67)
The decision surfaces are hyperquadrics, and can assume any of the general forms hyper-
quadric— hyperplanes, pairs of hyperplanes, hyperspheres, hyperellipsoids, hyperparaboloids,
and hyperhyperboloids of various types (Problem 29). Even in one dimension, for
arbitrary covariance the decision regions need not be simply connected (Fig. 2.13).
The two- and three-dimensional examples in Fig. 2.14 & 2.15 indicate how these
different forms can arise. These variances are indicated by the contours of constant
probability density.
The extension of these results to more than two categories is straightforward
though we need to keep clear which two of the total c categories are responsible for
any boundary segment. Figure 2.16 shows the decision surfaces for a four-category
case made up of Gaussian distributions. Of course, if the distributions are more com-
plicated, the decision regions can be even more complex, though the same underlying
theory holds there too.
-5 -2.5 2.5 5 7.5
0.1
0.2
0.3
0.4
x
p(x|ωi)
R2 R1R1
ω2
ω1
Figure 2.13: Non-simply connected decision regions can arise in one dimensions for
Gaussians having unequal variance.
26 CHAPTER 2. BAYESIAN DECISION THEORY
-10
0
10
20
-10
0
10
20
0
.01
.02
.03
.04
p
-10
0
10
20 -20
-10
0
10
20
-10
0
10
20
0
0.005
0.01
p
-20
-10
0
10
20
-10
0
10
20
-10
0
10
0
0.01
0.02
0.03
p
-10
0
10
20
-10
0
10
20
-10
0
10
0
0.01
0.02
0.03
0.04
0.05
p
-10
0
10
20
-10
0
10
20
-10
0
10
20
0
0.005
0.01
P
-10
0
10
20
0
0.005
0.01
p
-5
0
5
-5
0
5
0
0.05
0.1
0.15
p
-5
0
5
Figure 2.14: Arbitrary Gaussian distributions lead to Bayes decision boundaries that
are general hyperquadrics. Conversely, given any hyperquadratic, one can find two
Gaussian distributions whose Bayes decision boundary is that hyperquadric.
2.6. DISCRIMINANT FUNCTIONS FOR THE NORMAL DENSITY 27
0
2.5
5
7.5
-2.5
0
2.5
5
0
2
4
6
0
2.5
5
7.5
-20
0
20
0
20
40
-10
0
10
-20
0
20
0
5
10 -5
0
5
10
-2
0
2
4
6
0
5
10
0
5
10
0
5
10
0
5
10
0
5
10
-2
0
2
4
6
0
5
10
-4
-2
0
2
4
-2
0
2
4
6
-5
0
5 0
5
10
-5
0
5
10
-5
0
5
0
0.5
1
1.5
2
-1
-0.5
0
0.5
1
0
0.5
1
0
0.5
1
1.5
2
0
2
4
6
0
5
-1
0
1
2
3
0
2
4
6
0
1
2
3
4
-1
0
1
2
3
-1
0
1
2
0
1
2
3
4
-10
0
10 -20
-10
0
10
20
-5
0
5
-10
0
10
0
.5
1
1.5 -.2
0
.2
-.5
0
.5
D
ecision boundary
Figure 2.15: Arbitrary three-dimensional Gaussian distributions yield Bayes decision
boundaries that are two-dimensional hyperquadrics. There are even degenerate cases
in which the decision boundary is a line.
28 CHAPTER 2. BAYESIAN DECISION THEORY
ω1
ω2 ω4
ω3ω4
Figure 2.16: The decision regions for four normal distributions. Even with such a low
number of categories, the shapes of the boundary regions can be rather complex.
2.6. DISCRIMINANT FUNCTIONS FOR THE NORMAL DENSITY 29
Example 1: Decision regions for two-dimensional Gaussian data
To clarify these ideas, we explicitly calculate the decision boundary for the two-
category two-dimensional data in the Example figure. Let ω1 be the set of the four
black points, and ω2 the red points. Although we will spend much of the next chapter
understanding how to estimate the parameters of our distributions, for now we simply
assume that we need merely calculate the means and covariances by the discrete
versions of Eqs. 39 & 40; they are found to be:
µ1 =
[
3
6
]
; Σ1 =
(
1/2 0
0 2
)
and µ2 =
[
3
−2
]
; Σ2 =
(
2 0
0 2
)
.
The inverse matrices are then,
Σ−11 =
(
2 0
0 1/2
)
and Σ−12 =
(
1/2 0
0 1/2
)
.
We assume equal prior probabilities, P (ω1) = P (ω2) = 0.5, and substitute these into
the general form for a discriminant, Eqs. 64 – 67, setting g1(x) = g2(x) to obtain the
decision boundary:
x2 = 3.514− 1.125×1 + 0.1875×21.
This equation describes a parabola with vertex at
(
3
1.83
)
. Note that despite the
fact that the variance in the data along the x2 direction for both distributions is the
same, the decision boundary does not pass through the point
(
3
2
)
, midway between
the means, as we might have naively guessed. This is because for the ω1 distribution,
the probability distribution is “squeezed” in the x1-direction more so than for the ω2
distribution. Because the overall prior probabilities are the same (i.e., the integral over
space of the probability density), the distribution is increased along the x2 direction
(relative to that for the ω2 distribution). Thus the decision boundary lies slightly
lower than the point midway between the two means, as can be seen in the decision
boundary.
x1
x2
µ2
µ1
-2 2 4 6 8 10
-2.5
2.5
5
7.5
10
The computed Bayes decision boundary for two Gaussian distributions, each based
on four data points.
30 CHAPTER 2. BAYESIAN DECISION THEORY
2.7 Error Probabilities and Integrals
We can obtain additional insight into the operation of a general classifier — Bayes or
otherwise — if we consider the sources of its error. Consider first the two-category
case, and suppose the dichotomizer has divided the space into two regions R1 and R2
in a possibly non-optimal way. There are two ways in which a classification error can
occur; either an observation x falls in R2 and the true state of nature is ω1, or x falls
in R1 and the true state of nature is ω2. Since these events are mutually exclusive
and exhaustive, the probability of error is
P (error) = P (x ∈ R2, ω1) + P (x ∈ R1, ω2)
= P (x ∈ R2|ω1)P (ω1) + P (x ∈ R1|ω2)P (ω2)
=
∫
R2
p(x|ω1)P (ω1) dx +
∫
R1
p(x|ω2)P (ω2) dx. (68)
This result is illustrated in the one-dimensional case in Fig. 2.17. The two in-
tegrals in Eq. 68 represent the pink and the gray areas in the tails of the functions
p(x|ωi)P (ωi). Because the decision point x∗ (and hence the regions R1 and R2) were
chosen arbitrarily for that figure, the probability of error is not as small as it might
be. In particular, the triangular area marked “reducible error” can be eliminated if
the decision boundary is moved to xB . This is the Bayes optimal decision boundary
and gives the lowest probability of error. In general, if p(x|ω1)P (ω1) > p(x|ω2)P (ω2),
it is advantageous to classify x as in R1 so that the smaller quantity will contribute
to the error integral; this is exactly what the Bayes decision rule achieves.
ω2ω1
x
x* R2R1
p(x|ωi)P(ωi)
reducible
error
∫p(x|ω1)P(ω1)dx
R2
∫p(x|ω2)P(ω2)dx
R1
xB
Figure 2.17: Components of the probability of error for equal priors and (non-optimal)
decision point x∗. The pink area corresponds to the probability of errors for deciding
ω1 when the state of nature is in fact ω2; the gray area represents the converse, as
given in Eq. 68. If the decision boundary is instead at the point of equal posterior
probabilities, xB , then this reducible error is eliminated and the total shaded area is
the minimum possible — this is the Bayes decision and gives the Bayes error rate.
In the multicategory case, there are more ways to be wrong than to be right, and
it is simpler to compute the probability of being correct. Clearly
P (correct) =
c∑
i=1
P (x ∈ Ri, ωi)
2.8. *ERROR BOUNDS FOR NORMAL DENSITIES 31
=
c∑
i=1
P (x ∈ Ri|ωi)P (ωi)
=
c∑
i=1
∫
Ri
p(x|ωi)P (ωi) dx. (69)
The general result of Eq. 69 depends neither on how the feature space is partitioned
into decision regions nor on the form of the underlying distributions. The Bayes
classifier maximizes this probability by choosing the regions so that the integrand is
maximal for all x; no other partitioning can yield a smaller probability of error.
2.8 Error Bounds for Normal Densities
The Bayes decision rule guarantees the lowest average error rate, and we have seen
how to calculate the decision boundaries for normal densities. However, these results
do not tell us what the probability of error actually is. The full calculation of the error
for the Gaussian case would be quite difficult, especially in high dimensions, because
of the discontinuous nature of the decision regions in the integral in Eq. 69. However,
in the two-category case the general error integral of Eq. 5 can be approximated
analytically to give us an upper bound on the error.
2.8.1 Chernoff Bound
To derive a bound for the error, we need the following inequality:
min[a, b] ≤ aβb1−β for a, b ≥ 0 and 0 ≤ β ≤ 1. (70)
To understand this inequality we can, without loss of generality, assume a ≥ b. Thus
we need only show that b ≤ aβb1−β = (a
b
)βb. But this inequality is manifestly valid,
since (a
b
)β ≥ 1. Using Eqs. 7 & 1, we apply this inequality to Eq. 5 and get the bound:
P (error) ≤ P β(ω1)P 1−β(ω2)
∫
pβ(x|ω1)p1−β(x|ω2) dx for 0 ≤ β ≤ 1. (71)
Note especially that this integral is over all feature space — we do not need to impose
integration limits corresponding to decision boundaries.
If the conditional probabilities are normal, the integral in Eq. 71 can be evaluated
analytically (Problem 35), yielding:
∫
pβ(x|ω1)p1−β(x|ω2) dx = e−k(β) (72)
where
k(β) =
β(1− β)
2
(µ2 − µ1)t[βΣ1 + (1− β)Σ2]−1(µ2 − µ1) +
1
2
ln
|βΣ1 + (1− β)Σ2|
|Σ1|β |Σ2|1−β
. (73)
The graph in Fig. 2.18 shows a typical example of how e−k(β) varies with β. The
Chernoff bound, on P (error) is found by analytically or numerically finding the value
32 CHAPTER 2. BAYESIAN DECISION THEORY
of β that minimizes e−k(β), and substituting the results in Eq. 71. The key benefit
here is that this optimization is in the one-dimensional β space, despite the fact that
the distributions themselves might be in a space of arbitrarily high dimension.
0 0.25 0.75 1
0
0.2
0.4
0.6
0.8
1
Chernoff bound
Bhattacharyya bound
e-k(β)
β
0.5
β∗
Figure 2.18: The Chernoff error bound is never looser than the Bhattacharyya bound.
For this example, the Chernoff bound happens to be at β∗ = 0.66, and is slightly
tighter than the Bhattacharyya bound (β = 0.5).
2.8.2 Bhattacharyya Bound
The general dependence of the Chernoff bound upon β shown in Fig. 2.18 is typical
of a wide range of problems — the bound is loose for extreme values (i.e., β → 1 and
β → 0), and tighter for intermediate ones. While the precise value of the optimal
β depends upon the parameters of the distributions and the prior probabilities, a
computationally simpler, but slightly less tight bound can be derived by merely asing
the results for β = 1/2. This result is the so-called Bhattacharyya bound on the error,
where Eq. 71 then has the form
P (error) ≤
√
P (ω1)P (ω2)
∫ √
p(x|ω1)p(x|ω2) dx
=
√
P (ω1)P (ω2)e
−k(1/2), (74)
where by Eq. 73 we have for the Gaussian case:
k(1/2) = 1/8(µ2 − µ1)t
[Σ1 + Σ2
2
]−1
(µ2 − µ1) +
1
2
ln
∣∣∣Σ1+Σ22
∣∣∣√
|Σ1||Σ2|
. (75)
The Chernoff and Bhatacharyya bounds may still be used even if the underlying
distributions are not Gaussian. However, for distributions that deviate markedly from
a Gaussian, the bounds will not be informative (Problem 32).
2.8. *ERROR BOUNDS FOR NORMAL DENSITIES 33
Example 2: Error bounds for Gaussian distributions.
It is a straightforward matter to calculate the Bhattacharyya bound for the two-
dimensional data sets of Example 1. Substituting the means and covariances of Exam-
ple 1 into Eq. 75 we find k(1/2) = 4.11 and thus by Eqs. 74 & 75 the Bhattacharyya
bound on the error is P (error) ≤ 0.016382.
A tighter bound on the error can be approximated by searching numerically for the
Chernoff bound of Eq. 73, which for this problem gives 0.016380. One can get the best
estimate by numerically integrating the error rate directly Eq. 5, which gives 0.0021,
and thus the bounds here are not particularly tight. Such numerical integration is
often impractical for Gaussians in higher than two or three dimensions.
2.8.3 Signal Detection Theory and Operating Characteristics
Another measure of distance between two Gaussian distributions has found great
use in experimental psychology, radar detection and other fields. Suppose we are
interested in detecting a single weak pulse, such as a dim flash of light or a weak
radar reflection. Our model is, then, that at some point in the detector there is an
internal signal (such as a voltage) x, whose value has mean µ2 when the external signal
(pulse) is present, and mean µ1 when it is not present. Because of random noise —
within and outside the detector itself — the actual value is a random variable. We
assume the distributions are normal with different means but the same variance, i.e.,
p(x|ωi) ∼ N(µi, σ2), as shown in Fig. 2.19.
σσ
µ1 x* µ2
x
p(x|ωi) ω2ω1
Figure 2.19: During any instant when no external pulse is present, the probability
density for an internal signal is normal, i.e., p(x|ω1) ∼ N(µ1, σ2); when the external
signal is present, the density is p(x|ω2) ∼ N(µ2, σ2). Any decision threshold x∗ will
determine the probability of a hit (the red area under the ω2 curve, above x∗) and of
a false alarm (the black area under the ω1 curve, above x∗).
The detector (classifier) employs a threshold value x∗ for determining whether the
external pulse is present, but suppose we, as experimenters, do not have access to this
value (nor to the means and standard deviations of the distributions). We seek to
find some measure of the ease of discriminating whether the pulse is present or not, in
a form independent of the choice of x∗. Such a measure is the discriminability, which discrimin-
abilitydescribes the inherent and unchangeable properties due to noise and the strength of
the external signal, but not on the decision strategy (i.e., the actual choice of x∗).
This discriminability is defined as
34 CHAPTER 2. BAYESIAN DECISION THEORY
d′ =
|µ2 − µ1|
σ
. (76)
A high d′ is of course desirable.
While we do not know µ1, µ2, σ nor x∗, we assume here that we know the state
of nature and the decision of the system. Such information allows us to find d′. To
this end, we consider the following four probabilities:
• P (x > x∗|x ∈ ω2): a hit — the probability that the internal signal is above x∗
given that the external signal is present
• P (x > x∗|x ∈ ω1): a false alarm — the probability that the internal signal is
above x∗ despite there being no external signal is present
• P (x < x∗|x ∈ ω2): a miss — the probability that the internal signal is below x∗
given that the external signal is present
• P (x < x∗|x ∈ ω1): a correct rejection — the probability that the internal signal
is below x∗ given that the external signal is not present.
If we have a large number of trials (and we can assume x∗ is fixed, albeit at an
unknown value), we can determine these probabilities experimentally, in particular
the hit and false alarm rates. We plot a point representing these rates on a two-
dimensional graph. If the densities are fixed but the threshold x∗ is changed, then our
hit and false alarm rates will also change. Thus we see that for a given discriminability
d′, our point will move along a smooth curve — a receiver operating characteristic orreceiver
operating
character-
istic
ROC curve (Fig. 2.20).
P(x < x*|x ∈ ω2)
P
(x
>
x
*|
x
∈
ω
2)
d’=0
d’=1
d’=2
d’=3
1
1
false alarm
hit
Figure 2.20: In a receiver operating characteristic (ROC) curve, the abscissa is the
probability of false alarm, P (x > x∗|x ∈ ω1), and the ordinate the probability of hit,
P (x > x∗|x ∈ ω2). From the measured hit and false alarm rates (here corresponding
to x∗ in Fig. 2.19, and shown as the red dot), we can deduce that d′ = 3.
The great benefit of this signal detection framework is that we can distinguish
operationally between discriminability and decision bias — while the former is an
inherent property of the detector system, the latter is due to the receiver’s implied
but changeable loss matrix. Through any pair of hit and false alarm rates passes
one and only one ROC curve; thus, so long as neither rate is exactly 0 or 1, we
can determine the discriminability from these rates (Problem 38). Moreover, if the
Gaussian assumption holds, a determination of the discriminability (from an arbitrary
x∗) allows us to calculate the Bayes error rate — the most important property of any
2.8. *ERROR BOUNDS FOR NORMAL DENSITIES 35
classifier. If the actual error rate differs from the Bayes rate inferred in this way, we
should alter the threshold x∗ accordingly.
It is a simple matter to generalize the above discussion and apply it to two cate-
gories having arbitrary multidimensional distributions, Gaussian or not. Suppose we
have two distributions p(x|ω1) and p(x|ω2) which overlap, and thus have non-zero
Bayes classification error. Just as we saw above, any pattern actually from ω2 could
be properly classified as ω2 (a “hit”) or misclassified as ω1 (a “false alarm”). Unlike
the one-dimensional case above, however, there may be many decision boundaries
that give a particular hit rate, each with a different false alarm rate. Clearly here we
cannot determine a fundamental measure of discriminability without knowing more
about the underlying decision rule than just the hit and false alarm rates.
In a rarely attainable ideal, we can imagine that our measured hit and false alarm
rates are optimal, for example that of all the decision rules giving the measured hit
rate, the rule that is actually used is the one having the minimum false alarm rate.
If we constructed a multidimensional classifier — regardless of the distributions used
— we might try to characterize the problem in this way, though it would probably
require great computational resources to search for such optimal hit and false alarm
rates.
In practice, instead we eschew optimality, and simply vary a single parameter
controlling the decision rule and plot the resulting hit and false alarm rates — a
curve called merely an operating characteristic. Such a control parameter might be operating
character-
istic
the bias or nonlinearity in a discriminant function. It is traditional to choose a
control parameter that can yield, at extreme values, either a vanishing false alarm
or a vanishing hit rate, just as can be achieved with a very large or a very small x∗
in an ROC curve. We should note that since the distributions can be arbitrary, the
operating characteristic need not be symmetric (Fig. 2.21); in rare cases it need not
even be concave down at all points.
x
p(x|ωi) ω1
ω2
1
1
p(x < x* | x ω2)
p(
x
<
x
*
| x
ω
1)
hit
false alarm
Figure 2.21: In a general operating characteristic curve, the abscissa is the probability
of false alarm, P (x ∈ R2|x ∈ ω1), and the ordinate the probability of hit, P (x ∈
R2|x ∈ ω2). As illustrated here, operating characteristic curves are generally not
symmetric, as shown at the right.
Classifier operating curves are of value for problems where the loss matrix λij
might be changed. If the operating characteristic has been determined as a function
of the control parameter ahead of time, it is a simple matter, when faced with a new
loss function, to deduce the control parameter setting that will minimize the expected
risk (Problem 38).
36 CHAPTER 2. BAYESIAN DECISION THEORY
2.9 Bayes Decision Theory — Discrete Features
Until now we have assumed that the feature vector x could be any point in a d-
dimensional Euclidean space, Rd. However, in many practical applications the com-
ponents of x are binary-, ternary-, or higher integer valued, so that x can assume only
one of m discrete values v1, ...,vm. In such cases, the probability density function
p(x|ωj) becomes singular; integrals of the form
∫
p(x|ωj) dx (77)
must then be replaced by corresponding sums, such as
∑
x
P (x|ωj), (78)
where we understand that the summation is over all values of x in the discrete
distribution.∗ Bayes’ formula then involves probabilities, rather than probability den-
sities:
P (ωj |x) =
P (x|ωj)P (ωj)
P (x)
, (79)
where
P (x) =
c∑
j=1
P (x|ωj)P (ωj). (80)
The definition of the conditional risk R(α|x) is unchanged, and the fundamental
Bayes decision rule remains the same: To minimize the overall risk, select the action
αi for which R(αi|x) is minimum, or stated formally,
α∗ = arg max
i
R(αi|x). (81)
The basic rule to minimize the error-rate by maximizing the posterior probability is
also unchanged as are the discriminant functions of Eqs. 25 – 27, given the obvious
replacement of densities p(·) by probabilities P (·).
2.9.1 Independent Binary Features
As an example of a classification involving discrete features, consider the two-category
problem in which the components of the feature vector are binary-valued and condi-
tionally independent. To be more specific we let x = (x1, ..., xd)t, where the compo-
nents xi are either 0 or 1, with
pi = Prob (xi = 1|ω1) (82)
and
qi = Prob (xi = 1|ω2). (83)
∗ Technically speaking, Eq. 78 should be written as
∑
k
P (vk|ωj) where P (vk|ωj) is the conditional
probability that x = vk given that the state of nature is ωj .
2.9. BAYES DECISION THEORY — DISCRETE FEATURES 37
This is a model of a classification problem in which each feature gives us a yes/no
answer about the pattern. If pi > qi, we expect the ith feature to give a “yes” answer
more frequently when the state of nature is ω1 than when when it is ω2. (As an
example, consider two factories each making the same automobile, each of whose d
components could be functional or defective. If it was known how the factories differed
in their reliabilities for making each component, then this model could be used to judge
which factory manufactured a given automobile based on the knowledge of which
features are functional and which defective.) By assuming conditional independence
we can write P (x|ωi) as the product of the probabilities for the components of x.
Given this assumption, a particularly convenient way of writing the class-conditional
probabilities is as follows:
P (x|ω1) =
d∏
i=1
pxii (1− pi)
1−xi (84)
and
P (x|ω2) =
d∏
i=1
qxii (1− qi)
1−xi . (85)
Then the likelihood ratio is given by
P (x|ω1)
P (x|ω2)
=
d∏
i=1
(pi
qi
)xi(1− pi
1− qi
)1−xi
(86)
and consequently Eq. 30 yields the discriminant function
g(x) =
d∑
i=1
[
xi ln
pi
qi
+ (1− xi) ln
1− pi
1− qi
]
+ ln
P (ω1)
P (ω2)
. (87)
We note especially that this discriminant function is linear in the xi and thus we can
write
g(x) =
d∑
i=1
wixi + w0, (88)
where
wi = ln
pi(1− qi)
qi(1− pi)
i = 1, …, d (89)
and
w0 =
d∑
i=1
ln
1− pi
1− qi
+ ln
P (ω1)
P (ω2)
. (90)
Let us examine these results to see what insight they can give. Recall first that
we decide ω1 if g(x) > 0 and ω2 if g(x) ≤ 0. We have seen that g(x) is a weighted
combination of the components of x. The magnitude of the weight wi indicates the
relevance of a “yes” answer for xi in determining the classification. If pi = qi, xi gives
us no information about the state of nature, and wi = 0, just as we might expect.
If pi > qi, then 1 − pi < 1 − qi and wi is positive. Thus in this case a “yes” answer
38 CHAPTER 2. BAYESIAN DECISION THEORY
for xi contributes wi votes for ω1. Furthermore, for any fixed qi < 1, wi gets larger
as pi gets larger. On the other hand, if pi < qi, wi is negative and a “yes” answer
contributes |wi| votes for ω2.
The condition of feature independence leads to a very simple (linear) classifier;
of course if the features were not independent, a more complicated classifier would
be needed. We shall come across this again for systems with continuous features in
Chap. ??, but note here that the more independent we can make the features, the
simpler the classifier can be.
The prior probabilities P (ωi) appear in the discriminant only through the thresh-
old weight w0. Increasing P (ω1) increases w0 and biases the decision in favor of ω1,
whereas decreasing P (ω1) has the opposite effect. Geometrically, the possible values
for x appear as the vertices of a d-dimensional hypercube; the decision surface defined
by g(x) = 0 is a hyperplane that separates ω1 vertices from ω2 vertices.
Example 3: Bayesian decisions for three-dimensional binary features
Suppose two categories consist of independent binary features in three dimensions
with known feature probabilities. Let us construct the Bayesian decision boundary if
P (ω1) = P (ω2) = 0.5 and the individual components obey:
{
pi = 0.8
qi = 0.5
i = 1, 2, 3.
By Eqs. 89 & 90 we have that the weights are
wi = ln
.8(1− .5)
.5(1− .8)
= 1.3863
and the bias value is
w0 =
3∑
i=1
ln
1− .8
1− .5
+ ln
.5
.5
= 1.2.
0
.25
.5
.75
1
0
.25
.5
.75
1
0
.25
.5
.75
1
x1
x3
x2
x3
x1
x2
0
.25
.5
.75
1
0
.25
.5
.75
1
0
.25
.5
.75
1
The decision boundary for the Example involving three-dimensional binary features.
On the left we show the case pi = .8 and qi = .5. On the right we use the same values
except p3 = q3, which leads to w3 = 0 and a decision surface parallel to the x3 axis.
2.10. *MISSING AND NOISY FEATURES 39
The surface g(x) = 0 from Eq. 88 is shown on the left of the figure. Indeed, as we
might have expected, the boundary places points with two or more “yes” answers into
category ω1, since that category has a higher probability of having any feature have
value 1.
Suppose instead that while the prior probabilities remained the same, our individ-
ual components obeyed:
{
p1 = p2 = 0.8, p3 = 0.5
q1 = q2 = q3 = 0.5
In this case feature x3 gives us no predictive information about the categories, and
hence the decision boundary is parallel to the x3 axis. Note that in this discrete case
there is a large range in positions of the decision boundary that leaves the categoriza-
tion unchanged, as is particularly clear in the figure on the right.
2.10 Missing and Noisy Features
If we know the full probability structure of a problem, we can construct the (optimal)
Bayes decision rule. Suppose we develop a Bayes classifier using uncorrupted data,
but our input (test) data are then corrupted in particular known ways. How can we
classify such corrupted inputs to obtain a minimum error now?
There are two analytically solvable cases of particular interest: when some of the
features are missing, and when they are corrupted by a noise source with known
properties. In each case our basic approach is to recover as much information about
the underlying distribution as possible and use the Bayes decision rule.
2.10.1 Missing Features
Suppose we have a Bayesian (or other) recognizer for a problem using two features,
but that for a particular pattern to be classified, one of the features is missing.∗ For
example, we can easily imagine that the lightness can be measured from a portion of
a fish, but the width cannot because of occlusion by another fish.
We can illustrate with four categories a somewhat more general case (Fig. 2.22).
Suppose for a particular test pattern the feature x1 is missing, and the measured value
of x2 is x̂2. Clearly if we assume the missing value is the mean of all the x1 values,
i.e., x̄1, we will classify the pattern as ω3. However, if the priors are equal, ω2 would
be a better decision, since the figure implies that p(x̂2|ω2) is the largest of the four
likelihoods.
To clarify our derivation we let x = [xg,xb], where xg represents the known or
“good” features and xb represents the “bad” ones, i.e., either unknown or missing. We
seek the Bayes rule given the good features, and for that the posterior probabilities
are needed. In terms of the good features the posteriors are
P (ωi|xg) =
p(ωi,xg)
p(xg)
=
∫
p(ωi,xg,xb) dxb
p(xg)
∗ In practice, just determining that the feature is in fact missing rather than having a value of zero
(or the mean value) can be difficult in itself.
40 CHAPTER 2. BAYESIAN DECISION THEORY
ω1 ω4ω3
ω2
x2
x1
x1
_
x2
^
Figure 2.22: Four categories have equal priors and the class-conditional distributions
shown. If a test point is presented in which one feature is missing (here, x1) and the
other is measured to have value x̂2 (red dashed line), we want our classifier to classify
the pattern as category ω2, because p(x̂2|ω2) is the largest of the four likelihoods.
=
∫
P (ωi|xg,xb)p(xg,xb) dxb
p(xg)
=
∫
gi(x)p(x) dxb∫
p(x) dxb
, (91)
where gi(x) = gi(xg,xb) = P (ωi|xg,xb) is one form of our discriminant function.
We refer to
∫
p(ωi,xg,xb) dxb, as a marginal distribution; we say the full jointmarginal
distribution is marginalized over the variable xb. In short, Eq. 91 shows that we must
integrate (marginalize) the posterior probability over the bad features. Finally we
use the Bayes decision rule on the resulting posterior probabilities, i.e., choose ωi if
P (ωi|xg) > P (ωj |xg) for all i and j. We shall consider the Expectation-Maximization
(EM) algorithm in Chap. ??, which addresses a related problem involving missing
features.
2.10.2 Noisy Features
It is a simple matter to generalize the results of Eq. 91 to the case where a particular
feature has been corrupted by statistically independent noise.∗ For instance, in our
fish classification example, we might have a reliable measurement of the length, while
variability of the light source might degrade the measurement of the lightness. We
assume we have uncorrupted (good) features xg, as before, and a noise model, ex-
pressed as p(xb|xt). Here we let xt denote the true value of the observed xb features,
i.e., without the noise present; that is, the xb are observed instead of the true xt. We
assume that if xt were known, xb would be independent of ωi and xg. From such an
assumption we get:
P (ωi|xg,xb) =
∫
p(ωi,xg,xb,xt) dxt
p(xg,xb)
. (92)
∗ Of course, to tell the classifier that a feature value is missing, the feature extractor must be designed
to provide more than just a numerical value for each feature.
2.11. *COMPOUND BAYES DECISION THEORY AND CONTEXT 41
Now p(ωi,xg,xb,xt) = P (ωi|xg,xb,xt)p(xg,xb,xt), but by our independence assump-
tion, if we know xt, then xb does not provide any additional information about ωi.
Thus we have P (ωi|xg,xb,xt) = P (ωi|xg,xt). Similarly, we have p(xg,xb,xt) =
p(xb|xg,xt)p(xg,xt), and p(xb|xg,xt) = p(xb|xt). We put these together and thereby
obtain
P (ωi|xg,xb) =
∫
P (ωi|xg,xt)p(xg,xt)p(xb|xt) dxt∫
p(xg,xt)p(xb|xt) dxt
=
∫
gi(x)p(x)p(xb|xt) dxt∫
p(x)p(xb|xt) dxt
, (93)
which we use as discriminant functions for classification in the manner dictated by
Bayes.
Equation 93 differs from Eq. 91 solely by the fact that the integral is weighted
by the noise model. In the extreme case where p(xb|xt) is uniform over the entire
space (and hence provides no predictive information for categorization), the equation
reduces to the case of missing features — a satisfying result.
2.11 Compound Bayesian Decision Theory and Con-
text
Let us reconsider our introductory example of designing a classifier to sort two types
of fish. Our original assumption was that the sequence of types of fish was so unpre-
dictable that the state of nature looked like a random variable. Without abandoning
this attitude, let us consider the possibility that the consecutive states of nature might
not be statistically independent. We should be able to exploit such statistical depen-
dence to gain improved performance. This is one example of the use of context to aid
decision making.
The way in which we exploit such context information is somewhat different when
we can wait for n fish to emerge and then make all n decisions jointly than when
we must decide as each fish emerges. The first problem is a compound decision prob-
lem, and the second is a sequential compound decision problem. The former case is
conceptually simpler, and is the one we shall examine here.
To state the general problem, let ω = (ω(1), …, ω(n))t be a vector denoting the n
states of nature, with ω(i) taking on one of the c values ω1, …, ωc. Let P (ω) be the
prior probability for the n states of nature. Let X = (x1, …,xn) be a matrix giving
the n observed feature vectors, with xi being the feature vector obtained when the
state of nature was ω(i). Finally, let p(X|ω) be the conditional probability density
function for X given the true set of states of nature ω. Using this notation we see
that the posterior probability of ω is given by
P (ω|X) = p(X|ω)P (ω)
p(X)
=
p(X|ω)P (ω)∑
ω p(X|ω)P (ω)
. (94)
In general, one can define a loss matrix for the compound decision problem and
seek a decision rule that minimizes the compound risk. The development of this
theory parallels our discussion for the simple decision problem, and concludes that
the optimal procedure is to minimize the compound conditional risk. In particular, if
there is no loss for being correct, and if all errors are equally costly, then the procedure
42 CHAPTER 2. BAYESIAN DECISION THEORY
reduces to computing P (ω|X) for all ω and selecting the ω for which this posterior
probability is maximum.
While this provides the theoretical solution, in practice the computation of P (ω|X)
can easily prove to be an enormous task. If each component ω(i) can have one of
c values, there are cn possible values of ω to consider. Some simplification can be
obtained if the distribution of the feature vector xi depends only on the corresponding
state of nature ω(i), not on the values of the other feature vectors or the other states of
nature. In this case the joint density p(X|ω) is merely the product of the component
densities p(xi|ω(i)):
p(X|ω) =
n∏
i=1
p(xi|ω(i)). (95)
While this simplifies the problem of computing p(X|ω), there is still the problem
of computing the prior probabilities P (ω). This joint probability is central to the
compound Bayes decision problem, since it reflects the interdependence of the states
of nature. Thus it is unacceptable to simplify the problem of calculating P (ω) by
assuming that the states of nature are independent. In addition, practical applications
usually require some method of avoiding the computation of P (ω|X) for all cn possible
values of ω. We shall find some solutions to this problem in Chap. ??.
Summary
The basic ideas underlying Bayes decision theory are very simple. To minimize the
overall risk, one should always choose the action that minimizes the conditional risk
R(α|x). In particular, to minimize the probability of error in a classification problem,
one should always choose the state of nature that maximizes the posterior probability
P (ωj |x). Bayes’ formula allows us to calculate such probabilities from the prior prob-
abilities P (ωj) and the conditional densities p(x|ωj). If there are different penalties
for misclassifying patterns from ωi as if from ωj , the posteriors must be first weighted
according to such penalties before taking action.
If the underlying distributions are multivariate Gaussian, the decision boundaries
will be hyperquadrics, whose form and position depends upon the prior probabilities,
means and covariances of the distributions in question. The true expected error
can be bounded above by the Chernoff and computationally simpler Bhattacharyya
bounds. If an input (test) pattern has missing or corrupted features, we should form
the marginal distributions by integrating over such features, and then using Bayes
decision procedure on the resulting distributions. Receiver operating characteristic
curves describe the inherent and unchangeable properties of a classifier and can be
used, for example, to determine the Bayes rate.
For many pattern classification applications, the chief problem in applying these
results is that the conditional densities p(x|ωj) are not known. In some cases we may
know the form these densities assume, but may not know characterizing parameter
values. The classic case occurs when the densities are known to be, or can assumed
to be multivariate normal, but the values of the mean vectors and the covariance
matrices are not known. More commonly even less is known about the conditional
densities, and procedures that are less sensitive to specific assumptions about the
densities must be used. Most of the remainder of this book will be devoted to various
procedures that have been developed to attack such problems.
2.11. BIBLIOGRAPHICAL AND HISTORICAL REMARKS 43
Bibliographical and Historical Remarks
The power, coherence and elegance of Bayesian theory in pattern recognition make
it among the most beautiful formalisms in science. Its foundations go back to Bayes
himself, of course [3], but he stated his theorem (Eq. 1) for the case of uniform
priors. It was Laplace [25] who first stated it for the more general (but discrete) case.
There are several modern and clear descriptions of the ideas — in pattern recognition
and general decision theory — that can be recommended [7, 6, 26, 15, 13, 20, 27].
Since Bayesian theory rests on an axiomatic foundation, it is guaranteed to have
quantitative coherence; some other classification methods do not. Wald presents a
non-Bayesian perspective on these topics that can be highly recommended [36], and
the philosophical foundations of Bayesian and non-Bayesian methods are explored in
[16]. Neyman and Pearson provided some of the most important pioneering work
in hypothesis testing, and used the probability of error as the criterion [28]; Wald
extended this work by introducing the notions of loss and risk [35]. Certain conceptual
problems have always attended the use of loss functions and prior probabilities. In
fact, the Bayesian approach is avoided by many statisticians, partly because there are
problems for which a decision is made only once, and partly because there may be no
reasonable way to determine the prior probabilities. Neither of these difficulties seems
to present a serious drawback in typical pattern recognition applications: for nearly
all critical pattern recognition problems we will have training data; we will use our
recognizer more than once. For these reasons, the Bayesian approach will continue
to be of great use in pattern recognition. The single most important drawback of the
Bayesian approach is its assumption that the true probability distributions for the
problem can be represented by the classifier, for instance the true distributions are
Gaussian, and all that is unknown are parameters describing these Gaussians. This
is a strong assumption that is not always fulfilled and we shall later consider other
approaches that do not have this requirement.
Chow[10] was among the earliest to use Bayesian decision theory for pattern recog-
nition, and he later established fundamental relations between error and reject rate
[11]. Error rates for Gaussians have been explored by [18], and the Chernoff and
Bhattacharyya bounds were first presented in [9, 8], respectively and are explored in
a number of statistics texts, such as [17]. Computational approximations for bound-
ing integrals for Bayesian probability of error (the source for one of the homework
problems) appears in [2]. Neyman and Pearson also worked on classification given
constraints [28], and the analysis of minimax estimators for multivariate normals is
presented in [5, 4, 14]. Signal detection theory and receiver operating characteristics
are fully explored in [21]; a brief overview, targetting experimental psychologists, is
[34]. Our discussion of the missing feature problem follows closely the work of [1] while
the definitive book on missing features, including a great deal beyond our discussion
here, can be found in [30].
Entropy was the central concept in the foundation of information theory [31] and
the relation of Gaussians to entropy is explored in [33]. Readers requiring a review of
information theory [12], linear algebra [24, 23], calculus and continuous mathematics,
[38, 32] probability [29] calculus of variations and Lagrange multipliers [19] should
consult these texts and those listed in our Appendix.
44 CHAPTER 2. BAYESIAN DECISION THEORY
Problems
⊕
Section 2.1
1. In the two-category case, under the Bayes’ decision rule the conditional error
is given by Eq. 7. Even if the posterior densities are continuous, this form of the
conditional error virtually always leads to a discontinuous integrand when calculating
the full error by Eq. 5.
(a) Show that for arbitrary densities, we can replace Eq. 7 by P (error|x) = 2P (ω1|x)P (ω2|x)
in the integral and get an upper bound on the full error.
(b) Show that if we use P (error|x) = αP (ω1|x)P (ω2|x) for α < 2, then we are not
guaranteed that the integral gives an upper bound on the error.
(c) Analogously, show that we can use instead P (error|x) = P (ω1|x)P (ω2|x) and
get a lower bound on the full error.
(d) Show that if we use P (error|x) = βP (ω1|x)P (ω2|x) for β > 1, then we are not
guaranteed that the integral gives an lower bound on the error.
⊕
Section 2.2
2. Consider minimax criterion for the zero-one loss function, i.e., λ11 = λ22 = 0 and
λ12 = λ21 = 1.
(a) Prove that in this case the decision regions will satisfy
∫
R2
p(x|ω1)dx =
∫
R1
p(x|ω2)dx
(b) Is this solution always unique? If not, construct a simple counterexample.
3. Consider the minimax criterion for a two-category classification problem.
(a) Fill in the steps of the derivation of Eq. 22.
(b) Explain why the overall Bayes risk must be concave down as a function of the
prior P (ω1), as shown in Fig. 2.4.
(c) Assume we have one-dimensional Gaussian distributions p(x|ωi) ∼ N(µi, σ2i ),
i = 1, 2 but completely unknown prior probabilities. Use the minimax criterion
to find the optimal decision point x∗ in terms of µi and σi under a zero-one risk.
(d) For the decision point x∗ you found in (??), what is the overall minimax risk?
Express this risk in terms of an error function erf(·).
(e) Assume p(x|ω1) ∼ N(0, 1) and p(x|ω2) ∼ N(1/2, 1/4), under a zero-one loss.
Find x∗ and the overall minimax loss.
(f) Assume p(x|ω1) ∼ N(5, 1) and p(x|ω2) ∼ N(6, 1). Without performing any
explicit calculations, determine x∗ for the minimax criterion. Explain your
reasoning.
2.11. PROBLEMS 45
4. Generalize the minimax decision rule in order to classify patterns from three
categories having triangle densities as follows:
p(x|ωi) = T (µi, δi) ≡
{
(δi − |x− µi|)/δ2i for |x− µi| < δi
0 otherwise,
where δi > 0 is the half-width of the distribution (i = 1, 2, 3). Assume for convenience
that µ1 < µ2 < µ3, and make some minor simplifying assumptions about the δi’s as
needed, to answer the following:
(a) In terms of the priors P (ωi), means and half-widths, find the optimal decision
points x∗1 and x
∗
2 under a zero-one (categorization) loss.
(b) Generalize the minimax decision rule to two decision points, x∗1 and x
∗
2 for such
triangular distributions.
(c) Let {µi, δi} = {0, 1}, {.5, .5}, and {1, 1}. Find the minimax decision rule (i.e.,
x∗1 and x
∗
2) for this case.
(d) What is the minimax risk?
5. Consider the Neyman-Pearson criterion for two univariate normal distributions:
p(x|ωi) ∼ N(µi, σ2i ) and P (ωi) = 1/2 for i = 1, 2. Assume a zero-one error loss, and
for convenience µ2 > µ1.
(a) Suppose the maximum acceptable error rate for classifying a pattern that is
actually in ω1 as if it were in ω2 is E1. Determine the decision boundary in
terms of the variables given.
(b) For this boundary, what is the error rate for classifying ω2 as ω1?
(c) What is the overall error rate under zero-one loss?
(d) Apply your results to the specific case p(x|ω1) ∼ N(−1, 1) and p(x|ω2) ∼ N(1, 1)
and E1 = 0.05.
(e) Compare your result to the Bayes error rate (i.e., without the Neyman-Pearson
conditions).
6. Consider Neyman-Pearson criteria for two Cauchy distributions in one dimension
p(x|ωi) =
1
πb
· 1
1 +
(
x−ai
b
)2 , i = 1, 2.
Assume a zero-one error loss, and for simplicity a2 > a1, the same “width” b, and
equal priors.
(a) Suppose the maximum acceptable error rate for classifying a pattern that is
actually in ω1 as if it were in ω2 is E1. Determine the decision boundary in
terms of the variables given.
(b) For this boundary, what is the error rate for classifying ω2 as ω1?
(c) What is the overall error rate under zero-one loss?
(d) Apply your results to the specific case b = 1 and a1 = −1, a2 = 1 and E1 = 0.1.
46 CHAPTER 2. BAYESIAN DECISION THEORY
(e) Compare your result to the Bayes error rate (i.e., without the Neyman-Pearson
conditions).
⊕
Section 2.4
7. Let the conditional densities for a two-category one-dimensional problem be given
by the Cauchy distribution described in Problem 6.
(a) By explicit integration, check that the distributions are indeed normalized.
(b) Assuming P (ω1) = P (ω2), show that P (ω1|x) = P (ω2|x) if x = (a1 +a2)/2, i.e.,
the minimum error decision boundary is a point midway between the peaks of
the two distributions, regardless of b.
(c) Plot P (ω1|x) for the case a1 = 3, a2 = 5 and b = 1.
(d) How do P (ω1|x) and P (ω2|x) behave as x→ −∞? x→ +∞? Explain.
8. Use the conditional densities given in Problem 6, and assume equal prior proba-
bilities for the categories.
(a) Show that the minimum probability of error is given by
P (error) =
1
2
− 1
π
tan−1
∣∣∣a2 − a1
2b
∣∣∣.
(b) Plot this as a function of |a2 − a1|/b.
(c) What is the maximum value of P (error) and under which conditions can this
occur? Explain.
9. Consider the following decision rule for a two-category one-dimensional problem:
Decide ω1 if x > θ; otherwise decide ω2.
(a) Show that the probability of error for this rule is given by
P (error) = P (ω1)
θ∫
−∞
p(x|ω1) dx + P (ω2)
∞∫
θ
p(x|ω2) dx.
(b) By differentiating, show that a necessary condition to minimize P (error) is that
θ satisfy
p(θ|ω1)P (ω1) = p(θ|ω2)P (ω2).
(c) Does this equation define θ uniquely?
(d) Give an example where a value of θ satisfying the equation actually maximizes
the probability of error.
10. Consider
(a) True or false: In a two-category one-dimensional problem with continuous fea-
ture x, a monotonic transformation of x leave the Bayes error rate unchanged.
2.11. PROBLEMS 47
(b) True of false: In a two-category two-dimensional problem with continuous fea-
ture x, monotonic transformations of both x1 and x2 leave the Bayes error rate
unchanged.
11. Suppose that we replace the deterministic decision function α(x) with a ran-
domized rule, viz., the probability P (αi|x) of taking action αi upon observing x.
(a) Show that the resulting risk is given by
R =
∫ [ a∑
i=1
R(αi|x)P (αi|x)
]
p(x) dx.
(b) In addition, show that R is minimized by choosing P (αi|x) = 1 for the action
αi associated with the minimum conditional risk R(αi|x), thereby showing that
no benefit can be gained from randomizing the best decision rule.
(c) Can we benefit from randomizing a suboptimal rule? Explain.
12. Let ωmax(x) be the state of nature for which P (ωmax|x) ≥ P (ωi|x) for all i,
i = 1, …, c.
(a) Show that P (ωmax|x) ≥ 1/c.
(b) Show that for the minimum-error-rate decision rule the average probability of
error is given by
P (error) = 1−
∫
P (ωmax|x)p(x) dx.
(c) Use these two results to show that P (error) ≤ (c− 1)/c.
(d) Describe a situation for which P (error) = (c− 1)/c.
13. In many pattern classification problems one has the option either to assign the
pattern to one of c classes, or to reject it as being unrecognizable. If the cost for
rejects is not too high, rejection may be a desirable action. Let
λ(αi|ωj) =
0 i = j i, j = 1, …, c
λr i = c + 1
λs otherwise,
where λr is the loss incurred for choosing the (c+1)th action, rejection, and λs is the
loss incurred for making a substitution error. Show that the minimum risk is obtained
if we decide ωi if P (ωi|x) ≥ P (ωj |x) for all j and if P (ωi|x) ≥ 1− λr/λs, and reject
otherwise. What happens if λr = 0? What happens if λr > λs?
14. Consider the classification problem with rejection option.
(a) Use the results of Problem 13 to show that the following discriminant functions
are optimal for such problems:
gi(x) =
p(x|ωi)P (ωi) i = 1, …, c
λs−λr
λs
c∑
j=1
p(x|ωj)P (ωj) i = c + 1.
48 CHAPTER 2. BAYESIAN DECISION THEORY
(b) Plot these discriminant functions and the decision regions for the two-category
one-dimensional case having
• p(x|ω1) ∼ N(1, 1),
• p(x|ω2) ∼ N(−1, 1),
• P (ω1) = P (ω2) = 1/2, and
• λr/λs = 1/4.
(c) Describe qualitatively what happens as λr/λs is increased from 0 to 1.
(d) Repeat for the case having
• p(x|ω1) ∼ N(1, 1),
• p(x|ω2) ∼ N(0, 1/4),
• P (ω1) = 1/3, P (ω2) = 2/3, and
• λr/λs = 1/2.
⊕
Section 2.5
15. Confirm Eq. 45 for the volume of a d-dimensional hypersphere as follows:
(a) Verify that the equation is correct for a line (d = 1).
(b) Verify that the equation is correct for a disk (d = 2).
(c) Integrate the volume of a line over appropriate limits to obtain the volume of a
disk.
(d) Consider a general d-dimensional hypersphere. Integrate its volume to obtain
a formula (involving the ratio of gamma functions, Γ(·)) for the volume of a
(d + 1)-dimensional hypersphere.
(e) Apply your formula to find the volume of a hypersphere in an odd-dimensional
space by integrating the volume of a hypersphere in the lower even-dimensional
space, and thereby confirm Eq. 45 for odd dimensions.
(f) Repeat the above but for finding the volume of a hypersphere in even dimensions.
16. Derive the formula for the volume of a d-dimensional hypersphere in Eq. 45 as
follows:
(a) State by inspection the formula for V1.
(b) Follow the general procedure outlined in Problem 15 and integrate twice to find
Vd+2 as a function of Vd.
(c) Assume that the functional form of Vd is the same for all odd dimensions (and
likewise for all even dimensions). Use your integration results to determine the
formula for Vd for d odd.
(d) Use your intermediate integration results to determine Vd for d even.
(e) Explain why we should expect the functional form of Vd to be different in even
and in odd dimensions.
2.11. PROBLEMS 49
17. Derive the formula (Eq. 44) for the volume V of a hyperellipsoid of constant
Mahalanobis distance r (Eq. 43) for a Gaussian distribution having covariance Σ.
18. Consider two normal distributions in one dimension: N(µ1, σ21) and N(µ2, σ
2
2).
Imagine that we choose two random samples x1 and x2, one from each of the normal
distributions and calculate their sum x3 = x1 + x2. Suppose we do this repeatedly.
(a) Consider the resulting distribution of the values of x3. Show from first principles
that this is also a normal distribution.
(b) What is the mean, µ3, of your new distribution?
(c) What is the variance, σ23?
(d) Repeat the above with two distributions in a multi-dimensional space, i.e.,
N(µ1,Σ1) and N(µ2,Σ2).
19. Starting from the definition of entropy (Eq. 36), derive the general equation for
the maximum-entropy distribution given constraints expressed in the general form
∫
bk(x)p(x) dx = ak, k = 1, 2, …, q
as follows:
(a) Use Lagrange undetermined multipliers λ1, λ2, …, λq and derive the synthetic
function:
Hs = −
∫
p(x)
[
ln p(x)−
q∑
k=0
λkbk(x)
]
dx−
q∑
k=0
λkak.
State why we know a0 = 1 and b0(x) = 1 for all x.
(b) Take the derivative of Hs with respect to p(x). Equate the integrand to zero,
and thereby prove that the minimum-entropy distribution obeys
p(x) = exp
[
q∑
k=0
λkbk(x)− 1
]
,
where the q + 1 parameters are determined by the constraint equation.
20. Use the final result from Problem 19 for the following.
(a) Suppose we know only that a distribution is non-zero in the range xl ≤ x ≤ xu.
Prove that the maximum entropy distribution is uniform in that range, i.e.,
p(x) ∼ U(xl, xu) =
{
1/|xu − xl| xl ≤ x ≤ xu
0 otherwise.
50 CHAPTER 2. BAYESIAN DECISION THEORY
(b) Suppose we know only that a distribution is non-zero for x ≥ 0 and that its
mean is µ. Prove that the maximum entropy distribution is
p(x) =
{
1
µ
e−x/µ for x ≥ 0
0 otherwise.
(c) Now suppose we know solely that the distribution is normalized, has mean µ,
and standard deviation σ2, and thus from Problem 19 our maximum entropy
distribution must be of the form
p(x) = exp[λ0 − 1 + λ1x + λ2×2].
Write out the three constraints and solve for λ0, λ1, and λ2 and thereby prove
that the maximum entropy solution is a Gaussian, i.e.,
p(x) =
1√
2π
exp
[
−(x− µ)2
2σ2
]
.
21. Three distributions — a Gaussian, a uniform distribution, and a triangle dis-
tribution (cf., Problem 4) — each have mean zero and standard deviation σ2. Use
Eq. 36 to calculate and compare their entropies.
22. Calculate the entropy of a multidimensional Gaussian p(x) ∼ N(µ,Σ).
⊕
Section 2.6
23. Consider the three-dimensional normal distribution p(x|ω) ∼ N(µ,Σ) where
µ =
(
1
2
2
)
and Σ =
(
1 0 0
0 5 2
0 2 5
)
.
(a) Find the probability density at the point x0 = (.5, 0, 1)t.
(b) Construct the whitening transformation Aw. Show your Λ and Φ matrices.
Next, convert the distribution to one centered on the origin with covariance
matrix equal to the identity matrix, p(x|ω) ∼ N(0, I).
(c) Apply the same overall transformation to x0 to yield a transformed point xw.
(d) By explicit calculation, confirm that the Mahalanobis distance from x0 to the
mean µ in the original distribution is the same as for xw to 0 in the transformed
distribution.
(e) Does the probability density remain unchanged under a general linear transfor-
mation? In other words, is p(x0|N(µ,Σ)) = p(Ttx0|N(Ttµ,TtΣT)) for some
linear transform T? Explain.
(f) Prove that a general whitening transform Aw = ΦΛ
−1/2 when applied to a
Gaussian distribution insures that the final distribution has covariance propor-
tional to the identity matrix I. Check whether normalization is preserved by the
transformation.
2.11. PROBLEMS 51
24. Consider the multivariate normal density for which σij = 0 and σii = σ2i , i.e.,
Σ = diag(σ21 , σ
2
2 , …, σ
2
d).
(a) Show that the evidence is
p(x) =
1
d∏
i=1
√
2πσi
exp
[
−1
2
d∑
i=1
(
xi − µi
σi
)2]
.
(b) Plot and describe the contours of constant density.
(c) Write an expression for the Mahalanobis distance from x to µ.
25. Fill in the steps in the derivation from Eq. 57 to Eqs. 58–63.
26. Let p(x|ωi) ∼ N(µi,Σ) for a two-category d-dimensional problem with the
same covariances but arbitrary means and prior probabilities. Consider the squared
Mahalanobis distance
r2i = (x− µi)tΣ−1(x− µi).
(a) Show that the gradient of r2i is given by
∇r2i = 2Σ−1(x− µi).
(b) Show that at any position on a given line through µi the gradient ∇r2i points
in the same direction. Must this direction be parallel to that line?
(c) Show that ∇r21 and ∇r22 point in opposite directions along the line from µ1 to
µ2.
(d) Show that the optimal separating hyperplane is tangent to the constant prob-
ability density hyperellipsoids at the point that the separating hyperplane cuts
the line from µ1 to µ2.
(e) True of False: For a two-category problem involving normal densities with ar-
bitrary means and covariances, and P (ω1) = P (ω2) = 1/2, the Bayes decision
boundary consists of the set of points of equal Mahalanobis distance from the
respective sample means. Explain.
27. Suppose we have two normal distributions with the same covariances but different
means: N(µ1,Σ) and N(µ2,Σ). In terms of their prior probabilities P (ω1) and
P (ω2), state the condition that the Bayes decision boundary not pass between the
two means.
28. Two random variables x and y are called “statistically independent” if p(x,y|ω) =
p(x|ω)p(y|ω).
(a) Prove that if xi − µi and xj − µj are statistically independent (for i �= j) then
σij as defined in Eq. 42 is 0.
(b) Prove that the converse is true for the Gaussian case.
(c) Show by counterexample that this converse is not true in the general case.
52 CHAPTER 2. BAYESIAN DECISION THEORY
29. Consider the Bayes decision boundary for two-category classification in d dimen-
sions.
(a) Prove that for any arbitrary hyperquadratic in d dimensions, there exist normal
distributions p(x|ωi) ∼ N(µi,Σi) and priors P (ωi), i = 1, 2, that possess this
hyperquadratic as their Bayes decision boundary.
(b) Is the above also true if the priors are held fixed and non-zero, e.g., P (ω1) =
P (ω2) = 1/2?
⊕
Section 2.7
30. Let p(x|ωi) ∼ N(µi, σ2) for a two-category one-dimensional problem with
P (ω1) = P (ω2) = 1/2.
(a) Show that the minimum probability of error is given by
Pe =
1√
2π
∞∫
a
e−u
2/2 du,
where a = |µ2 − µ1|/(2σ).
(b) Use the inequality
Pe =
1√
2π
∞∫
a
e−t
2/2 dt ≤ 1√
2πa
e−a
2/2
to show that Pe goes to zero as |µ2 − µ1|/σ goes to infinity.
31. Let p(x|ωi) ∼ N(µi, σ2I) for a two-category d-dimensional problem with P (ω1) =
P (ω2) = 1/2.
(a) Show that the minimum probability of error is given by
Pe =
1√
2π
∞∫
a
e−u
2/2 du,
where a = ‖µ2 − µ1‖/(2σ).
(b) Let µ1 = 0 and µ = (µ1, …, µd)
t. Use the inequality from Problem 30 to show
that Pe approaches zero as the dimension d approaches infinity.
(c) Express the meaning of this result in words.
32. Show that if the densities in a two-category classification problem differ markedly
from Gaussian, the Chernoff and Bhattacharyya bounds are not likely to be informa-
tion by considering the following one-dimensional examples. Consider a number of
problems in which the mean and variance are the same (and thus the Chernoff bound
and the Bhattacharyya bound remain the same), but nevertheless have a wide range
in Bayes error. For definiteness, assume the distributions have means at µ1 = −µ and
µ2 = +µ, and σ21 = σ
2
2 = µ
2.
2.11. PROBLEMS 53
(a) Use the equations in the text to calculate the Chernoff and the Bhattacharyya
bounds on the error.
(b) Suppose the distributions are both Gaussian. Calculate explicitly the Bayes
error. Express it in terms of an error function erf(·) and as a numerical value.
(c) Now consider a another case, in which half the density for ω1 is concentrated
at a point x = −2µ and half at x = 0; likewise (symmetrically) the density for
ω2 has half its mass at x = +2µ and half at x = 0. Show that the means and
variance remain as desired, but that now the Bayes error is 0.5.
(d) Now consider yet another case, in which half the density for ω1 is concentrated
near x = −2 and half at x = −1, where 1 is an infinitessimally small positive
distance; likewise (symmetrically) the density for ω2 has half its mass near
x = +2µ and half at +1. Show that by making 1 sufficiently small, the means
and variances can be made arbitrarily close to µ and µ2, respectively. Show,
too, that now the Bayes error is zero.
(e) Compare your errors in (b), (c) and (d) to your Chernoff and Bhattacharyya
bounds of (a) and explain in words why those bounds are unlikely to be of much
use if the distributions differ markedly from Gaussians.
33. Suppose we know exactly two arbitrary distributions p(x|ωi) and priors P (ωi)
in a d-dimensional feature space.
(a) Prove that the true error cannot decrease if we first project the distributions to
a lower dimensional space and then classify them.
(b) Despite this fact, suggest why in an actual pattern recognition application we
might not want to include an arbitrarily high number of feature dimensions.
⊕
Section 2.8
34. Show for non-pathological cases that if we include more feature dimensions
in a Bayesian classifier for multidimensional Gaussian distributions then the Bhat-
tacharyya bound decreases. Do this as follows: Let Pd(P (ω1),µ1,Σ1, P (ω2),µ2,Σ2),
or simply Pd, be the Bhattacharyya bound if we consider the distributions restricted
to d dimensions.
(a) Using general properties of a covariance matrix, prove that k(1/2) of Eq. 75
must increase as we increase from d to d + 1 dimensions, and hence the error
bound must decrease.
(b) Explain why this general result does or does not depend upon which dimension
is added.
(c) What is a “pathological” case in which the error bound does not decrease, i.e.,
for which Pd+1 = Pd?
(d) Is it ever possible that the true error could increase as we go to higher dimension?
(e) Prove that as d → ∞, Pd → 0 for non-pathological distributions. Describe
pathological distributions for which this infinite limit does not hold.
54 CHAPTER 2. BAYESIAN DECISION THEORY
(f) Given that the Bhattacharyya bound decreases for the inclusion of a particular
dimension, does this guarantee that the true error will decrease? Explain.
35. Derive Eqs. 72 & 73 from Eq. 71 by the following steps:
(a) Substitute the normal distributions into the integral and gather the terms de-
pendent upon x and those that are not dependent upon x.
(b) Factor the term independent of x from the integral.
(c) Integrate explicitly the term dependent upon x.
36. Consider a two-category classification problem in two dimensions with p(x|ω1) ∼
N(0, I), p(x|ω2) ∼ N
((
1
1
)
, I
)
, and P (ω1) = P (ω2) = 1/2.
(a) Calculate the Bayes decision boundary.
(b) Calculate the Bhattacharyya error bound.
(c) Repeat the above for the same prior probabilities, but p(x|ω1) ∼ N
(
0,
(
2 .5
.5 2
))
and p(x|ω2) ∼ N
((
1
1
)
,
(
5 4
4 5
))
.
37. Derive the Bhattacharyya error bound without the need for first examining the
Chernoff bound. Do this as follows:
(a) If a and b are nonnegative numbers, show directly that min[a,b] ≤
√
ab.
(b) Use this to show that the error rate for a two-category Bayes classifier must
satisfy
P (error) ≤
√
P (ω1)P (ω2) ρ ≤ ρ/2,
where ρ is the so-called Bhattacharyya coefficient
ρ =
∫ √
p(x|ω1) p(x|ω2) dx.
38. Use the signal detection theory, the notation and basic Gaussian assumptions
described in the text to address the following.
(a) Prove that P (x > x∗|x ∈ ω2) and P (x < x∗|x ∈ ω2), taken together, uniquely
determine the discriminability d′.
(b) Use error functions erf(·) to express d′ in terms of the hit and false alarm rates.
Estimate d′ if P (x > x∗|x ∈ ω2) = 0.8 and P (x < x∗|x ∈ ω2) = 0.3. Repeat for
P (x > x∗|x ∈ ω2) = 0.7 and P (x < x∗|x ∈ ω2) = 0.4.
(c) Given that the Gaussian assumption is valid, calculate the Bayes error for both
the cases in (b).
(d) Determine by a trivial one-line computation, which case has the higher d′:
case A: P (x > x∗|x ∈ ω2) = 0.8, P (x < x∗|x ∈ ω2) = 0.3 or
case B: P (x > x∗|x ∈ ω2) = 0.9, P (x < x∗|x ∈ ω2) = 0.7.
2.11. PROBLEMS 55
Explain your logic.
39. Suppose in our signal detection framework we had two Gaussians, but with dif-
ferent variances (cf., Fig. 2.20), that is, p(x|ω1 ∼ N(µ1, σ21) and p(x|ω2) ∼ N(µ2, σ22)
for µ2 > µ1 and σ22 �= σ21 . In that case the resulting ROC curve would no longer be
symmetric.
(a) Suppose in this asymmetric case we modified the definition of the discriminabil-
ity to be d′a = |µ2−µ1|/
√
σ1σ2. Show by non-trivial counterexample or analysis
that one cannot determine d′a uniquely based on a single pair of hit and false
alarm rates.
(b) Assume we measure the hit and false alarm rates for two different, but unknown,
values of the threshold x∗. Derive a formula for d′a based on measurements.
(c) State and explain all pathological values for which your formula does not give
a meaningful value for d′a.
(d) Plot several ROC curves for the case p(x|ω1) ∼ N(0, 1) and p(x|ω2) ∼ N(1, 2).
40. Consider two one-dimensional triangle distributions having different means, but
the same width:
p(x|ωi) = T (µi, δ) =
{
(δ − |x− µi|)/δ2 for |x− µi| < δ 0 otherwise, with µ2 > µ1. We define a new discriminability here as d′T = (µ2 − µ1)/δi.
(a) Write an analytic function, parameterized by d′T , for the operating characteristic
curves.
(b) Plot these novel operating characteristic curves for d′T = {.1, .2, …, 1.0}. Inter-
pret your answer for the case d′T = 1.0.
(c) Suppose we measure P (x > x∗|x ∈ ω2) = .4 and P (x > x∗|x ∈ ω1) = .2. What
is d′T ? What is the Bayes error rate?
(d) Infer the decision rule. That is, express x∗ in terms of the variables given in the
problem.
(e) Suppose we measure P (x > x∗|x ∈ ω2) = .9 and (x > x∗|x ∈ ω1) = .3. What is
d′T ? What is the Bayes error rate?
(f) Infer the decision rule. That is, express x∗ in terms of the variables given in the
problem.
41. Equation 70 can be used to obtain an upper bound on the error. One can
also derive tighter analytic bounds in the two-category case — both upper and lower
bounds — analogous to Eq. 71 for general distributions. If we let p ≡ p(x|ω1), then
we seek tighter bounds on Min[p, 1− p] (which has discontinuous derivative).
(a) Prove that
bL(p) =
1
β
ln
[
1 + e−β
e−βp + e−β(1−p)
]
for any β > 0 is a lower bound on Min[p, 1− p].
56 CHAPTER 2. BAYESIAN DECISION THEORY
(b) Prove that one can choose β in (a) to give an arbitrarily tight lower bound.
(c) Repeat (a) and (b) for the upper bound given by
bU (p) = bL(p) + [1− 2gL(0.5)]bG(p)
where bG(p) is any upper bound that obeys
bG(p) ≥ Min[p, 1− p]
bG(p) = bG(1− p)
bG(0) = bG(1) = 0
bG(0.5) = 0.5.
(d) Confirm that bG(p) = 1/2 sin[πp] obeys the conditions in (c).
(e) Let bG(p) = 1/2 sin[πp], and plot your upper and lower bounds as a function of
p, for 0 ≤ p ≤ 1 and β = 1, 10, 50.
⊕
Section 2.9
42. Let the components of the vector x = (x1, …, xd)t be binary valued (0 or 1) and
P (ωj) be the prior probability for the state of nature ωj and j = 1, …, c. Now define
pij = Prob(xi = 1|ωj)
i = 1, …, d
j = 1, …, c,
with the components of xi being statistically independent for all x in ωj .
(a) Interpret in words the meaning of pij .
(b) Show that the minimum probability of error is achieved by the following decision
rule: Decide ωk if gk(x) ≥ gj(x) for all j and k, where
gj(x) =
d∑
i=1
xi ln
pij
1− pij
+
d∑
i=1
ln (1− pij) + ln P (ωj).
43. Let the components of the vector x = (x1, …, xd)t be ternary valued (1, 0 or
−1), with
pij = Prob(xi = 1 |ωj)
qij = Prob(xi = 0 |ωj)
rij = Prob(xi = −1|ωj),
and with the components of xi being statistically independent for all x in ωj .
(a) Show that a minimum probability of error decision rule can be derived that
involves discriminant functions gj(x) that are quadratic function of the compo-
nents xi.
(b) Suggest a generalization to more categories of your answers to this and Prob-
lem 42.
2.11. PROBLEMS 57
44. Let x be distributed as in Problem 42 with c = 2, d odd, and
pi1 = p > 1/2 i = 1, …d
pi2 = 1− p i = 1, …d
and P (ω1) = P (ω2) = 1/2.
(a) Show that the minimum-error-rate decision rule becomes:
Decide ω1 if
d∑
i=1
xi > d/2 and ω2 otherwise.
(b) Show that the minimum probability of error is given by
Pe(d, p) =
(d−1)/2∑
k=0
(
d
k
)
pk(1− p)d−k.
where
(
d
k
)
= d!/(k!(d− k)!) is the binomial coefficient.
(c) What is the limiting value of Pe(d, p) as p→ 1/2? Explain.
(d) Show that Pe(d, p) approaches zero as d→∞. Explain.
45. Under the natural assumption concerning losses, i.e., that λ21 > λ11 and λ12 >
λ22, show that the general minimum risk discriminant function for the independent
binary case described in Sect. 2.9.1 is given by g(x) = wtx+w0, where w is unchanged,
and
w0 =
d∑
i=1
ln
1− pi
1− qi
+ ln
P (ω1)
P (ω2)
+ ln
λ21 − λ11
λ12 − λ22
.
46. The Poisson distribution for a discrete variable x = 0, 1, 2, … and real parameter
λ is
P (x|λ) = e−λλ
x
x!
.
(a) Prove that the mean of such a distribution is E [x] = λ.
(b) Prove that the variance of such a distribution is E [x− x̄] = λ.
(c) The mode of a distribution is the value of x that has the maximum probability.
Prove that the mode of a Poisson distribution is the greatest integer that does
not exceed λ, i.e., the mode is �λ�. (If λ is an integer, then both λ and λ − 1
are modes.)
(d) Consider two equally probable categories having Poisson distributions but with
differing parameters; assume for definiteness λ1 > λ2. What is the Bayes clas-
sification decision?
(e) What is the Bayes error rate?
58 CHAPTER 2. BAYESIAN DECISION THEORY
⊕
Section 2.10
47. Suppose we have three categories in two dimensions with the following underlying
distributions:
• p(x|ω1) ∼ N(0, I)
• p(x|ω2) ∼ N
((
1
1
)
, I
)
• p(x|ω3) ∼ 12N
((
.5
.5
)
, I
)
+ 1
2
N
((−.5
.5
)
, I
)
with P (ωi) = 1/3, i = 1, 2, 3.
(a) By explicit calculation of posterior probabilities, classify the point x =
(
.3
.3
)
for
minimum probability of error.
(b) Suppose that for a particular test point the first feature is missing. That is,
classify x =
(∗
.3
)
.
(c) Suppose that for a particular test point the second feature is missing. That is,
classify x =
(
.3
∗
)
.
(d) Repeat all of the above for x =
(
.2
.6
)
.
48. Show that Eq. 93 reduces to Bayes rule when the true feature is µi and
p(xb|xt) ∼ N(xt,Σ). Interpret this answer in words.
⊕
Section 2.11
49. Suppose we have three categories with P (ω1) = 1/2, P (ω2) = P (ω3) = 1/4 and
the following distributions
• p(x|ω1) ∼ N(0, 1)
• p(x|ω2) ∼ N(.5, 1)
• p(x|ω3) ∼ N(1, 1) ,
and that we sample the following four points: x = 0.6, 0.1, 0.9, 1.1.
(a) Calculate explicitly the probability that the sequence actually came from ω1, ω3, ω3, ω2.
Be careful to consider normalization.
(b) Repeat for the sequence ω1, ω2, ω2, ω3.
(c) Find the sequence having the maximum probability.
2.11. COMPUTER EXERCISES 59
Computer exercises
Several of the computer exercises will rely on the following data.
ω1 ω2 ω3
sample x1 x2 x3 x1 x2 x3 x1 x2 x3
1 −5.01 −8.12 −3.68 −0.91 −0.18 −0.05 5.35 2.26 8.13
2 −5.43 −3.48 −3.54 1.30 −2.06 −3.53 5.12 3.22 −2.66
3 1.08 −5.52 1.66 −7.75 −4.54 −0.95 −1.34 −5.31 −9.87
4 0.86 −3.78 −4.11 −5.47 0.50 3.92 4.48 3.42 5.19
5 −2.67 0.63 7.39 6.14 5.72 −4.85 7.11 2.39 9.21
6 4.94 3.29 2.08 3.60 1.26 4.36 7.17 4.33 −0.98
7 −2.51 2.09 −2.59 5.37 −4.63 −3.65 5.75 3.97 6.65
8 −2.25 −2.13 −6.94 7.18 1.46 −6.66 0.77 0.27 2.41
9 5.56 2.86 −2.26 −7.39 1.17 6.30 0.90 −0.43 −8.71
10 1.03 −3.33 4.33 −7.50 −6.32 −0.31 3.52 −0.36 6.43
⊕
Section 2.2
1. You may need the following procedures for several exercises below.
(a) Write a procedure to generate random samples according to a normal distribu-
tion N(µ,Σ) in d dimensions.
(b) Write a procedure to calculate the discriminant function (of the form given in
Eq. 47) for a given normal distribution and prior probability P (ωi).
(c) Write a procedure to calculate the Euclidean distance between two arbitrary
points.
(d) Write a procedure to calculate the Mahalanobis distance between the mean µ
and an arbitrary point x, given the covariance matrix Σ.
⊕
Section 2.5
2. Use your classifier from Problem ?? to classify the following 10 samples from
the table above in the following way. Assume that the underlying distributions are
normal.
(a) Assume that the prior probabilities for the first two categories are equal (P (ω1) =
P (ω2) = 1/2 and P (ω3) = 0) and design a dichotomizer for those two categories
using only the x1 feature value.
(b) Determine the empirical training error on your samples, i.e., the percentage of
points misclassified.
(c) Use the Bhattacharyya bound to bound the error you will get on novel patterns
drawn from the distributions.
(d) Repeat all of the above, but now use two feature values, x1, and x2.
(e) Repeat, but use all three feature values.
(f) Discuss your results. In particular, is it ever possible for a finite set of data that
the empirical error might be larger for more data dimensions?
60 CHAPTER 2. BAYESIAN DECISION THEORY
3. Repeat Computer exercise 2 but for categories ω1 and ω3.
4. Repeat Computer exercise 2 but for categories ω2 and ω3.
5. Consider the three categories in Computer exercise 2, and assume P (ωi) = 1/3.
(a) What is the Mahalanobis distance between each of the following test points and
each of the category means in Computer exercise 2: (1, 2, 1)t, (5, 3, 2)t, (0, 0, 0)t,
(1, 0, 0)t.
(b) Classify those points.
(c) Assume instead that P (ω1) = 0.8, and P (ω2) = P (ω3) = 0.1 and classify the
test points again.
6. Illustrate the fact that the average of a large number of independent random
variables will approximate a Gaussian by the following:
(a) Write a program to generate n random integers from a uniform distribution
U(xl, xu). (Some computer systems include this as a single, compiled function
call.)
(b) Now write a routine to choose xl and xu randomly, in the range −100 ≤ xl <
xu ≤ +100, and n (the number of samples) randomly in the range 0 < n ≤ 1000.
(c) Generate and plot a histogram of the accumulation of 104 points sampled as
just described.
(d) Calculate the mean and standard deviation of your histogram, and plot it
(e) Repeat the above for 105 and for 106. Discuss your results.
⊕
Section 2.8
7. Explore how the empirical error does or does not approach the Bhattacharyya
bound as follows:
(a) Write a procedure to generate sample points in d dimensions with a normal
distribution having mean µ and covariance matrix Σ.
(b) Consider p(x|ω1) ∼ N
((
1
0
)
, I
)
and p(x|ω2) ∼ N
((−1
0
)
, I
)
with P (ω1) = P (ω2) =
1/2. By inspection, state the Bayes decision boundary.
(c) Generate n = 100 points (50 for ω1 and 50 for ω2) and calculate the empirical
error.
(d) Repeat for increasing values of n, 100 ≤ n ≤ 1000, in steps of 100 and plot your
empirical error.
(e) Discuss your results. In particular, is it ever possible that the empirical error is
greater than the Bhattacharyya or Chernoff bound?
8. Consider two one-dimensional normal distributions p(x|ω1) ∼ N(−.5, 1) and
p(x|ω2) ∼ N(+.5, 1) and P (ω1) = P (ω2) = 0.5.
(a) Calculate the Bhattacharyya bound for the error of a Bayesian classifier.
(b) Express the true error rate in terms of an error function, erf(·).
2.11. COMPUTER EXERCISES 61
(c) Evaluate this true error to four significant figures by numerical integration (or
other routine).
(d) Generate 10 points each for the two categories and determine the empirical error
using your Bayesian classifier. (You should recalculate the decision boundary
for each of your data sets.)
(e) Plot the empirical error as a function of the number of points from either dis-
tribution by repeating the previous part for 50, 100, 200, 500 and 1000 sample
points from each distribution. Compare your asymptotic empirical error to the
true error and the Bhattacharyya error bound.
9. Repeat Computer exercise 8 with the following conditions:
(a) p(x|ω1) ∼ N(−.5, 2) and p(x|ω2) ∼ N(.5, 2), P (ω1) = 2/3 and P (ω2) = 1/3.
(b) p(x|ω1) ∼ N(−.5, 2) and p(x|ω2) ∼ N(.5, 2) and P (ω1) = P (ω2) = 1/2.
(c) p(x|ω1) ∼ N(−.5, 3) and p(x|ω2) ∼ N(.5, 1) and P (ω1) = P (ω2) = 1/2.
62 CHAPTER 2. BAYESIAN DECISION THEORY
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Index
λ(·, ·), see loss
ω, see state of nature, 3
d′, 34, see discriminability
Rd, see Euclidean space
Ri, see decision, region
action, 7
action (α), 7
average, see mean
Bayes
decision rule, 6
Bayes’ formula, 4, 7
Bayesian decision theory, see decision
theory, Bayesian
Bhattacharyya
bound, 32
coefficient (ρ), 54
bias, 21, 38
binary feature, see feature, binary
binomial coefficient, 57
bit, 16
bound
Bhattacharyya, see Bhattacharyya,
bound
boundary, see decision, boundary
category symbol (ω), 3
Central Limit Theorem, 17
Chernoff bound, 31
class-conditional probability, see prob-
ability, class-conditional
classification
fish example, 3
classifier
linear, 38
coefficient
Bhattacharyya, see Bhattacharyya,
coefficient
conditional independence, 36, 37
conditional probability, see probabil-
ity, conditional
conditional risk, see risk, conditional
constraint
risk, 12
context
statistical dependence, 41
correct rejection, see rejection, correct
covariance, 17
covariance matrix, see matrix, covari-
ance
criterion
Neyman-Pearson, see Neyman-Pearson
criterion
decision, 7
Bayes, 6, 8, 36
binary features, 36
bias, 34
boundary, 14
hyperquadratic, 25
compound, 41
missing feature, 39
noisy feature, 39
randomized, 47
region, 14
rule, see rule
sequential, 41
decision theory
Bayes, 7
discrete features, 36
Bayesian, 3
continuous features, 7
dichotomizer, 15
discriminability, 54
discriminability (d′), 33, see receiver
operating characteristic
discriminant function, 37
discrete, 36
distance
Euclidean, 20
Mahalanobis, 18
distribution
66
INDEX 67
and missing data, 39
marginal, 40
Poisson, 57
triangle, 55
dot product, see inner product
entropy, 16
error
Bayes, 5
probability, 4
discrete case, 38
minimal, 5
error function, 44
Euclidean norm, see distance, Euclidean
Euclidean space (Rd), 36
evidence, 6
Expectation-Maximization algorithm, 40
expected value, 15, see mean
feature, 16
false alarm, 34
feature
binary, 36
good (uncorrupted), 40
independence, 38
integer valued, 36
missing, 39
noisy, 39–42
space, 7
ternary, 36
vector
binary, 36
continuous, 7
fish
classification example, 41
occlusion, 39
game theory, 11
Gaussian
distribution, 42
multidimensional, 17
one-dimensional, 16
univariate, 16
hit, 34
hypercube, 38
hyperellipsoid, 25
hyperparaboloid, 25
hyperplane, 25, 38
hyperquadric, 25
hypersphere, 25
independence
conditional, see conditional inde-
pendence
statistical, 17, 41
inner product, 17
joint probability, see probability, joint
knowledge
prior, 3
likelihood, 5, 37
ratio, 37
loss
classification, 9
expected, 7
function, 7
matrix, 7, 34
minimal, 8
symmetric, 9
zero-one, 9
Mahalanobis distance, see distance, Ma-
halanobis
marginal distribution, see distribution,
marginal
marginalize, 40
matching
template, see template matching
matrix
covariance, 17
mean, 16, 39
minimax risk, see risk, minimax
miss, 34
mode, 57
nat, 16
Neyman-Pearson criterion, 12
noise
model, 40, 41
norm, see distance or metric
omega (ω), see state of nature or cat-
egory symbol
operating characteristic, 33–35
Poisson distribution, see distribution,
Poisson
polychotomizer, 15
posterior probability, see probability,
posterior
68 INDEX
prior, 3, 4, 7, 38
probability, 36
a posteriori, 5
a priori, 3
class-conditional, 37
conditional, 4, 6
density, 4, 36
singular, 36
state-conditional, 7
joint, 4
prior, 4
random variable, see variable, random
randomized decision rule, see decision
rule, randomized
receiver operating characteristic (ROC),
34
reject option, 7, 47
rejection
correct, 34
risk, 8
conditional, 8, 36
minimax, 11
overall, 36
ROC, see receiver operating character-
istic
rule
decision, 4
signal detection theory, 33
space
Euclidean (Rd), 7
state of nature (ω), 3
state-conditional probability density, see
probability density, see prob-
ability density, state conditional
statistical
dependence, 41
independence, 17
noise, 40
template matching, 22
ternary feature, see feature, ternary
threshold, 21, see bias
threshold weight, 38
transform
whitening, 18
variable
random, 41
variance, 16
whitening transform, see transform, whiten-
ing
zero-one loss, see loss, zero-one
Contents
3 Maximum likelihood and Bayesian estimation 3
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . 4
3.2.1 The General Principle . . . . . . . . . . . . . . . . . . . . . . . 4
3.2.2 The Gaussian Case: Unknown µ . . . . . . . . . . . . . . . . . 7
3.2.3 The Gaussian Case: Unknown µ and Σ . . . . . . . . . . . . . 7
3.2.4 Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Bayesian estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3.1 The Class-Conditional Densities . . . . . . . . . . . . . . . . . 10
3.3.2 The Parameter Distribution . . . . . . . . . . . . . . . . . . . . 10
3.4 Bayesian Parameter Estimation: Gaussian Case . . . . . . . . . . . . . 11
3.4.1 The Univariate Case: p(µ|D) . . . . . . . . . . . . . . . . . . . 11
3.4.2 The Univariate Case: p(x|D) . . . . . . . . . . . . . . . . . . . 14
3.4.3 The Multivariate Case . . . . . . . . . . . . . . . . . . . . . . . 14
3.5 Bayesian Parameter Estimation: General Theory . . . . . . . . . . . . 16
Example 1: Recursive Bayes learning and maximum likelihood . . . . . 17
3.5.1 When do Maximum Likelihood and Bayes methods differ? . . . 19
3.5.2 Non-informative Priors and Invariance . . . . . . . . . . . . . . 20
3.6 *Sufficient Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Theorem 3.1: Factorization . . . . . . . . . . . . . . . . . . . . . . . . 22
3.6.1 Sufficient Statistics and the Exponential Family . . . . . . . . . 24
3.7 Problems of Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . 27
3.7.1 Accuracy, Dimension, and Training Sample Size . . . . . . . . . 27
3.7.2 Computational Complexity . . . . . . . . . . . . . . . . . . . . 28
3.7.3 Overfitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.8 *Expectation-Maximization (EM) . . . . . . . . . . . . . . . . . . . . . 32
Algorithm 1: Expectation-Maximization . . . . . . . . . . . . . . . . . 33
Example 2: Expectation-Maximization for a 2D normal model . . . . . 33
3.9 *Bayesian Belief Networks . . . . . . . . . . . . . . . . . . . . . . . . . 36
Example 3: Belief network for fish . . . . . . . . . . . . . . . . . . . . 39
3.10 *Hidden Markov Models . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.10.1 First-order Markov models . . . . . . . . . . . . . . . . . . . . 42
3.10.2 First-order hidden Markov models . . . . . . . . . . . . . . . . 43
3.10.3 Hidden Markov Model Computation . . . . . . . . . . . . . . . 44
3.10.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Algorithm 2: Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Algorithm 3: Backward . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Example 4: Hidden Markov Model . . . . . . . . . . . . . . . . . . . . 47
1
2 CONTENTS
3.10.5 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Algorithm 4: HMM decode . . . . . . . . . . . . . . . . . . . . . . . . . 49
Example 5: HMM decoding . . . . . . . . . . . . . . . . . . . . . . . . 50
3.10.6 Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Algorithm 5: Forward-Backward . . . . . . . . . . . . . . . . . . . . . . 52
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Bibliographical and Historical Remarks . . . . . . . . . . . . . . . . . . . . 54
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Computer exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Chapter 3
Maximum likelihood and
Bayesian parameter
estimation
3.1 Introduction
I n Chap. ?? we saw how we could design an optimal classifier if we knew the priorprobabilities P (ωi) and the class-conditional densities p(x|ωi). Unfortunately, in
pattern recognition applications we rarely if ever have this kind of complete knowledge
about the probabilistic structure of the problem. In a typical case we merely have
some vague, general knowledge about the situation, together with a number of design
samples or training data — particular representatives of the patterns we want to training
dataclassify. The problem, then, is to find some way to use this information to design or
train the classifier.
One approach to this problem is to use the samples to estimate the unknown prob-
abilities and probability densities, and to use the resulting estimates as if they were
the true values. In typical supervised pattern classification problems, the estimation
of the prior probabilities presents no serious difficulties (Problem 3). However, es-
timation of the class-conditional densities is quite another matter. The number of
available samples always seems too small, and serious problems arise when the di-
mensionality of the feature vector x is large. If we know the number of parameters in
advance and our general knowledge about the problem permits us to parameterize the
conditional densities, then the severity of these problems can be reduced significantly.
Suppose, for example, that we can reasonably assume that p(x|ωi) is a normal density
with mean µi and covariance matrix Σi, although we do not know the exact values
of these quantities. This knowledge simplifies the problem from one of estimating an
unknown function p(x|ωi) to one of estimating the parameters µi and Σi.
The problem of parameter estimation is a classical one in statistics, and it can be
approached in several ways. We shall consider two common and reasonable proce-
dures, maximum likelihood estimation and Bayesian estimation. Although the results maximum
likelihood
Bayesian
estimation
obtained with these two procedures are frequently nearly identical, the approaches
3
4 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
are conceptually quite different. Maximum likelihood and several other methods view
the parameters as quantities whose values are fixed but unknown. The best estimate
of their value is defined to be the one that maximizes the probability of obtaining
the samples actually observed. In contrast, Bayesian methods view the parameters as
random variables having some known a priori distribution. Observation of the sam-
ples converts this to a posterior density, thereby revising our opinion about the true
values of the parameters. In the Bayesian case, we shall see that a typical effect of
observing additional samples is to sharpen the a posteriori density function, causing
it to peak near the true values of the parameters. This phenomenon is known as
Bayesian learning. In either case, we use the posterior densities for our classificationBayesian
learning rule, as we have seen before.
It is important to distinguish between supervised learning and unsupervised learn-
ing. In both cases, samples x are assumed to be obtained by selecting a state of nature
ωi with probability P (ωi), and then independently selecting x according to the proba-
bility law p(x|ωi). The distinction is that with supervised learning we know the state
of nature (class label) for each sample, whereas with unsupervised learning we do not.
As one would expect, the problem of unsupervised learning is the more difficult one.
In this chapter we shall consider only the supervised case, deferring consideration of
unsupervised learning to Chap. ??.
3.2 Maximum Likelihood Estimation
Maximum likelihood estimation methods have a number of attractive attributes.
First, they nearly always have good convergence properties as the number of train-
ing samples increases. Further, maximum likelihood estimation often can be simpler
than alternate methods, such as Bayesian techniques or other methods presented in
subsequent chapters.
3.2.1 The General Principle
Suppose that we separate a collection of samples according to class, so that we have c
sets, D1, ...,Dc, with the samples in Dj having been drawn independently according to
the probability law p(x|ωj). We say such samples are i.i.d. — independent identicallyi.i.d.
distributed random variables. We assume that p(x|ωj) has a known parametric form,
and is therefore determined uniquely by the value of a parameter vector θj . For
example, we might have p(x|ωj) ∼ N(µj ,Σj), where θj consists of the components of
µj and Σj . To show the dependence of p(x|ωj) on θj explicitly, we write p(x|ωj) as
p(x|ωj ,θj). Our problem is to use the information provided by the training samples
to obtain good estimates for the unknown parameter vectors θ1, ...,θc associated with
each category.
To simplify treatment of this problem, we shall assume that samples in Di give no
information about θj if i �= j — that is, we shall assume that the parameters for the
different classes are functionally independent. This permits us to work with each class
separately, and to simplify our notation by deleting indications of class distinctions.
With this assumption we thus have c separate problems of the following form: Use a
set D of training samples drawn independently from the probability density p(x|θ) to
estimate the unknown parameter vector θ.
Suppose that D contains n samples, x1, ...,xn. Then, since the samples were drawn
independently, we have
3.2. MAXIMUM LIKELIHOOD ESTIMATION 5
p(D|θ) =
n∏
k=1
p(xk|θ). (1)
Recall from Chap. ?? that, viewed as a function of θ, p(D|θ) is called the likelihood
of θ with respect to the set of samples. The maximum likelihood estimate of θ is, by
definition, the value θ̂ that maximizes p(D|θ). Intuitively, this estimate corresponds
to the value of θ that in some sense best agrees with or supports the actually observed
training samples (Fig. 3.1).
1 2 3 4 5 6 7
x
1 2 3 4 5 6 7
-100
-60
-40
-20
θ
l(θ)
θ̂
1 2 3 4 5 6 7
.4 x 10-7
.8 x 10-7
1.2 x 10-7
θ
p(D|θ)
θ̂
Figure 3.1: The top graph shows several training points in one dimension, known or
assumed to be drawn from a Gaussian of a particular variance, but unknown mean.
Four of the infinite number of candidate source distributions are shown in dashed
lines. The middle figures shows the likelihood p(D|θ) as a function of the mean. If
we had a very large number of training points, this likelihood would be very narrow.
The value that maximizes the likelihood is marked θ̂; it also maximizes the logarithm
of the likelihood — i.e., the log-likelihood l(θ), shown at the bottom. Note especially
that the likelihood lies in a different space from p(x|θ̂), and the two can have different
functional forms.
For analytical purposes, it is usually easier to work with the logarithm of the like-
lihood than with the likelihood itself. Since the logarithm is monotonically increasing,
the θ̂ that maximizes the log-likelihood also maximizes the likelihood. If p(D|θ) is a
well behaved, differentiable function of θ, θ̂ can be found by the standard methods of
differential calculus. If the number of parameters to be set is p, then we let θ denote
6 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
the p-component vector θ = (θ1, ..., θp)t, and ∇θ be the gradient operator
∇θ ≡
∂
∂θ1
...
∂
∂θp
. (2)
We define l(θ) as the log-likelihood function∗log-
likelihood
l(θ) ≡ ln p(D|θ). (3)
We can then write our solution formally as the argument θ that maximizes the log-
likelihood, i.e.,
θ̂ = arg max
θ
l(θ), (4)
where the dependence on the data set D is implicit. Thus we have from Eq. 1
l(θ) =
n∑
k=1
ln p(xk|θ) (5)
and
∇θl =
n∑
k=1
∇θ ln p(xk|θ). (6)
Thus, a set of necessary conditions for the maximum likelihood estimate for θ can be
obtained from the set of p equations
∇θl = 0. (7)
A solution θ̂ to Eq. 7 could represent a true global maximum, a local maximum or
minimum, or (rarely) an inflection point of l(θ). One must be careful, too, to check
if the extremum occurs at a boundary of the parameter space, which might not be
apparent from the solution to Eq. 7. If all solutions are found, we are guaranteed
that one represents the true maximum, though we might have to check each solution
individually (or calculate second derivatives) to identify which is the global optimum.
Of course, we must bear in mind that θ̂ is an estimate; it is only in the limit of an
infinitely large number of training points that we can expect that our estimate will
equal to the true value of the generating function (Sec. 3.5.1).
We note in passing that a related class of estimators — maximum a posteriori ormaximum a
posteriori MAP estimators — find the value of θ that maximizes l(θ)p(θ). Thus a maximum
likelihood estimator is a MAP estimator for the uniform or “flat” prior. As such,
a MAP estimator finds the peak, or mode of a posterior density. The drawback ofmode
MAP estimators is that if we choose some arbitrary nonlinear transformation of the
parameter space (e.g., an overall rotation), the density will change, and our MAP
solution need no longer be appropriate (Sec. 3.5.2).
∗ Of course, the base of the logarithm can be chosen for convenience, and in most analytic problems
base e is most natural. For that reason we will generally use ln rather than log or log2.
3.2. MAXIMUM LIKELIHOOD ESTIMATION 7
3.2.2 The Gaussian Case: Unknown µ
To see how maximum likelihood methods results apply to a specific case, suppose
that the samples are drawn from a multivariate normal population with mean µ and
covariance matrix Σ. For simplicity, consider first the case where only the mean is
unknown. Under this condition, we consider a sample point xk and find
ln p(xk|µ) = −
1
2
ln
[
(2π)d|Σ|
]
− 1
2
(xk − µ)tΣ−1(xk − µ) (8)
and
∇θ ln p(xk|µ) = Σ
−1(xk − µ). (9)
Identifying θ with µ, we see from Eq. 9 that the maximum likelihood estimate for µ
must satisfy
n∑
k=1
Σ−1(xk − µ̂) = 0, (10)
that is, each of the d components of µ̂ must vanish. Multiplying by Σ and rearranging,
we obtain
µ̂ =
1
n
n∑
k=1
xk. (11)
This is a very satisfying result. It says that the maximum likelihood estimate for
the unknown population mean is just the arithmetic average of the training samples
— the sample mean, sometimes written µ̂n to clarify its dependence on the number sample
meanof samples. Geometrically, if we think of the n samples as a cloud of points, the
sample mean is the centroid of the cloud. The sample mean has a number of desirable
statistical properties as well, and one would be inclined to use this rather obvious
estimate even without knowing that it is the maximum likelihood solution.
3.2.3 The Gaussian Case: Unknown µ and Σ
In the more general (and more typical) multivariate normal case, neither the mean µ
nor the covariance matrix Σ is known. Thus, these unknown parameters constitute
the components of the parameter vector θ. Consider first the univariate case with
θ1 = µ and θ2 = σ2. Here the log-likelihood of a single point is
ln p(xk|θ) = −
1
2
ln 2πθ2 −
1
2θ2
(xk − θ1)2 (12)
and its derivative is
∇θl = ∇θ ln p(xk|θ) =
[
1
θ2
(xk − θ1)
− 1
2θ2
+ (xk−θ1)
2
2θ22
]
. (13)
Applying Eq. 7 to the full log-likelihood leads to the conditions
n∑
k=1
1
θ̂2
(xk − θ̂1) = 0 (14)
8 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
and
−
n∑
k=1
1
θ̂2
+
n∑
k=1
(xk − θ̂1)2
θ̂22
= 0, (15)
where θ̂1 and θ̂2 are the maximum likelihood estimates for θ1 and θ2, respectively. By
substituting µ̂ = θ̂1, σ̂2 = θ̂2 and doing a little rearranging, we obtain the following
maximum likelihood estimates for µ and σ2:
µ̂ =
1
n
n∑
k=1
xk (16)
and
σ̂2 =
1
n
n∑
k=1
(xk − µ̂)2. (17)
While the analysis of the multivariate case is basically very similar, considerably
more manipulations are involved (Problem 6). Just as we would predict, though, the
result is that the maximum likelihood estimates for µ and Σ are given by
µ̂ =
1
n
n∑
k=1
xk (18)
and
Σ̂ =
1
n
n∑
k=1
(xk − µ̂)(xk − µ̂)t. (19)
Thus, once again we find that the maximum likelihood estimate for the mean
vector is the sample mean. The maximum likelihood estimate for the covariance
matrix is the arithmetic average of the n matrices (xk − µ̂)(xk − µ̂)t. Since the true
covariance matrix is the expected value of the matrix (x− µ̂) (x− µ̂)t, this is also a
very satisfying result.
3.2.4 Bias
The maximum likelihood estimate for the variance σ2 is biased; that is, the expectedbias
value over all data sets of size n of the sample variance is not equal to the true
variance:∗
E
[
1
n
n∑
i=1
(xi − x̄)2
]
=
n− 1
n
σ2 �= σ2. (20)
We shall return to a more general consideration of bias in Chap. ??, but for the
moment we can verify Eq. 20 for an underlying distribution with non-zero variance,
σ2, in the extreme case of n = 1, in which the expectation value E [·] = 0 �= σ2. The
maximum likelihood estimate of the covariance matrix is similarly biased.
Elementary unbiased estimators for σ2 and Σ are given by
∗ There should be no confusion over this use of the statistical term bias, and that for an offset in
neural networks and many other places.
3.3. BAYESIAN ESTIMATION 9
E
[
1
n− 1
n∑
i=1
(xi − x̄)2
]
= σ2 and (21)
C =
1
n− 1
n∑
k=1
(xk − µ̂)(xk − µ̂)t, (22)
where C is the so-called sample covariance matrix, as explored in Problem 33. If sample
covariancean estimator is unbiased for all distributions, as for example the variance estimator
in Eq. 21, then it is called absolutely unbiased. If the estimator tends to become
absolutely
unbiased
unbiased as the number of samples becomes very large, as for instance Eq. 20, then
the estimator is asymptotically unbiased. In many pattern recognition problems with
asymptot-
ically
unbiased
large training data sets, asymptotically unbiased estimators are acceptable.
Clearly, Σ̂ = [(n−1)/n]C, and Σ̂ is asymptotically unbiased — these two estimates
are essentially identical when n is large. However, the existence of two similar but
nevertheless distinct estimates for the covariance matrix may be disconcerting, and it
is natural to ask which one is “correct.” Of course, for n > 1 the answer is that these
estimates are neither right nor wrong — they are just different. What the existence of
two actually shows is that no single estimate possesses all of the properties we might
desire. For our purposes, the most desirable property is rather complex — we want
the estimate that leads to the best classification performance. While it is usually both
reasonable and sound to design a classifier by substituting the maximum likelihood
estimates for the unknown parameters, we might well wonder if other estimates might
not lead to better performance. Below we address this question from a Bayesian
viewpoint.
If we have a reliable model for the underlying distributions and their dependence
upon the parameter vector θ, the maximum likelihood classifier will give excellent
results. But what if our model is wrong — do we nevertheless get the best classifier in
our assumed set of models? For instance, what if we assume that a distribution comes
from N(µ, 1) but instead it actually comes from N(µ, 10)? Will the value we find for
θ = µ by maximum likelihood yield the best of all classifiers of the form derived from
N(µ, 1)? Unfortunately, the answer is “no,” and an illustrative counterexample is
given in Problem 7 where the so-called model error is large indeed. This points out
the need for reliable information concerning the models — if the assumed model is
very poor, we cannot be assured that the classifier we derive is the best, even among
our model set. We shall return to the problem of choosing among candidate models
in Chap. ??.
3.3 Bayesian estimation
We now consider the Bayesian estimation or Bayesian learning approach to pattern
classification problems. Although the answers we get by this method will generally
be nearly identical to those obtained by maximum likelihood, there is a conceptual
difference: whereas in maximum likelihood methods we view the true parameter vector
we seek, θ, to be fixed, in Bayesian learning we consider θ to be a random variable,
and training data allows us to convert a distribution on this variable into a posterior
probability density.
10 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
3.3.1 The Class-Conditional Densities
The computation of the posterior probabilities P (ωi|x) lies at the heart of Bayesian
classification. Bayes’ formula allows us to compute these probabilities from the prior
probabilities P (ωi) and the class-conditional densities p(x|ωi), but how can we proceed
when these quantities are unknown? The general answer to this question is that the
best we can do is to compute P (ωi|x) using all of the information at our disposal.
Part of this information might be prior knowledge, such as knowledge of the functional
forms for unknown densities and ranges for the values of unknown parameters. Part
of this information might reside in a set of training samples. If we again let D denote
the set of samples, then we can emphasize the role of the samples by saying that our
goal is to compute the posterior probabilities P (ωi|x,D). From these probabilities we
can obtain the Bayes classifier.
Given the sample D, Bayes’ formula then becomes
P (ωi|x,D) =
p(x|ωi,D)P (ωi|D)
c∑
j=1
p(x|ωj ,D)P (ωj |D)
. (23)
As this equation suggests, we can use the information provided by the training samples
to help determine both the class-conditional densities and the a priori probabilities.
Although we could maintain this generality, we shall henceforth assume that the
true values of the a priori probabilities are known or obtainable from a trivial calcu-
lation; thus we substitute P (ωi) = P (ωi|D). Furthermore, since we are treating the
supervised case, we can separate the training samples by class into c subsets D1, …,Dc,
with the samples in Di belonging to ωi. As we mentioned when addressing maximum
likelihood methods, in most cases of interest (and in all of the cases we shall consider),
the samples in Di have no influence on p(x|ωj ,D) if i �= j. This has two simplifying
consequences. First, it allows us to work with each class separately, using only the
samples in Di to determine p(x|ωi,D). Used in conjunction with our assumption that
the prior probabilities are known, this allows us to write Eq. 23 as
P (ωi|x,D) =
p(x|ωi,Di)P (ωi)
c∑
j=1
p(x|ωj ,Dj)P (ωj)
. (24)
Second, because each class can be treated independently, we can dispense with need-
less class distinctions and simplify our notation. In essence, we have c separate prob-
lems of the following form: use a set D of samples drawn independently according to
the fixed but unknown probability distribution p(x) to determine p(x|D). This is the
central problem of Bayesian learning.
3.3.2 The Parameter Distribution
Although the desired probability density p(x) is unknown, we assume that it has a
known parametric form. The only thing assumed unknown is the value of a parameter
vector θ. We shall express the fact that p(x) is unknown but has known parametric
form by saying that the function p(x|θ) is completely known. Any information we
might have about θ prior to observing the samples is assumed to be contained in a
known prior density p(θ). Observation of the samples converts this to a posterior
density p(θ|D), which, we hope, is sharply peaked about the true value of θ.
3.4. BAYESIAN PARAMETER ESTIMATION: GAUSSIAN CASE 11
Note that we are changing our supervised learning problem into an unsupervised
density estimation problem. To this end, our basic goal is to compute p(x|D), which
is as close as we can come to obtaining the unknown p(x). We do this by integrating
the joint density p(x,θ|D) over θ. That is,
p(x|D) =
∫
p(x,θ|D) dθ, (25)
where the integration extends over the entire parameter space. Now as discussed in
Problem 12 we can write p(x,θ|D) as the product p(x|θ,D)p(θ|D). Since the selection
of x and that of the training samples in D is done independently, the first factor is
merely p(x|θ). That is, the distribution of x is known completely once we know the
value of the parameter vector. Thus, Eq. 25 can be rewritten as
p(x|D) =
∫
p(x|θ)p(θ|D) dθ. (26)
This key equation links the desired class-conditional density p(x|D) to the posterior
density p(θ|D) for the unknown parameter vector. If p(θ|D) peaks very sharply
about some value θ̂, we obtain p(x|D)
p(x|θ̂), i.e., the result we would obtain by
substituting the estimate θ̂ for the true parameter vector. This result rests on the
assumption that p(x|θ) is smooth, and that the tails of the integral are not important.
These conditions are typically but not invariably the case, as we shall see in Sect. ??.
In general, if we are less certain about the exact value of θ, this equation directs us to
average p(x|θ) over the possible values of θ. Thus, when the unknown densities have
a known parametric form, the samples exert their influence on p(x|D) through the
posterior density p(θ|D). We should also point out that in practice, the integration
in Eq. 26 is often performed numerically, for instance by Monte-Carlo simulation.
3.4 Bayesian Parameter Estimation: Gaussian Case
In this section we use Bayesian estimation techniques to calculate the a posteri-
ori density p(θ|D) and the desired probability density p(x|D) for the case where
p(x|µ) ∼ N(µ,Σ).
3.4.1 The Univariate Case: p(µ|D)
Consider the case where µ is the only unknown parameter. For simplicity we treat
first the univariate case, i.e.,
p(x|µ) ∼ N(µ, σ2), (27)
where the only unknown quantity is the mean µ. We assume that whatever prior
knowledge we might have about µ can be expressed by a known prior density p(µ).
Later we shall make the further assumption that
p(µ) ∼ N(µ0, σ20), (28)
where both µ0 and σ20 are known. Roughly speaking, µ0 represents our best a priori
guess for µ, and σ20 measures our uncertainty about this guess. The assumption
that the prior distribution for µ is normal will simplify the subsequent mathematics.
12 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
However, the crucial assumption is not so much that the prior distribution for µ is
normal, but that it is known.
Having selected the a priori density for µ, we can view the situation as follows.
Imagine that a value is drawn for µ from a population governed by the probability
law p(µ). Once this value is drawn, it becomes the true value of µ and completely
determines the density for x. Suppose now that n samples x1, …, xn are independently
drawn from the resulting population. Letting D = {x1, …, xn}, we use Bayes’ formula
to obtain
p(µ|D) = p(D|µ)p(µ)∫
p(D|µ)p(µ) dµ
= α
n∏
k=1
p(xk|µ)p(µ), (29)
where α is a normalization factor that depends on D but is independent of µ. This
equation shows how the observation of a set of training samples affects our ideas about
the true value of µ; it relates the prior density p(µ) to an a posteriori density p(µ|D).
Since p(xk|µ) ∼ N(µ, σ2) and p(µ) ∼ N(µ0, σ20), we have
p(µ|D) = α
n∏
k=1
p(xk|µ)︷ ︸︸ ︷
1√
2πσ
exp
[
− 1
2
(xk − µ
σ
)2]
p(µ)︷ ︸︸ ︷
1√
2πσ0
exp
[
− 1
2
(µ− µ0
σ0
)2]
= α′ exp
[
−1
2
(
n∑
k=1
(
µ− xk
σ
)2
+
(
µ− µ0
σ0
)2) ]
= α′′ exp
[
− 1
2
[( n
σ2
+
1
σ20
)
µ2 − 2
(
1
σ2
n∑
k=1
xk +
µ0
σ20
)
µ
] ]
, (30)
where factors that do not depend on µ have been absorbed into the constants α,
α′, and α′′. Thus, p(µ|D) is an exponential function of a quadratic function of µ,
i.e., is again a normal density. Since this is true for any number of training samples,
p(µ|D) remains normal as the number n of samples is increased, and p(µ|D) is said
to be a reproducing density and p(µ) is said to be a conjugate prior. If we writereproducing
density p(µ|D) ∼ N(µn, σ2n), then µn and σ2n can be found by equating coefficients in Eq. 30
with corresponding coefficients in the generic Gaussian of the form
p(µ|D) = 1√
2πσn
exp
[
−1
2
(
µ− µn
σn
)2]
. (31)
Identifying coefficients in this way yields
1
σ2n
=
n
σ2
+
1
σ20
(32)
and
µn
σ2n
=
n
σ2
x̄n +
µ0
σ20
, (33)
where x̄n is the sample mean
3.4. BAYESIAN PARAMETER ESTIMATION: GAUSSIAN CASE 13
x̄n =
1
n
n∑
k=1
xk. (34)
We solve explicitly for µn and σ2n and obtain
µn =
(
nσ20
nσ20 + σ
2
)
x̄n +
σ2
nσ20 + σ
2
µ0 (35)
and
σ2n =
σ20σ
2
nσ20 + σ
2
. (36)
These equations show how the prior information is combined with the empirical
information in the samples to obtain the a posteriori density p(µ|D). Roughly speak-
ing, µn represents our best guess for µ after observing n samples, and σ2n measures
our uncertainty about this guess. Since σ2n decreases monotonically with n — ap-
proaching σ2/n as n approaches infinity — each additional observation decreases our
uncertainty about the true value of µ. As n increases, p(µ|D) becomes more and
more sharply peaked, approaching a Dirac delta function as n approaches infinity.
This behavior is commonly known as Bayesian learning (Fig. 3.2). Bayesian
learning
1
-4 -2 2 4
µ
1
30
20
5
10
p(µ|x1, … , xn)
0
-2
-1
0
1
2
-4
-3
-2
-1
0
1
0
1
2
3
-2
-1
0
1
2
50
12
24
5
1
p(µ|x1, … , xn)
Figure 3.2: Bayesian learning of the mean of normal distributions in one and two di-
mensions. The posterior distribution estimates are labelled by the number of training
samples used in the estimation.
In general, µn is a linear combination of x̄n and µ0, with coefficients that are
non-negative and sum to one. Thus µn always lies somewhere between x̄n and µ0. If
σ �= 0, µn approaches the sample mean as n approaches infinity. If σ0 = 0, we have
a degenerate case in which our a priori certainty that µ = µ0 is so strong that no
number of observations can change our opinion. At the other extreme, if σ0 � σ, we
are so uncertain about our a priori guess that we take µn = x̄n, using only the samples
to estimate µ. In general, the relative balance between prior knowledge and empirical
data is set by the ratio of σ2 to σ20 , which is sometimes called the dogmatism. If the dogmatism
dogmatism is not infinite, after enough samples are taken the exact values assumed
for µ0 and σ20 will be unimportant, and µn will converge to the sample mean.
14 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
3.4.2 The Univariate Case: p(x|D)
Having obtained the a posteriori density for the mean, p(µ|D), all that remains is to
obtain the “class-conditional” density for p(x|D).∗ From Eqs. 26, 27 & 31 we have
p(x|D) =
∫
p(x|µ)p(µ|D) dµ
=
∫
1√
2πσ
exp
[
− 1
2
(x− µ
σ
)2] 1√
2πσn
exp
[
− 1
2
(µ− µn
σn
)2]
dµ
=
1
2πσσn
exp
[
− 1
2
(x− µn)2
σ2 + σ2n
]
f(σ, σn), (37)
where
f(σ, σn) =
∫
exp
[
− 1
2
σ2 + σ2n
σ2σ2n
(
µ− σ
2
nx + σ
2µn
σ2 + σ2n
)2]
dµ.
That is, as a function of x, p(x|D) is proportional to exp[−(1/2)(x−µn)2/(σ2 +σ2n)],
and hence p(x|D) is normally distributed with mean µn and variance σ2 + σ2n:
p(x|D) ∼ N(µn, σ2 + σ2n). (38)
In other words, to obtain the class-conditional density p(x|D), whose parametric
form is known to be p(x|µ) ∼ N(µ, σ2), we merely replace µ by µn and σ2 by σ2 +σ2n.
In effect, the conditional mean µn is treated as if it were the true mean, and the
known variance is increased to account for the additional uncertainty in x resulting
from our lack of exact knowledge of the mean µ. This, then, is our final result:
the density p(x|D) is the desired class-conditional density p(x|ωj ,Dj), and together
with the prior probabilities P (ωj) it gives us the probabilistic information needed to
design the classifier. This is in contrast to maximum likelihood methods that only
make points estimates for µ̂ and σ̂2, rather that estimate a distribution for p(x|D).
3.4.3 The Multivariate Case
The treatment of the multivariate case in which Σ is known but µ is not, is a di-
rect generalization of the univariate case. For this reason we shall only sketch the
derivation. As before, we assume that
p(x|µ) ∼ N(µ,Σ) and p(µ) ∼ N(µ0,Σ0), (39)
where Σ, Σ0, and µ0 are assumed to be known. After observing a set D of n inde-
pendent samples x1, …,xn, we use Bayes’ formula to obtain
p(µ|D) = α
n∏
k=1
p(xk|µ)p(µ) (40)
= α′exp
[
−1
2
(
µt(nΣ−1 + Σ−10 )µ− 2µ
t
(
Σ−1
n∑
k=1
xk + Σ
−1
0 µ0
))]
,
∗ Recall that for simplicity we dropped class distinctions, but that all samples here come from the
same class, say ωi, and hence p(x|D) is really p(x|ωi,Di).
3.4. BAYESIAN PARAMETER ESTIMATION: GAUSSIAN CASE 15
which has the form
p(µ|D) = α′′exp
[
−1
2
(µ− µn)tΣ−1n (µ− µn)
]
. (41)
Thus, p(µ|D) ∼ N(µn,Σn), and once again we have a reproducing density. Equating
coefficients, we obtain the analogs of Eqs. 35 & 36,
Σ−1n = nΣ
−1 + Σ−10 (42)
and
Σ−1n µn = nΣ
−1µ̂n + Σ
−1
0 µ0, (43)
where µ̂n is the sample mean
µ̂n =
1
n
n∑
k=1
xk. (44)
The solution of these equations for µ and Σn is simplified by knowledge of the matrix
identity
(A−1 + B−1)−1 = A(A + B)−1B = B(A + B)−1A, (45)
which is valid for any pair of nonsingular, d-by-d matrices A and B. After a little
manipulation (Problem 16), we obtain the final results:
µn = Σ0
(
Σ0 +
1
n
Σ
)−1
µ̂n +
1
n
Σ
(
Σ0 +
1
n
Σ
)−1
µ0 (46)
(which, as in the univariate case, is a linear combination of µ̂n and µ0) and
Σn = Σ0
(
Σ0 +
1
n
Σ
)−1 1
n
Σ. (47)
The proof that p(x|D) ∼ N(µn,Σ+Σn) can be obtained as before by performing
the integration
p(x|D) =
∫
p(x|µ)p(µ|D) dµ. (48)
However, this result can be obtained with less effort by observing that x can be viewed
as the sum of two mutually independent random variables, a random vector µ with
p(µ|D) ∼ N(µn,Σn) and an independent random vector y with p(y) ∼ N(0,Σ).
Since the sum of two independent, normally distibuted vectors is again a normally
distributed vector whose mean is the sum of the means and whose covariance matrix
is the sum of the covariance matrices (Chap. ?? Problem ??), we have
p(x|D) ∼ N(µn,Σ + Σn), (49)
and the generalization is complete.
16 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
3.5 Bayesian Parameter Estimation: General Theory
We have just seen how the Bayesian approach can be used to obtain the desired density
p(x|D) in a special case — the multivariate Gaussian. This approach can be gener-
alized to apply to any situation in which the unknown density can be parameterized.
The basic assumptions are summarized as follows:
• The form of the density p(x|θ) is assumed to be known, but the value of the
parameter vector θ is not known exactly.
• Our initial knowledge about θ is assumed to be contained in a known a priori
density p(θ).
• The rest of our knowledge about θ is contained in a set D of n samples x1, …,xn
drawn independently according to the unknown probability density p(x).
The basic problem is to compute the posterior density p(θ|D), since from this we
can use Eq. 26 to compute p(x|D):
p(x|D) =
∫
p(x|θ)p(θ|D) dθ. (50)
By Bayes’ formula we have
p(θ|D) = p(D|θ)p(θ)∫
p(D|θ)p(θ) dθ
, (51)
and by the independence assumption
p(D|θ) =
n∏
k=1
p(xk|θ). (52)
This constitutes the solution to the problem, and Eqs. 51 & 52 illuminate its
relation to the maximum likelihood solution. Suppose that p(D|θ) reaches a sharp
peak at θ = θ̂. If the prior density p(θ) is not zero at θ = θ̂ and does not change
much in the surrounding neighborhood, then p(θ|D) also peaks at that point. Thus,
Eq. 26 shows that p(x|D) will be approximately p(x|θ̂), the result one would obtain
by using the maximum likelihood estimate as if it were the true value. If the peak
of p(D|θ) is very sharp, then the influence of prior information on the uncertainty in
the true value of θ can be ignored. In this and even the more general case, though,
the Bayesian solution tells us how to use all the available information to compute the
desired density p(x|D).
While we have obtained the formal Bayesian solution to the problem, a number
of interesting questions remain. One concerns the difficulty of carrying out these
computations. Another concerns the convergence of p(x|D) to p(x). We shall discuss
the matter of convergence briefly, and later turn to the computational question.
To indicate explicitly the number of samples in a set for a single category, we shall
write Dn = {x1, …,xn}. Then from Eq. 52, if n > 1
p(Dn|θ) = p(xn|θ)p(Dn−1|θ). (53)
Substituting this in Eq. 51 and using Bayes’ formula, we see that the posterior density
satisfies the recursion relation
3.5. BAYESIAN PARAMETER ESTIMATION: GENERAL THEORY 17
p(θ|Dn) = p(xn|θ)p(θ|D
n−1)∫
p(xn|θ)p(θ|Dn−1) dθ
. (54)
With the understanding that p(θ|D0) = p(θ), repeated use of this equation pro-
duces the sequence of densities p(θ), p(θ|x1), p(θ|x1,x2), and so forth. (It should be
obvious from Eq. 54 that p(θ|Dn) depends only on the points in Dn, not the sequence
in which they were selected.) This is called the recursive Bayes approach to param- recursive
Bayeseter estimation. This is, too, our first example of an incremental or on-line learning
incremental
learning
method, where learning goes on as the data is collected. When this sequence of den-
sities converges to a Dirac delta function centered about the true parameter value —
Bayesian learning (Example 1). We shall come across many other, non-incremental
learning schemes, where all the training data must be present before learning can take
place.
In principle, Eq. 54 requires that we preserve all the training points in Dn−1 in
order to calculate p(θ|Dn) but for some distributions, just a few parameters associated
with p(θ|Dn−1) contain all the information needed. Such parameters are the sufficient
statistics of those distributions, as we shall see in Sect. 3.6. Some authors reserve the
term recursive learning to apply to only those cases where the sufficient statistics are
retained — not the training data — when incorporating the information from a new
training point. We could call this more restrictive usage true recursive Bayes learning.
Example 1: Recursive Bayes learning
Suppose we believe our one-dimensional samples come from a uniform distribution
p(x|θ) ∼ U(0, θ) =
{
1/θ 0 ≤ x ≤ θ
0 otherwise,
but initially we know only that our parameter is bounded. In particular we assume
0 < θ ≤ 10 (a non-informative or “flat prior” we shall discuss in Sect. 3.5.2). We
will use recursive Bayes methods to estimate θ and the underlying densities from the
data D = {4, 7, 2, 8}, which were selected randomly from the underlying distribution.
Before any data arrive, then, we have p(θ|D0) = p(θ) = U(0, 10). When our first data
point x1 = 4 arrives, we use Eq. 54 to get an improved estimate:
p(θ|D1) ∝ p(x|θ)p(θ|D0) =
{
1/θ for 4 ≤ θ ≤ 10
0 otherwise,
where throughout we will ignore the normalization. When the next data point x2 = 7
arrives, we have
p(θ|D2) ∝ p(x|θ)p(θ|D1) =
{
1/θ2 for 7 ≤ θ ≤ 10
0 otherwise,
and similarly for the remaining sample points. It should be clear that since each
successive step introduces a factor of 1/θ into p(x|θ), and the distribution is nonzero
only for x values above the largest data point sampled, the general form of our solution
is p(θ|Dn) ∝ 1/θn for max
x
[Dn] ≤ θ ≤ 10, as shown in the figure. Given our full data
18 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
2 4 6 8 10
0.2
0.4
0.6
θ
0
1
2
3
4
p(θ|Dn)
The posterior p(θ|Dn) for the model and n points in the data set in this Example.
The posterior begins p(θ) ∼ U(0, 10), and as more points are incorporated it becomes
increasingly peaked at the value of the highest data point.
set, the maximum likelihood solution here is clearly θ̂ = 8, and this implies a uniform
p(x|D) ∼ U(0, 8).
According to our Bayesian methodology, which requires the integration in Eq. 50,
the density is uniform up to x = 8, but has a tail at higher values — an indication
that the influence of our prior p(θ) has not yet been swamped by the information in
the training data.
2 4 6 8 10
0.1
0.2
0
x
p(x|D)
ML
Bayes
Given the full set of four points, the distribution based on the maximum likelihood
solution is p(x|θ̂) ∼ U(0, 8), whereas the distribution derived from Bayesian methods
has a small tail above x = 8, reflecting the prior information that values of x near 10
are possible.
Whereas the maximum likelihood approach estimates a point in θ space, the
Bayesian approach instead estimates a distribution. Technically speaking, then, we
cannot directly compare these estimates. It is only when the second stage of inference
is done — that is, we compute the distributions p(x|D), as shown in the above figure
— that the comparison is fair.
For most of the typically encountered probability densities p(x|θ), the sequence of
posterior densities does indeed converge to a delta function. Roughly speaking, this
implies that with a large number of samples there is only one value for θ that causes
p(x|θ) to fit the data, i.e., that θ can be determined uniquely from p(x|θ). When this
is the case, p(x|θ) is said to be identifiable. A rigorous proof of convergence underidentifi-
ability these conditions requires a precise statement of the properties required of p(x|θ) and
p(θ) and considerable care, but presents no serious difficulties (Problem 21).
There are occasions, however, when more than one value of θ may yield the same
value for p(x|θ). In such cases, θ cannot be determined uniquely from p(x|θ), and
p(x|Dn) will peak near all of the values of θ that explain the data. Fortunately, this
ambiguity is erased by the integration in Eq. 26, since p(x|θ) is the same for all of
3.5. BAYESIAN PARAMETER ESTIMATION: GENERAL THEORY 19
these values of θ. Thus, p(x|Dn) will typically converge to p(x) whether or not p(x|θ)
is identifiable. While this might make the problem of identifiabilty appear to be moot,
we shall see in Chap. ?? that identifiability presents a genuine problem in the case of
unsupervised learning.
3.5.1 When do Maximum Likelihood and Bayes methods differ?
In virtually every case, maximum likelihood and Bayes solutions are equivalent in the
asymptotic limit of infinite training data. However since practical pattern recognition
problems invariably have a limited set of training data, it is natural to ask when
maximum likelihood and Bayes solutions may be expected to differ, and then which
we should prefer.
There are several criteria that will influence our choice. One is computational
complexity (Sec. 3.7.2), and here maximum likelhood methods are often to be pref-
ered since they require merely differential calculus techniques or gradient search for θ̂,
rather than a possibly complex multidimensional integration needed in Bayesian esti-
mation. This leads to another consideration: interpretability. In many cases the max-
imum likelihood solution will be easier to interpret and understand since it returns the
single best model from the set the designer provided (and presumably understands).
In contrast Bayesian methods give a weighted average of models (parameters), often
leading to solutions more complicated and harder to understand than those provided
by the designer. The Bayesian approach reflects the remaining uncertainty in the
possible models.
Another consideration is our confidence in the prior information, such as in the
form of the underlying distribution p(x|θ). A maximum likelihood solution p(x|θ̂)
must of course be of the assumed parametric form; not so for the Bayesian solution.
We saw this difference in Example 1, where the Bayes solution was not of the para-
metric form originally assumed, i.e., a uniform p(x|D). In general, through their use
of the full p(θ|D) distribution Bayesian methods use more of the information brought
to the problem than do maximum likelihood methods. (For instance, in Example 1
the addition of the third training point did not change the maximum likelihood so-
lution, but did refine the Bayesian estimate.) If such information is reliable, Bayes
methods can be expected to give better results. Further, general Bayesian methods
with a “flat” or uniform prior (i.e., where no prior information is explicitly imposed)
are equivalent to maximum likelihood methods. If there is much data, leading to a
strongly peaked p(θ|D), and the prior p(θ) is uniform or flat, then the MAP estimate
is essentially the same as the maximum likelihood estimate.
When p(θ|D) is broad, or asymmetric around θ̂, the methods are quite likely to
yield p(x|D) distributions that differ from one another. Such a strong asymmetry
(when not due to rare statistical irregularities in the selection of the training data)
generally convey some information about the distribution, just as did the asymmetric
role of the threshold θ in Example 1. Bayes methods would exploit such information;
not so maximum likelihood ones (at least not directly). Further, Bayesian methods
make more explicit the crucial problem of bias and variance tradeoffs — roughly
speaking the balance between the accuracy of the estimation and its variance, which
depend upon the amount of traning data. This important matter was irrelevant in
Chap. ??, where there was no notion of a finite training set, but it will be crucial in
our considerations of the theory of machine learning in Chap. ??.
When designing a classifier by either of these methods, we determine the posterior
densities for each category, and classify a test point by the maximum posterior. (If
20 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
there are costs, summarized in a cost matrix, these can be incorporated as well.)
There are three sources of classification error in our final system:
Bayes or indistinguisability error: the error due to overlapping densities p(x|ωi
for different values of i. This error is an inherent property of the problem and
can never be eliminated.
Model error: the error due to having an incorrect model. This error can only be
eliminated if the designer specifies a model that includes the true model which
generated the data. Designers generally choose the model based on knowledge
of the problem domain rather than on the subsequent estimation method, and
thus the model error in maximum likelihood and Bayes methods rarely differ.
Estimation error: the error arising from the fact that the parameters are estimated
from a finite sample. This error can best be reduced by increasing the training
data, a topic we shall revisit in greater detail in Chap. ??.
The relative contributions of these sources depend upon problem, of course. In the
limit of infinite training data, the estimation error vanishes, and the total classification
error will be the same for both maximum likelihodd and Bayes methods.
In summary, there are strong theoretical and methodological arguments supporting
Bayesian estimation, though in practice maximum likelihood estimation is simpler,
and when used for designing classifiers, can lead to classifiers nearly as accurate.
3.5.2 Non-informative Priors and Invariance
Generally speaking, the information about the prior p(θ) derives from the designer’s
knowledge of the problem domain and as such is beyond our study of the design of
classifiers. Nevertheless in some cases we have guidence in how to create priors that
do not impose structure when we believe none exists, and this leads us to the notion
of non-informative priors.
Recall our discussion of the role of prior category probabilities in Chap. ??, where
in the absense of other information, we assumed each of c categories equally likely.
Analogously, in a Bayesian framework we can have a “non-informative” prior over a
parameter for a single category’s distribution. Suppose for instance that we are using
Bayesian methods to infer from data the mean and variance of a Gaussian. What
prior might we put on these parameters? Surely the unit of spatial measurement —
meters, feet, inches — is an historical accident and irrelevant to the functional form
of the prior. Thus there is an implied scale invariance, formally stated asscale
invariance
p(θ) = αp(θ/α) (55)
for some constant α. Such scale invariance here leads to priors such as p(µ) ∝ µ−k
for some undermined constant k (Problem 20). (Such a prior is improper; it doesimproper
prior not integrate to unity, and hence cannot strictly be interpreted as representing our
actual prior belief.) In general, then, if there is known or assumed invariance — such
as translation, or for discrete distributions invariance to the sequential order of data
selection — there will be constraints on the form of the prior. If we can find a prior
that satisfies such constraints, the resulting prior is “non-informative” with respect
to that invariance.
It is tempting to assert that the use of non-informative priors is somehow “ob-
jective” and lets the data speak for themselves, but such a view is a bit naive. For
3.6. *SUFFICIENT STATISTICS 21
example, we may seek a non-informative prior when estimating the standard deviation
σ of a Gaussian. But this requirement might not lead to the non-informative prior
for estimating the variance, σ2. Which should we use? In fact, the greatest benefit
of this approach is that it forces the designer to acknowledge and be clear about the
assumed invariance — the choice of which generally lies outside our methodology. It
may be more difficult to accommodate such arbitrary transformations in a maximum
a posteriori (MAP) estimator (Sec. 3.2.1), and hence considerations of invariance are
of greatest use in Bayesian estimation, or when the posterior is very strongly peaked
and the mode not influenced by transformations of the density (Problem 19).
3.6 *Sufficient Statistics
From a practical viewpoint, the formal solution provided by Eqs. 26, 51 & 52 is not
computationally attractive. In pattern recognition applications it is not unusual to
have dozens or hundreds of parameters and thousands of training samples, which
makes the direct computation and tabulation of p(D|θ) or p(θ|D) quite out of the
question. We shall see in Chap. ?? how neural network methods avoid many of the
difficulties of setting such a large number of parameters in a classifier, but for now we
note that the only hope for an analytic, computationally feasible maximum likelihood
solution lies in being able to find a parametric form for p(x|θ) that on the one hand
matches the characteristics of the problem and on the other hand allows a reasonably
tractable solution.
Consider the simplification that occurred in the problem of learning the parameters
of a multivariate Gaussian density. The basic data processing required was merely
the computation of the sample mean and sample covariance. This easily computed
and easily updated statistic contained all the information in the samples relevant to
estimating the unknown population mean and covariance. One might suspect that
this simplicity is just one more happy property of the normal distribution, and that
such good fortune is not likely to occur in other cases. While this is largely true,
there are distributions for which computationally feasible solutions can be obtained,
and the key to their simplicity lies in the notion of a sufficient statistic.
To begin with, any function of the samples is a statistic. Roughly speaking, a
sufficient statistic is a (possibly vector-valued) function s of the samples D that con-
tains all of the information relevant to estimating some parameter θ. Intuitively, one
might expect the definition of a sufficient statistic to involve the requirement that
p(θ|s,D) = p(θ|s). However, this would require treating θ as a random variable,
limiting the definition to a Bayesian domain. To avoid such a limitation, the conven-
tional definition is as follows: A statistic s is said to be sufficient for θ if p(D|s,θ) is
independent of θ. If we think of θ as a random variable, we can write
p(θ|s,D) = p(D|s,θ)p(θ|s)
p(D|s)
, (56)
whereupon it becomes evident that p(θ|s,D) = p(θ|s) if s is sufficient for θ. Con-
versely, if s is a statistic for which p(θ|s,D) = p(θ|s), and if p(θ|s) �= 0, it is easy to
show that p(D|s,θ) is independent of θ (Problem 27). Thus, the intuitive and the
conventional definitions are basically equivalent. As one might expect, for a Gaussian
distribution the sample mean and covariance, taken together, represent a sufficient
statistic for the true mean and covariance; if these are known, all other statistics
22 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
such as the mode, range, higher-order moments, number of data points, etc., are
superfluous when estimating the true mean and covariance.
A fundamental theorem concerning sufficient statistics is the Factorization Theo-
rem, which states that s is sufficient for θ if and only if p(D|θ) can be factored into
the product of two functions, one depending only on s and θ, and the other depend-
ing only on the training samples. The virtue of the Factorization Theorem is that it
allows us to shift our attention from the rather complicated density p(D|s,θ), used
to define a sufficient statistic, to the simpler function
p(D|θ) =
n∏
k=1
p(xk|θ). (57)
In addition, the Factorization Theorem makes it clear that the characteristics of a
sufficient statistic are completely determined by the density p(x|θ), and have nothing
to do with a felicitous choice of an a priori density p(θ). A proof of the Factorization
Theorem in the continuous case is somewhat tricky because degenerate situations are
involved. Since the proof has some intrinsic interest, however, we include one for the
simpler discrete case.
Theorem 3.1 (Factorization) A statistic s is sufficient for θ if and only if the
probability P (D|θ) can be written as the product
P (D|θ) = g(s,θ)h(D), (58)
for some function h(·).
Proof:
(a) We begin by showing the “if” part of the theorem. Suppose first that s is sufficient
for θ, so that P (D|s,θ) is independent of θ. Since we want to show that P (D|θ) can
be factored, our attention is directed toward computing P (D|θ) in terms of P (D|s,θ).
We do this by summing the joint probability P (D, s|θ) over all values of s:
P (D|θ) =
∑
s
P (D, s|θ)
=
∑
s
P (D|s,θ)P (s|θ). (59)
But since s = ϕ(D) for some ϕ(·), there is only one possible value for s for the given
data, and thus
P (D|θ) = P (D|s,θ)P (s|θ). (60)
Moreover, since by hypothesis P (D|s,θ) is independent of θ, the first factor depends
only on D. Identifying P (s|θ) with g(s,θ), we see that P (D|θ) factors, as desired.
(b) We now consider the “only if” part of the theorem. To show that the ability to
factor P (D|θ) as the product g(s,θ)h(D) implies that s is sufficient for θ, we must
show that such a factoring implies that the conditional probability P (D|s,θ) is inde-
pendent of θ. Because s = ϕ(D), specifying a value for s constrains the possible sets
of samples to some set D̄. Formally, D̄ = {D|ϕ(D) = s}. If D̄ is empty, no assignment
of values to the samples can yield that value of s, and P (s|θ) = 0. Excluding such
cases, i.e., considering only values of s that can arise, we have
3.6. *SUFFICIENT STATISTICS 23
P (D|s,θ) = P (D, s|θ)
P (s|θ)
. (61)
The denominator can be computed by summing the numerator over all values of D.
Since the numerator will be zero if D /∈ D̄, we can restrict the summation to D ∈ D̄.
That is,
P (D|s,θ) = P (D|s,θ)∑
D∈D̄
P (D|s,θ)
=
P (D|θ)∑
D∈D̄
P (D|θ)
=
g(s,θ)h(D)∑
D∈D̄
g(s,θ)h(D)
=
h(D)∑
D∈D̄
h(D)
, (62)
which is independent of θ. Thus, by definition, s is sufficient for θ.
It should be pointed out that there are trivial ways of constructing sufficient
statistics. For example we can define s to be a vector whose components are the
n samples themselves: x1, ...,xn. In that case g(s,θ) = p(D|θ) and h(D) = 1. One
can even produce a scalar sufficient statistic by the trick of interleaving the digits
in the decimal expansion of the components of the n samples. Sufficient statistics
such as these are of little interest, since they do not provide us with simpler results.
The ability to factor p(D|θ) into a product g(s,θ)h(D) is interesting only when the
function g and the sufficient statistic s are simple. It should be noted that sufficiency
is an integral notion. That is, if s is a sufficient statistic for θ, this does not necessarily
imply that their corresponding components are sufficient, i.e., that s1 is sufficient for
θ1, or s2 for θ2, and so on (Problem 26).
An obvious fact should also be mentioned: the factoring of p(D|θ) into g(s,θ)h(D)
is not unique. If f(s) is any function of s, then g′(s,θ) = f(s)g(s,θ) and h′(D) =
h(D)/f(s) are equivalent factors. This kind of ambiguity can be eliminated by defining
the kernel density kernel
density
ḡ(s,θ) =
g(s,θ)∫
g(s,θ) dθ
(63)
which is invariant to this kind of scaling.
What is the importance of sufficient statistics and kernel densities for parameter
estimation? The general answer is that the most practical applications of classical
parameter estimation to pattern classification involve density functions that possess
simple sufficient statistics and simple kernel densities. Moreover, it can be shown
that for any clasification rule, we can find another based solely on sufficient statistics
that has equal or better performance. Thus — in principle at least — we need only
consider decisions based on sufficient statistics. It is, in essence, the ultimate in data
reduction: we can reduce an extremely large data set down to a few numbers — the
sufficient statistics — confident that all relevant information has been preserved. This
means, too, that we can always create the Bayes classifier from sufficient statistics, as
for example our Bayes classifiers for Gaussian distributions were functions solely of
the sufficient statistics, estimates of µ and Σ.
In the case of maximum likelihood estimation, when searching for a value of θ
that maximizes p(D|θ) = g(s,θ)h(D), we can restrict our attention to g(s,θ). In this
case, the normalization provided by Eq. 63 is of no particular value unless ḡ(s,θ) is
simpler than g(s,θ). The significance of the kernel density is revealed however in the
Bayesian case. If we substitute p(D|θ) = g(s,θ)h(D) in Eq. 51, we obtain
24 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
p(θ|D) = g(s,θ)p(θ)∫
g(s,θ)p(θ) dθ
. (64)
If our prior knowledge of θ is very vague, p(θ) will tend to be uniform, or changing
very slowly as a function of θ. For such an essentially uniform p(θ), Eq. 64 shows
that p(θ|D) is approximately the same as the kernel density. Roughly speaking, the
kernel density is the posterior distribution of the parameter vector when the prior
distribution is uniform. Even when the a priori distribution is far from uniform, the
kernel density typically gives the asymptotic distribution of the parameter vector. In
particular, when p(x|θ) is identifiable and when the number of samples is large, g(s,θ)
usually peaks sharply at some value θ = θ̂. If the a priori density p(θ) is continuous
at θ = θ̂ and if p(θ̂) is not zero, p(θ|D) will approach the kernel density ḡ(s,θ).
3.6.1 Sufficient Statistics and the Exponential Family
To see how the Factorization Theorem can be used to obtain sufficient statistics,
consider once again the familiar d-dimensional normal case with fixed covariance but
unknown mean, i.e., p(x|θ) ∼ N(θ,Σ). Here we have
p(D|θ) =
n∏
k=1
1
(2π)d/2|Σ|1/2
exp
[
− 1
2
(xk − θ)tΣ−1(xk − θ)
]
=
1
(2π)nd/2|Σ|n/2
exp
[
− 1
2
n∑
k=1
(θtΣ−1θ − 2θtΣ−1xk + xtkΣ−1xk)
]
= exp
[
−n
2
θtΣ−1θ + θtΣ−1
(
n∑
k=1
xk
)]
× 1
(2π)nd/2|Σ|n/2
exp
[
−1
2
n∑
k=1
xtkΣ
−1xk
]
. (65)
This factoring isolates the θ dependence of p(D|θ) in the first term, and hence from
the Factorization Theorem we conclude that
∑n
k=1 xk is sufficient for θ. Of course,
any one-to-one function of this statistic is also sufficient for θ; in particular, the sample
mean
µ̂n =
1
n
n∑
k=1
xk (66)
is also sufficient for θ. Using this statistic, we can write
g(µ̂n,θ) = exp
[
− n
2
(
θtΣ−1θ − 2θtΣ−1µ̂n
)]
. (67)
From using Eq. 63, or by completing the square, we can obtain the kernel density:
ḡ(µ̂n,θ) =
1
(2π)d/2| 1
n
Σ|1/2
exp
[
− 1
2
(θ − µ̂n)t
( 1
n
Σ
)−1
(θ − µ̂n)
]
. (68)
These results make it immediately clear that µ̂n is the maximum likelihood estimate
for θ. The Bayesian posterior density can be obtained from ḡ(µ̂n,θ) by performing
3.6. *SUFFICIENT STATISTICS 25
the integration indicated in Eq. 64. If the a priori density is essentally uniform,
p(θ|D) = ḡ(µ̂n,θ).
This same general approach can be used to find sufficient statistics for other density
functions. In particular, it applies to any member of the exponential family, a group
of probability and probability density functions that possess simple sufficient statis-
tics. Members of the exponential family include the Gaussian, exponential, Rayleigh,
Poisson, and many other familiar distributions. They can all be written in the form
p(x|θ) = α(x) exp [a(θ) + b(θ)tc(x)]. (69)
If we multiply n terms of the form in Eq. 69 we find
p(D|θ) = exp
[
na(θ) + b(θ)t
n∑
k=1
c(xk)
] n∏
k=1
α(xk) = g(s,θ)h(D), (70)
where we can take
s =
1
n
n∑
k=1
c(x),
g(s,θ) = exp [n{a(θ) + b(θ)ts}],
and
h(D) =
n∏
k=1
α(xk).
The distributions, sufficient statistics, and unnormalized kernels for a number of
commonly encountered members of the exponential family are given in Table ??.
It is a fairly routine matter to derive maximum likelihood estimates and Bayesian
a posteriori distributions from these solutions. With two exceptions, the solutions
given are for univariate cases, though they can be used in multivariate situations if
statistical independence can be assumed. Note that a few well-known probability
distributions, such as the Cauchy, do not have sufficient statistics, so that the sample
mean can be a very poor estimator of the true mean (Problem 28).
26 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
Table 3.1: Common Exponential Distributions and their Sufficient Statistics.
Name Distribution Domain s [g(s,θ)]1/n
Normal
p(x|θ) =√
θ2
2π
e−(1/2)θ2(x−θ1)
2 θ2 > 0
x
θ1
θ2 √2/
1
n
n∑
k=1
xk
1
n
n∑
k=1
x2k
√
θ2e
− 12 θ2(s2−2θ1s1+θ
2
1)
Multi-
variate
Normal
p(x|θ) =
|Θ2|1/2
(2π)d/2
e−(1/2)(x−θ1)
tΘ2(x−θ1)
Θ2
positive
definite
x1
x2
1
n
n∑
k=1
xk
1
n
n∑
k=1
xkxtk
|Θ2|
1/2e−
1
2 [trΘ2s2
−2θt1Θ2s1+θ
t
1Θ2θ1]
Exponential
p(x|θ) ={
θe−θx x ≥ 0
0 otherwise
θ > 0
0 1 2 3 4 5
x
0.2
0.4
0.6
θ
1
n
n∑
k=1
xk θe
−θs
Rayleigh
p(x|θ) ={
2θxe−θx
2
x ≥ 0
0 otherwise
θ > 0
1 2 3 4
x
0.2
0.4
0.6
θ
1
n
n∑
k=1
x2k θe
−θs
Maxwell
p(x|θ) ={
4√
π
θ3/2x2e−θx
2
x ≥ 0
0 otherwise
θ > 0
1 2 3 4
x
0.2
0.4
0.6
θ
1
n
n∑
k=1
x2k θ
3/2e−θs
Gamma
p(x|θ) ={
θ
θ1+1
2
Γ(θ1+1)
xθ1e−θ2x x ≥ 0
0 otherwise
θ1 > −1
θ2 > 0
1 2 3 4
x
0.5
1
1.5
θ2
θ1 = -.2
(
n∏
k=1
xk
)1/n
1
n
n∑
k=1
xk
θ
θ1+1
2
Γ(θ1+1)
sθ11 e
−θ2s2
Beta
p(x|θ) =
Γ(θ1+θ2+2)
Γ(θ1+1)Γ(θ2+1)
xθ1(1− x)θ2
0 ≤ x ≤ 1
0 otherwise
θ1 > −1
θ2 > −1
0 0.2 0.4 0.6 0.8 1
x
1
2
3
θ2
θ1 = -.2
(
n∏
k=1
xk
)1/n
(
n∏
k=1
(1− xk)
)1/n
Γ(θ1+θ2+2)
Γ(θ1+1)Γ(θ2+1)
sθ11 s
θ2
2
Poisson P (x|θ) = θ
x
x!
e−θ x = 0, 1, 2, … θ > 0
5 10 15 20
x
.05
.1
.15
θ
1
n
n∑
k=1
xk θ
se−θ
Bernoulli P (x|θ) = θx(1− θ)1−x x = 0, 1 0 < θ < 1
0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
θ
1 - θ
x
1
n
n∑
k=1
xk θ
s(1− θ)1−s
Binomial
P (x|θ) =
m!
x!(m−x)!θ
x(1− θ)m−x
x = 0, 1, ...,m
0 < θ < 1
2 4 6 8
x
0.1
0.2
mθ
m = 10
1
n
n∑
k=1
xk θ
s(1− θ)m−s
Multinomial
P (x|θ) =
m!
d∏
i=1
θ
xi
i
d∏
i=1
xi!
xi = 0, 1, ...,m
d∑
i=1
xi = m
0 < θi < 1
d∑
i=1
θi = 1
x1
x2
P(x|θ)
0
3
6
9
12 0
3
6
9
12
0
.03
.06
0
3
9
12x3 = m - x1 - x2
m = 12
1
n
n∑
k=1
xk
d∏
i=1
θsii
3.7. PROBLEMS OF DIMENSIONALITY 27
3.7 Problems of Dimensionality
In practical multicategory applications, it is not at all unusual to encounter problems
involving fifty or a hundred features, particularly if the features are binary valued.
We might typically believe that each feature is useful for at least some of the discrim-
inations; while we may doubt that each feature provides independent information,
intentionally superfluous features have not been included. There are two issues that
must be confronted. The most important is how classification accuracy depends upon
the dimensionality (and amount of training data); the second is the computational
complexity of designing the classifier.
3.7.1 Accuracy, Dimension, and Training Sample Size
If the features are statistically independent, there are some theoretical results that
suggest the possibility of excellent performance. For example, consider the two-class
multivariate normal case with the same covariance where p(x|ωj) ∼ N(µj ,Σ), j =
1, 2. If the a priori probabilities are equal, then it is not hard to show (Chap. ??,
Problem ??) that the Bayes error rate is given by
P (e) =
1√
2π
∞∫
r/2
e−u
2/2 du, (71)
where r2 is the squared Mahalanobis distance (Chap. ??, Sect. ??):
r2 = (µ1 − µ2)tΣ−1(µ1 − µ2). (72)
Thus, the probability of error decreases as r increases, approaching zero as r ap-
proaches infinity. In the conditionally independent case, Σ = diag(σ21 , ..., σ
2
d), and
r2 =
d∑
i=1
(µi1 − µi2
σi
)2
. (73)
This shows how each feature contributes to reducing the probability of error.
Naturally, the most useful features are the ones for which the difference between the
means is large relative to the standard deviations. However no feature is useless if its
means for the two classes differ. An obvious way to reduce the error rate further is to
introduce new, independent features. Each new feature need not add much, but if r
can be increased without limit, the probability of error can be made arbitrarily small.
In general, if the performance obtained with a given set of features is inadequate,
it is natural to consider adding new features, particularly ones that will help separate
the class pairs most frequently confused. Although increasing the number of features
increases the cost and complexity of both the feature extractor and the classifier, it
is often reasonable to believe that the performance will improve. After all, if the
probabilistic structure of the problem were completely known, the Bayes risk could
not possibly be increased by adding new features. At worst, the Bayes classifer would
ignore the new features, but if the new features provide any additional information,
the performance must improve (Fig. 3.3).
Unfortunately, it has frequently been observed in practice that, beyond a certain
point, the inclusion of additional features leads to worse rather than better perfor-
mance. This apparent paradox presents a genuine and serious problem for classifier
28 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
x1
x3
x2
Figure 3.3: Two three-dimensional distributions have nonoverlapping densities, and
thus in three dimensions the Bayes error vanishes. When projected to a subspace —
here, the two-dimensional x1−x2 subspace or a one-dimensional x1 subspace — there
can be greater overlap of the projected distributions, and hence greater Bayes errors.
design. The basic source of the difficulty can always be traced to the fact that we
have the wrong model — e.g., the Gaussian assumption or conditional assumption
are wrong — or the number of design or training samples is finite and thus the dis-
tributions are not estimated accurately. However, analysis of the problem is both
challenging and subtle. Simple cases do not exhibit the experimentally observed phe-
nomena, and more realistic cases are difficult to analyze. In an attempt to provide
some rigor, we shall return to topics related to problems of dimensionality and sample
size in Chap. ??.
3.7.2 Computational Complexity
We have mentioned that one consideration affecting our design methodology is that of
the computational difficulty, and here the technical notion of computational complex-
ity can be useful. First, we will will need to understand the notion of the order of aorder
function f(x): we say that the f(x) is “of the order of h(x)” — written f(x) = O(h(x))
big oh and generally read “big oh of h(x)” — if there exist constants c0 and x0 such that
|f(x)| ≤ c0|h(x)| for all x > x0. This means simply that for sufficiently large x,
an upper bound on the function grows no worse than h(x). For instance, suppose
f(x) = a0 + a1x + a2x2; in that case we have f(x) = O(x2) because for sufficiently
large x, the constant, linear and quadratic terms can be “overcome” by proper choice
of c0 and x0. The generalization to functions of two or more variables is straightfor-
ward. It should be clear that by the definition above, the big oh order of a function is
not unique. For instance, we can describe our particular f(x) as being O(x2), O(x3),
O(x4), O(x2 ln x).
Because of the non-uniqueness of the big oh notation, we occasionally need to be
3.7. PROBLEMS OF DIMENSIONALITY 29
more precise in describing the order of a function. We say that f(x) = Θ(h(x)) “big
theta of h(x)” if there are constants x0, c1 and c2 such that for x > x0, f(x) always
lies between c1h(x) and c2h(x). Thus our simple quadratic function above would obey
f(x) = Θ(x2), but would not obey f(x) = Θ(x3). (A fuller explanation is provided
in the Appendix.)
In describing the computational complexity of an algorithm we are generally inter-
ested in the number of basic mathematical operations, such as additions, multiplica-
tions and divisions it requires, or in the time and memory needed on a computer. To
illustrate this concept we consider the complexity of a maximum likelihood estimation
of the parameters in a classifier for Gaussian priors in d dimensions, with n training
samples for each of c categories. For each category it is necessary to calculate the
discriminant function of Eq. 74, below. The computational complexity of finding the
sample mean µ̂ is O(nd), since for each of the d dimensions we must add n component
values. The required division by n in the mean calculation is a single computation,
independent of the number of points, and hence does not affect this complexity. For
each of the d(d + 1)/2 independent components of the sample covariance matrix Σ̂
there are n multiplications and additions (Eq. 19), giving a complexity of O(d2n).
Once Σ̂ has been computed, its determinant is an O(d2) calculation, as we can easily
verify by counting the number of operations in matrix “sweep” methods. The inverse
can be calculated in O(d3) calculations, for instance by Gaussian elimination.∗ The
complexity of estimating P (ω) is of course O(n). Equation 74 illustrates these indi-
vidual components for the problem of setting the parameters of normal distributions
via maximum lielihood:
g(x) = −1
2
(x−
O(dn)
↑
µ̂ )t
O(nd2)︷︸︸︷
Σ̂
−1
(x− µ̂)−
O(1)︷ ︸︸ ︷
d
2
ln 2π−
O(d2n)︷ ︸︸ ︷
1
2
ln |Σ̂|+
O(n)︷ ︸︸ ︷
ln P (ω) . (74)
Naturally we assume that n > d (otherwise our covariance matrix will not have a
well defined inverse), and thus for large problems the overall complexity of calculating
an individual discriminant function is dominated by the O(d2n) term in Eq. 74. This
is done for each of the categories, and hence our overall computational complexity
for learning in this Bayes classifer is O(cd2n). Since c is typically a constant much
smaller than d2 or n, we can call our complexity O(d2n). We saw in Sect. 3.7 that it
was generally desirable to have more training data from a larger dimensional space;
our complexity analysis shows the steep cost in so doing.
We next reconsider the matter of estimating a covariance matrix in a bit more
detail. This requires the estimation of d(d+1)/2 parameters — the d diagonal elements
and d(d−1)/2 independent off-diagonal elements. We observe first that the appealing
maximum likelihood estimate
Σ̂ =
1
n
n∑
k=1
(xk −mn)(xk −mn)t, (75)
is an O(nd2) calculation, is the sum of n−1 independent d-by-d matrices of rank one,
and thus is guaranteed to be singular if n ≤ d. Since we must invert Σ̂ to obtain the
discriminant functions, we have an algebraic requirement for at least d + 1 samples.
To smooth our statistical fluctuations and obtain a really good estimate, it would not
be surprising if several times that number of samples were needed.
∗ We mention for the afficionado that there are more complex matrix inversion algorithms that are
O(d2.376…), and there may be algorithms with even lower complexity yet to be discovered.
30 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
The computational complexity for classification is less, of course. Given a test
point x we must compute (x − µ̂), an O(d) calculation. Moreover, for each of the
categories we must multiply the inverse covariance matrix by the separation vector,
an O(d2) calculation. The maxigi(x) decision is a separate O(c) operation. For small
c then, recall is an O(d2) operation. Here, as throughout virtually all pattern clas-
sification, recall is much simpler (and faster) than learning. The complexity of the
corresponding case for Bayesian learning, summarized in Eq. 49, yields the same com-
putational complexity as in maximum likelihood. More generally, however, Bayesian
learning has higher complexity as a consequence of integrating over model parameters
θ.
Such a rough analysis did not tell us the constants of proportionality. For a finite
size problem it is possible (though not particularly likely) that a particular O(n3)
algorithm is simpler than a particular O(n2) algorithm, and it is occasionally necessary
for us to determine these constants to find which of several implemementations is the
simplest. Nevertheless, big oh and big theta analyses, as just described, are generally
the best way to describe the computational complexity of an algorithm.
Sometimes we stress space and time complexities, which are particularly relevant
when contemplating parallel implementations. For instance, the sample mean of a
category could be calculated with d separate processors, each adding n sample values.
Thus we can describe this implementation as O(d) in space (i.e., the amount of memoryspace
complexity or possibly the number of processors) and O(n) in time (i.e., number of sequential
steps). Of course for any particular algorithm there may be a number of time-space
time
complexity
tradeoffs, for instance using a single processor many times, or using many processors
in parallel for a shorter time. Such tradeoffs are important considerations can be
important in neural network implementations, as we shall see in Chap. ??.
A common qualitative distinction is made between polynomially complex and ex-
ponentially complex algorithms — O(ak) for some constant a and aspect or variable k
of the problem. Exponential algorithms are generally so complex that for reasonable
size cases we avoid them altogether, and resign ourselves to approximate solutions
that can be found by polynomially complex algorithms.
3.7.3 Overfitting
It frequently happens that the number of available samples is inadequate, and the
question of how to proceed arises. One possibility is to reduce the dimensionality,
either by redesigning the feature extractor, by selecting an appropriate subset of the
existing features, or by combining the existing features in some way (Chap ??). An-
other possibility is to assume that all c classes share the same covariance matrix, and
to pool the available data. Yet another alternative is to look for a better estimate for
Σ. If any reasonable a priori estimate Σ0 is available, a Bayesian or pseudo-Bayesian
estimate of the form λΣ0 + (1 − λ)Σ̂ might be employed. If Σ0 is diagonal, this
diminishes the troublesome effects of “accidental” correlations. Alternatively, one can
remove chance correlations heuristically by thresholding the sample covariance matrix.
For example, one might assume that all covariances for which the magnitude of the
correlation coefficient is not near unity are actually zero. An extreme of this approach
is to assume statistical independence, thereby making all the off-diagonal elements be
zero, regardless of empirical evidence to the contrary — an O(nd) calculation. Even
though such assumptions are almost surely incorrect, the resulting heuristic estimates
sometimes provide better performance than the maximum likelihood estimate of the
full parameter space.
3.7. PROBLEMS OF DIMENSIONALITY 31
Here we have another apparent paradox. The classifier that results from assuming
independence is almost certainly suboptimal. It is understandable that it will perform
better if it happens that the features actually are independent, but how can it provide
better performance when this assumption is untrue? The answer again involves the
problem of insufficient data, and some insight into its nature can be gained from
considering an analogous problem in curve fitting. Figure 3.4 shows a set of ten data
points and two candidate curves for fitting them. The data points were obtained
by adding zero-mean, independent noise to a parabola. Thus, of all the possible
polynomials, presumably a parabola would provide the best fit, assuming that we are
interested in fitting data obtained in the future as well as the points at hand. Even
a straight line could fit the training data fairly well. The parabola provides a better
fit, but one might wonder whether the data are adequate to fix the curve. The best
parabola for a larger data set might be quite different, and over the interval shown
the straight line could easily be superior. The tenth-degree polynomial fits the given
data perfectly. However, we do not expect that a tenth-degree polynomial is required
here. In general, reliable interpolation or extrapolation can not be obtained unless
the solution is overdetermined, i.e., there are more points than function parameters
to be set.
2 4 6 8
x
-10
-5
5
10
f(x)
Figure 3.4: The “training data” (black dots) were selected from a quadradic function
plus Gaussian noise, i.e., f(x) = ax2 + bx + c + % where p(%) ∼ N(0, σ2). The 10th
degree polynomial shown fits the data perfectly, but we desire instead the second-order
function f(x), since it would lead to better predictions for new samples.
In fitting the points in Fig. 3.4, then, we might consider beginning with a high-
order polynomial (e.g., 10th order), and successively smoothing or simplifying our
model by eliminating the highest-order terms. While this would in virtually all cases
lead to greater error on the “training data,” we might expect the generalization to
improve.
Analogously, there are a number of heuristic methods that can be applied in
the Gaussian classifier case. For instance, suppose we wish to design a classifier
for distributions N(µ1,Σ1) and N(µ2,Σ2) and we have reason to believe that we
have insufficient data for accurately estimating the parameters. We might make the
simplification that they have the same covariance, i.e., N(µ1,Σ) and N(µ2,Σ), and
estimate Σ accordingly. Such estimation requires proper normalization of the data
32 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
(Problem 36).
An intermediate approach is to assume a weighted combination of the equal and
individual covariances, a technique known as shrinkage, (also called regularized dis-shrinkage
criminant analysis) since the individual covariances “shrink” toward a common one.
If i is an index on the c categories in question, we have
Σi(α) =
(1− α)niΣi + αnΣ
(1− α)ni + αn
, (76)
for 0 < α < 1. Additionally, we could “shrink” the estimate of the (assumed) common
covariance matrix toward the identity matrix, as
Σ(β) = (1− β)Σ + βI, (77)
for 0 < β < 1 (Computer exercise 8). (Such methods for simplifying classifiers have
counterparts in regression, generally known as ridge regression.)
Our short, intuitive descussion here will have to suffice until Chap. ??, where we
will explore the crucial issue of controlling the complexity or expressive power of a
classifer for optimum performance.
3.8 *Expectation-Maximization (EM)
We saw in Chap. ?? Sec. ?? how we could classify a test point even when it has miss-
ing features. We can now extend our application of maximum likelihood techniques
to permit the learning of parameters governing a distribution from training points,
some of which have missing features. If we had uncorrupted data, we could use maxi-
mum likelihood, i.e., find θ̂ that maximized the log-likelihood l(θ). The basic idea in
the expectation maximization or EM algorithm, is to iteratively estimate the likeli-
hood given the data that is present. The method has precursors in the Baum-Welch
algorithm we will consider in Sec. 3.10.6.
Consider a full sample D = {x1, ...,xn} of points taken from a single distribution.
Suppose, though, that here some features are missing; thus any sample point can
be written as xk = {xkg,xkb}, i.e., comprising the “good” features and the missing,
or “bad” ones (Chapt. ??, Sect. ??). For notational convenience we separate these
individual features (not samples) into two sets, Dg and Db with D = Dg ∪ Db being
the union of such features.
Next we form the function
Q(θ; θi) = EDb [ln p(Dg,Db; θ)|Dg; θ
i], (78)
where the use of the semicolon denotes, for instance on the left hand side, that
Q(θ; θi) is a function of θ with θi assumed fixed; on the right hand side it de-
notes that the expected value is over the missing features assuming θi are the true
parameters describing the (full) distribution. The simplest way to interpret this, the
central equation in expectation maximization, is the following. The parameter vector
θi is the current (best) estimate for the full distribution; θ is a candidate vector for
an improved estimate. Given such a candidate θ, the right hand side of Eq. 78 calcu-
lates the likelihood of the data, including the unknown feature Db marginalized with
respect to the current best distribution, which is described by θi. Different candidate
θs will of course lead to different such likelihoods. Our algorithm will select the best
such candidate θ and call it θi+1 — the one corresponding to the greatest Q(θ; θi).
3.8. *EXPECTATION-MAXIMIZATION (EM) 33
If we continue to let i be an interation counter, and now let T be a preset conver-
gence criterion, our algorithm is as follows and illustrated in Fig. 3.5:
Algorithm 1 (Expectation-Maximization)
1 begin initialize θ0, T, i = 0
2 do i← i + 1
3 E step : compute Q(θ; θi)
5 M step : θi+1 ← arg max
θ
Q(θ; θi)
6 until Q(θi+1; θi)−Q(θi; θi−1) ≤ T
7 return θ̂ ← θi+1
8 end
Q(θi+1; θi)
θ1
θ0
θ2
θ4
θ3
Figure 3.5: The search for the best model via the EM algorithm starts with some
initial value of the model parameters, θ0. Then, via the M step the optimal θ1
is found. Next, θ1 is held constant and the value θ2 found which optimizes Q(·, ·).
This process iterates until no value of θ can be found that will increase Q(·, ·). Note
in particular that this is different from a gradient search. For example here θ1 is
the global optimum (given fixed θ0), and would not necessarily have been found via
gradient search. (In this illustration, Q(·, ·) is shown symmetric in its arguments; this
need not be the case in general, however.)
This so-called Expectation-Maximization or EM algorithm is most useful when the
optimization of Q(·, ·) is simpler than that of l(·). Most importantly, the algorithm
guarantees that the log-likelihood of the good data (with the bad data marginalized)
will increase monotonically, as explored in Problem 37. This is not the same as
finding the particular value of the bad data that gives the maximum likelihood of the
full (completed) data, as can be seen in Example 2.
Example 2: Expectation-Maximization for a 2D normal model
Suppose our data consists of four points in two dimensions, one point of which
is missing a feature: D = {x1,x2,x3,x4} =
{(
0
2
)
,
(
1
0
)
,
(
2
2
)
,
(∗
4
)}
, where ∗ represents
the unknown value of the first feature of point x4. Thus our bad data Db consists of
34 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
the single feature x41, and the good data Dg all the rest. We assume our model is a
Gaussian with diagonal covariance and arbitrary mean, and thus can be described by
the parameter vector
θ =
µ1
µ2
σ21
σ22
.
We take our initial guess to be a Gaussian centered on the origin having Σ = I, that
is:
θ0 =
0
0
1
1
.
In finding our first improved estimate, θ1, we must calculate Q(θ,θ0) or, by Eq. 78,
Q(θ; θ0) = Ex41 [ln p(xg,xb; θ|θ
0; Dg)]
=
∞∫
−∞
[
3∑
k=1
lnp(xk|θ) + lnp(x4|θ)
]
p(x41|θ0; x42 = 4) dx41
=
3∑
k=1
[lnp(xk|θ)] +
∞∫
−∞
lnp
((
x41
4
)∣∣∣∣∣θ
)
p
((
x41
4
)
|θ0
)
∞∫
−∞
p
((
x′41
4
)∣∣∣∣∣θ
0
)
dx′41
︸ ︷︷ ︸
≡K
dx41,
where x41 is the unknown first feature of point x4, and K is a constant that can be
brought out of the integral. We focus on the integral, substitute the equation for a
general Gaussian, and find
Q(θ; θ0) =
3∑
k=1
[ln p(xk|θ)] +
1
K
∞∫
−∞
ln p
((
x41
4
)∣∣∣∣∣θ
)
1
2π
∣∣(1 0
0 1
)∣∣exp
[
−1
2
(x241 + 4
2)
]
dx41
=
3∑
k=1
[ln p(xk|θ)]−
1 + µ21
2σ21
− (4− µ2)
2
2σ22
− ln (2πσ1σ2).
This completes the expectation or E step. Through a straightforward calculation,
we find the values of θ (that is, µ1, µ2, σ1 and σ2 that maximize Q(, ·), to get the next
estimate:
θ1 =
0.75
2.0
0.938
2.0
.
3.8. *EXPECTATION-MAXIMIZATION (EM) 35
This new mean and the 1/e ellipse of the new covariance matrix are shown in the figure.
Subsequent iterations are conceptually the same, but require a bit more extensive
calculation. The mean will remain at µ2 = 2. After three iterations the algorithm
converges at the solution µ =
(
1.0
2.0
)
, and Σ =
(
0.667 0
0 2.0
)
.
2
4
10
0
1 2 3
x1
x2
The four data points, one of which is missing the value of x1 component, are shown
in red. The initial estimate is a circularly symmetric Gaussian, centered on the
origin (gray). (A better initial estimate could have been derived from the three
known points.) Each iteration leads to an improved estimate, labelled by the iteration
number i; here, after three iterations the algorithm has converged.
We must be careful and note that the EM algorithm leads to the greatest log-
likelihood of the good data, with the bad data marginalized. There may be particular
values of the bad data that give a different solution and an even greater log-likelihood.
For instance, in this Example if the missing feature had value x41 = 2, so that
x4 =
(
2
4
)
, we would have a solution
θ =
1.0
2.0
0.5
2.0
and a log-likelihood for the full data (good plus bad) that is greater than for the good
alone. Such an optimization, however, is not the goal of the canonical EM algorithm.
Note too that if no data is missing, the calculation of Q(θ; θi) is simple since no
integrals are involved.
Generalized Expectation-Maximization or GEM algorithms are a bit more lax than generalized
Expectation-
Maximization
the EM algorithm, and require merely that an improved θi+1 be set in the M step
(line 5) of the algorithm — not necessarily the optimal. Naturally, convergence will
not be as rapid as for a proper EM algorithm, but GEM algorithms afford greater
freedom to choose computationally simpler steps. One version of GEM is to find the
maximum likelihood value of unknown features at each iteration step, then recalculate
θ in light of these new values — if indeed they lead to a greater likelihood.
36 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
In practice, the term Expectation-Maximization has come to mean loosely any
iterative scheme in which the likelihood of some data increases with each step, even if
such methods are not, technically speaking, the true EM algorithm as presented here.
3.9 Bayesian Belief Networks
The methods we have described up to now are fairly general — all that we assumed,
at base, was that we could parameterize the distributions by a feature vector θ. If we
had prior information about the distribution of θ, this too could be used. Sometimes
our knowledge about a distribution is not directly of this type, but instead about
the statistical dependencies (or independencies) among the component features. Re-
call that for some multidimensional distribution p(x), if for two features we have
p(xi, xj) = p(xi)p(xj), we say those variables are statistically independent (Fig. 3.6).
x1
x2
x3
Figure 3.6: A three-dimensional distribution which obeys p(x1, x3) = p(x1)p(x3); thus
here x1 and x3 are statistically independent but the other feature pairs are not.
There are many cases where we know or can safely assume which variables are
or are not independent, even without sampled data. Suppose for instance we are
describing the state of an automobile — temperature of the engine, pressures of the
fluids and in the tires, voltages in the wires, and so on. Our basic knowledge of cars
includes the fact that the oil pressure in the engine and the air pressure in a tire are
functionally unrelated, and hence can be safely assumed to be statistically indepen-
dent. However the oil temperature and engine temperature are not independent (but
could be conditionally independent). Furthermore we may know several variables that
might influence another: the coolant temperature is affected by the engine tempera-
ture, the speed of the radiator fan (which blows air over the coolant-filled radiator),
and so on.
We will represent these dependencies graphically, by means of Bayesian belief nets,
also called causal networks, or simply belief nets. They take the topological form of a
directed acyclic graph (DAG), where each link is directional, and there are no loops.
(More general networks permit such loops, however.) While such nets can represent
continuous multidimensional distributions, they have enjoyed greatest application and
3.9. *BAYESIAN BELIEF NETWORKS 37
success for discrete variables. For this reason, and because the formal properties are
simpler, we shall concentrate on the discrete case.
C D
A B
E
F G
P(c|a) P(d|b)
P(c|d)
P(e|c)
P(g|e)P(f|e)
P(a) P(b)
P(g|f)
Figure 3.7: A belief network consists of nodes (labelled with upper case bold letters)
and their associated discrete states (in lower-case). Thus node A has states a1, a2,
..., denoted simply a; node B has states b1, b2, ..., denoted b, and so forth. The
links between nodes represent conditional probabilities. For example, P (c|a) can be
described by a matrix whose entries are P (ci|aj).
Each node (or unit) represents one of the system variables, and here takes on node
discrete values. We will label nodes with A, B, ..., and the variables at each node
by the corresponding lower-case letter. Thus, while there are a discrete number of
possible values of node A — here two, a1 and a2 — there may be continuous-valued
probabilities on these discrete states. For example, if node A represents the state of
a binary lamp switch — a1 = on, a2 = off — we might have P (a1) = 0.739, P (a2) =
0.261, or indeed any other probabilities. A link joining node A to node C in Fig. 3.7
is directional, and represents the conditional probabilities P (ci|aj), or simply P (c|a).
For the time being we shall not be concerned with how these conditional probabilities
are determined, except to note that in some cases human experts provide the values.
Suppose we have a belief net, complete with conditional probabilities, and know
the values or probabilities of some of the states. Through careful application of Bayes
rule or Bayesian inference, we will be able to determine the maximum posterior value
of the unknown variables in the net. We first consider how to determine the state
of just one node from the states in units with which it is connected. The connected
nodes are the only ones we need to consider directly — the others are conditionally
independent. This is, at base, the simplification provided by our knowledge of the
dependency structure of the system.
In considering a single node X in the simple net of Fig. 3.8, it is extremely useful
to distinguish the set of nodes before X — called its parents P — and the set of those parent
after it — called its children C. When we evaluate the probabilities at X, we must
childtreat the parents of X differently from its children. Thus, in Fig. 3.8, A and B are in
P of X while C and D are in C.
The belief of a set of propositions x = (x1, x2, ...) on node X describes the relative belief
probabilities of the variables given all the evidence e throughout the rest of the net-
work, i.e., P (x|e).∗ We can divide the dependency of the belief upon the parents and
∗ While this is sometimes denoted BEL(x), we keep a notation that clarifies the dependencies and
is more similar to that in our previous discussions.
38 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
A B
X
C D
P(x|b)P(x|a)
P(d|x)P(c|x)
Parents of X
Children of X
Figure 3.8: A portion of a belief network, consistsing of a node X, having variable
values (x1, x2, ...), its parents (A and B), and its children (C and D).
the children in the following way:
P (x|e) ∝ P (eC |x)P (x|eP), (79)
where e represents all evidence (i.e., values of variables on nodes other than X), eP
the evidence on the parent nodes, and eC the children nodes. In Eq. 79 we show only
a proportionality — at the end of our calculation we will normalize the probabilities
over the states at X.
The first term in Eq. 79 is quite simple, and is a manifestation of Bayes’ formula.
We can expand the dependency upon the evidence of the children nodes as follows:
P (eC |x) = P (eC1 , eC2 , ..., eC|C| |x)
= P (eC1 |x)P (eC2 |x) · · ·P (eC|C| |x)
=
|C|∏
j=1
P (eCj |x), (80)
where Cj represents the jth child node and eCj the values of the probabilities of
its states. Note too our convention that |C| denotes the cardinality of set C — thecardinality
number of elements in the set — a convenient notation for indicating the full range of
summations or products. In the last step of Eq. 80 we used our knowledge that since
the child nodes cannot be joined by a line, then they are conditionally independent
given x. Equation 80 simply states that the probability of a given set of states
throughout all the children nodes of X is the product of the (independent) probabilities
in the individual children nodes. For instance, in the simple example in Fig. 3.8, we
have
P (eC, eD|x) = P (eC|x)P (eD|x). (81)
Incorporating evidence from parent nodes is a bit more subtle. We have:
P (x|eP) = P (x|eP1 , eP2 , ..., eP|P|)
=
∑
all i,j,...,k
P (x|P1i,P2j , ...,P|P|k)P (P1i,P2j , ...,P|P|k|eP1 , ..., eP|P|)
=
∑
all i,j,...,k
P (x|P1i,P2j , ...,P|P|k)P (P1i|eP1) · · ·P (P|P|k|eP|P|k), (82)
3.9. *BAYESIAN BELIEF NETWORKS 39
where the summation is over all possible configurations of values on the different
parent nodes. Here Pmn denotes a particular value for state n on parent node Pm.
In the last step of Eq. 82 we have again used our assumption that the (unconnected)
parent nodes are statistically independent.
While Eq. 82 and its unavoidable notational complexities may appear intimidating,
it is actually just a logical consequence of Bayes’ rule. For the purposes of clarity and
for computing x, each term at the extreme right, P (P1i|eP1) can be considered to be
P (P1i) — the probability of state i on the first parent node. Our notation shows that
this probability depends upon the evidence at P1, including from its parents, but for
the sake of computing the probabilities at X we temporarily ignore the dependencies
beyond the parents and children of X.
Thus we rewrite Eq. 82 as
P (x|eP) =
∑
all Pmn
P (x|Pmn)
|P|∏
i=1
P (Pi|ePi) (83)
We put these results together for the general case with |P| parent nodes and |C|
children nodes, Eqs. 80 & 83, and find
P (x|e) ∝
|C|∏
j=1
P (eCj |x)
︸ ︷︷ ︸
P (eC|x)
∑
all Pmn
P (x|Pmn)
|P|∏
i=1
P (Pi|ePi)
︸ ︷︷ ︸
P (x|eP)
. (84)
In words, Eq. 84 states that the probability of a particular values for node X is
the product of two factors. The first is due to the children (the product of their
independent likelihoods). The second is the sum over all possible configurations of
states on the parent nodes of the prior probabilities of their values and the conditional
probabilities of the x variables given those parent values. The final values must be
normalized to represent probabilities.
Example 3: Belief network for fish
Suppose we are again interested in classifying fish, but now we want to use more
information. Imagine that a human expert has constructed the simple belief network
in the figure, where node A represents the time of year, and can have four values:
a1 = winter, a2 = spring, a3 = summer and a4 = autumn. Node B represents the
geographical area where the fish was caught: b1 = north Atlantic and b2 = south
Atlantic. A and B are the parents of the node X, which represents the fish and has
just two possible values: x1 = salmon and x2 = sea bass. Similarly, our expert tells
us that the children nodes represent lightness, C, with c1 = dark, c2 = medium and
c3 = light as well as thickness, D, with d1 = thick and d2 = thin. The direction of the
links (from A and B to X and likewise from X to C and D) is meant to describe the
influences among the variables, as shown in the figure.
40 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
A
season
P(x|b)P(x|a)
P(d|x)P(c|x)
B
locale
X
fish
C
light-
ness
D
thick-
ness
a
2
= spr ng
a
3
= summer
a
4
= autumn
1
b
2
= south Atlantic
x
1
= salmon
x
2
= sea bass
c
1
= light
c
2
= medium
c
3
= dark
d
1
= wide
d
2
= thin
A simple belief net for the fish example. The season and the fishing locale are statisti-
cally independent, but the type of fish caught does depend on these factors. Further,
the width of the fish and its color depend upon the fish.
The following probability matrixes (here, given by an expert) describe the influence
of time of year and fishing area on the identity of the fish:
P (xi|aj) :
salmon sea bass
winter .9 .1
spring .3 .7
summer .4 .6
autumn .8 .2
, P (xi|bj) :
( salmon sea bass
north .65 .35
south .25 .75
)
Thus salmon are best found in the north fishing areas in the winter and autumn,
sea bass in the south fishing areas in the spring and summer, and so forth. Recall
that in our belief networks the variables are discrete, and all influences are cast as
probabilites, rather than probability densities. Given that we have any particular
feature value on a parent node, we must have some fish; thus each row is normalized,
as for instance P (x1|a1) + P (x2|a1) = 1.
Suppose our expert tells us that the conditional probabilities for the variables in
the children nodes are as follows:
P (ci|xj) :
( light medium dark
salmon .33 .33 .34
sea bass .8 .1 .1
)
, P (di|xj) :
( wide thin
salmon .4 .6
sea bass .95 .05
)
Thus salmon come in the full range of lightnesses, while sea bass are primarily light
in color and are primarily wide.
Now we turn to the problem of using such a belief net to infer the identity
of a fish. We have no direct information about the identity of the fish, and thus
P (x1) = P (x2) = 0.5. This might be a reasonable starting point, expressing our lack
of knowledge of the identity of the fish. Our goal now is to estimate the probabilities
P (x1|e) and P (x2|e). Note that without any evidence we have
P (x1) =
∑
i,j,k,l
P (x1, ai, bj , ck, dl)
=
∑
i,j,k,l
P (ai)P (bj)P (x1|ai, bj)P (ck|x1)P (dl|x1)
=
∑
i,j
P (ai)P (bj)P (x1|ai, bj)
= (0.25)(0.5)
∑
i,j
P (x1|ai, bj)
= (0.25)(0.5)(0.9 + 0.3 + 0.4 + 0.7 + 0.8 + 0.2 + 0.1 + 0.6)
= 0.5,
and thus P (x1) = P (x2), as we would expect.
3.9. *BAYESIAN BELIEF NETWORKS 41
Now we collect evidence for each node, {eA, eB, eC, eD}, assuming they are in-
dependent of each other. Suppose we know that it is winter, i.e., P (a1|eA) = 1 and
P (ai|eA) = 0 for i = 2, 3, 4. Suppose we do not know which fishing area the boat came
from but found that the particular fishing crew prefers to fish in the south Atlantic;
we assume, then, that P (b1|eB) = 0.2 and P (b2|eB) = 0.8. We measure the fish and
find that it is fairly light, and set by hand to be P (eC|c1) = 1, P (eC|c2) = 0.5, and
P (eC|c3) = 0. Suppose that due to occlusion, we cannot measure the width of the
fish; we thus set P (eD|d1) = P (eD|d2).
By Eq. 82, we have the estimated probability of each fish due to the parents P is,
in full expanded form
PP(x1) ∝ P (x1|a1, b1)P (a1)P (b1)
+P (x1|a1, b2)P (a1)P (b2)
+P (x1|a2, b1)P (a2)P (b1)
+P (x1|a2, b2)P (a2)P (b2)
+P (x1|a3, b1)P (a3)P (b1)
+P (x1|a3, b2)P (a3)P (b2)
+P (x1|a4, b1)P (a4)P (b1)
+P (x1|a4, b2)P (a4)P (b2)
= 0.82.
A similar calculation gives PP(x2) = 0.18.
We now turn to the children nodes and find by Eq. 84
PC(x1) ∝ P (eC|x1)P (eD|x1)
= [P (eC|c1)P (c1|x1) + P (eC |c2)P (c2|x1) + P (eC|c3)P (c3|x1)]
×[P (eD|d1)P (d1|x1) + P (eD|d2)P (d2|x1)]
= [(1.0)(0.33) + (0.5)(0.33) + (0)(0.34)]× [(1.0)(0.4) + (1.0)(0.6)]
= 0.495.
A similar calculation gives PC(x2) ∝ 0.85. We put these estimates together by Eq. 79
as products P (xi) ∝ PC(xi)PP(xi) and renormalize (i.e., divide by their sum). Thus
our final estimates for node X are
P (x1|e) =
(0.82)(0.495)
(0.82)(0.495) + (0.18)(0.85)
= 0.726
P (x2|e) =
(0.18)(0.85)
(0.82)(0.495) + (0.18)(0.85)
= 0.274.
Thus given all the evidence throughout the belief net, the most probable outcome is
x1 = salmon.
A given belief net can be used to infer any of the unknown variables. In Example
3, we used information about the time of year, fishing location and some measured
42 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
properties of the fish to infer its identity (salmon or sea bass). The same network could
instead be used to infer the probability that a fish is thin, or dark in color, based on
probabilities of the identity of the fish, time of year, and so on (Problem 42).
When the dependency relationships among the features used by a classifier are
unknown, we generally proceed by taking the simplest assumption, i.e., that the
features are conditionally independent given the category, i.e.,
p(ωk|x) ∝
d∏
i=1
p(xi|ωk). (85)
In practice, this so-called naive Bayes rule or idiot Bayes rule often works quite wellnaive
Bayes
rule
in practice, and can be expressed by a very simple belief net (Problem 43).
In Example 3 our entire belief net consisted of X, its parents and children, and
we needed to update only the values on X. In the more general case, where the
network is large, there may be many nodes whose values are unknown. In that case
we may have to visit nodes randomly and update the probabilites until the entire
configuration of probabilities is stable. It can be shown that under weak conditions,
this process will converge to consistent values of the variables throughout the entire
network (Problem 44).
Belief nets have found increasing use in complicated problems such as medical
diagnosis. Here the upper-most nodes (ones without their own parents) represent a
fundamental biological agent such as the presence of a virus or bacteria. Intermediate
nodes then describe diseases, such as flu or emphysema, and the lower-most nodes
the symptoms, such as high temperature or coughing. A physician enters measured
values into the net and finds the most likely disease or cause. Such networks can be
used in a somewhat more sophisticated way, automatically computing which unknown
variable (node) should be measured to best reveal the identity of the disease.
We will return in Chap. ?? to address the problem of learning in such belief net
models.
3.10 Hidden Markov Models
While belief nets are a powerful method for representing the dependencies and inde-
pendencies among variables, we turn now to the problem of representing a particular
but extremely important dependencies. In problems that have an inherent temporal-
ity — that is, consist of a process that unfolds in time — we may have states at time
t that are influenced directly by a state at t − 1. Hidden Markov models (HMMs)
have found greatest use in such problems, for instance speech recognition or gesture
recognition. While the notation and description is aunavoidably more complicated
than the simpler models considered up to this point, we stress that the same underly-
ing ideas are exploited. Hidden Markov models have a number of parameters, whose
values are set so as to best explain training patterns for the known category. Later, a
test pattern is classified by the model that has the highest posterior probability, i.e.,
that best “explains” the test pattern.
3.10.1 First-order Markov models
We consider a sequence of states at successive times; the state at any time t is denoted
ω(t). A particular sequence of length T is denoted by ωT = {ω(1), ω(2), ..., ω(T )} as
3.10. *HIDDEN MARKOV MODELS 43
for instance we might have ω6 = {ω1, ω4, ω2, ω2, ω1, ω4}. Note that the system can
revisit a state at different steps, and not every state need be visited.
Our model for the production of any sequence is described by transition probabil-
ities P (ωj(t + 1)|ωi(t)) = aij — the time-independent probability of having state ωj transition
probabilityat step t + 1 given that the state at time t was ωi. There is no requirement that the
transition probabilities be symmetric (aij �= aji, in general) and a particular state
may be visited in succession (aii �= 0, in general), as illustrated in Fig. 3.9.
ω3
ω2
ω1
a12
a21
a23
a32
a31
a13
a33a11
a22
Figure 3.9: The discrete states, ωi, in a basic Markov model are represented by nodes,
and the transition probabilities, aij , by links. In a first-order discrete time Markov
model, at any step t the full system is in a particular state ω(t). The state at step
t+1 is a random function that depends solely on the state at step t and the transition
probabilities.
Suppose we are given a particular model θ — that is, the full set of aij — as well
as a particular sequence ωT . In order to calculate the probability that the model
generated the particular sequence we simply multiply the successive probabilities.
For instance, to find the probability that a particular model generated the sequence
described above, we would have P (ωT |θ) = a14a42a22a21a14. If there is a prior
probability on the first state P (ω(1) = ωi), we could include such a factor as well; for
simplicity, we will ignore that detail for now.
Up to here we have been discussing a Markov model, or technically speaking, a
first-order discrete time Markov model, since the probability at t+1 depends only on
the states at t. For instance, in a Markov model for the production of spoken words,
we might have states representing phonemes, and a Markov model for the production
of a spoken work might have states representing phonemes. Such a Markov model for
the word “cat” would have states for /k/, /a/ and /t/, with transitions from /k/ to
/a/; transitions from /a/ to /t/; and transitions from /t/ to a final silent state.
Note however that in speech recognition the perceiver does not have access to the
states ω(t). Instead, we measure some properties of the emitted sound. Thus we will
have to augment our Markov model to allow for visible states — which are directly
accessible to external measurement — as separate from the ω states, which are not.
3.10.2 First-order hidden Markov models
We continue to assume that at every time step t the system is in a state ω(t) but now
we also assume that it emits some (visible) symbol v(t). While sophisticated Markov
models allow for the emission of continuous functions (e.g., spectra), we will restrict
ourselves to the case where a discrete symbol is emitted. As with the states, we define
44 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
a particular sequence of such visible states as VT = {v(1), v(2), ..., v(T )} and thus we
might have V6 = {v5, v1, v1, v5, v2, v3}.
Our model is then that in any state ω(t) we have a probability of emitting a par-
ticular visible state vk(t). We denote this probability P (vk(t)|ωj(t)) = bjk. Because
we have access only to the visible states, while the ωi are unobservable, such a full
model is called a hidden Markov model (Fig. 3.10)
ω3
ω2
ω1
a12
a21
a23
a32
a31
a13
a33a11
a22
v1 v2 v3 v4
b21
b22 b23 b24
v1 v2 v3 v4
b11 b12 b13
b14
v1 v2 v3 v4
b31 b32 b33
b34
Figure 3.10: Three hidden units in an HMM and the transitions between them are
shown in black while the visible states and the emission probabilities of visible states
are shown in red. This model shows all transitions as being possible; in other HMMs,
some such candidate transitions are not allowed.
3.10.3 Hidden Markov Model Computation
Now we define some new terms and clarify our notation. In general networks such as
those in Fig. 3.10 are finite-state machines, and when they have associated transition
probabilities, they are called Markov networks. They are strictly causal — the prob-
abilities depend only upon previous states. A Markov model is called ergodic if every
one of the states has a non-zero probability of occuring given some starting state. A
final or absorbing state ω0 is one which, if entered, is never left (i.e., a00 = 1).absorbing
state As mentioned, we denote the transition probabilities aij among hidden states and
for the probability bjk of the emission of a visible state:
aij = P (ωj(t + 1)|ωi(t))
bjk = P (vk(t)|ωj(t)). (86)
We demand that some transition occur from step t→ t + 1 (even if it is to the same
state), and that some visible symbol be emitted after every step. Thus we have the
normalization conditions:
∑
j
aij = 1 for all i and
3.10. *HIDDEN MARKOV MODELS 45
∑
k
bjk = 1 for all j, (87)
where the limits on the summations are over all hidden states and all visible symbols,
respectively.
With these preliminaries behind us, we can now focus on the three central issues
in hidden Markov models:
The Evaluation problem. Suppose we have an HMM, complete with transition
probabilites aij and bjk. Determine the probability that a particular sequence
of visible states VT was generated by that model.
The Decoding problem. Suppose we have an HMM as well as a set of observations
VT . Determine the most likely sequence of hidden states ωT that led to those
observations.
The Learning problem. Suppose we are given the coarse structure of a model (the
number of states and the number of visible states) but not the probabilities aij
and bjk. Given a set of training observations of visible symbols, determine these
parameters.
We consider each of these problems in turn.
3.10.4 Evaluation
The probability that the model produces a sequence VT of visible states is:
P (VT ) =
rmax∑
r=1
P (VT |ωTr )P (ωTr ), (88)
where each r indexes a particular sequence ωTr = {ω(1), ω(2), ..., ω(T )} of T hidden
states. In the general case of c hidden states, there will be rmax = cT possible
terms in the sum of Eq. 88, corresponding to all possible sequences of length T . Thus,
according to Eq. 88, in order to compute the probability that the model generated the
particular sequence of T visible states VT , we should take each conceivable sequence
of hidden states, calculate the probability they produce VT , and then add up these
probabilities. The probability of a particular visible sequence is merely the product
of the corresponding (hidden) transition probabilities aij and the (visible) output
probabilities bjk of each step.
Because we are dealing here with a first-order Markov process, the second factor
in Eq. 88, which describes the transition probability for the hidden states, can be
rewritten as:
P (ωTr ) =
T∏
t=1
P (ω(t)|ω(t− 1)) (89)
that is, a product of the aij ’s according to the hidden sequence in question. In
Eq. 89, ω(T ) = ω0 is some final absorbing state, which uniquely emits the visible state
v0. In speech recognition applications, ω0 typically represents a null state or lack of
utterance, and v0 is some symbol representing silence. Because of our assumption
that the output probabilities depend only upon the hidden state, we can write the
first factor in Eq. 88 as
46 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
P (VT |ωTr ) =
T∏
t=1
P (v(t)|ω(t)), (90)
that is, a product of bjk’s according to the hidden state and the corresponding visible
state. We can now use Eqs. 89 & 90 to express Eq. 88 as
P (VT ) =
rmax∑
r=1
T∏
t=1
P (v(t)|ω(t))P (ω(t)|ω(t− 1)). (91)
Despite its formal complexity, Eq. 91 has a straightforward interpretation. The
probability that we observe the particular sequence of T visible states VT is equal to
the sum over all rmax possible sequences of hidden states of the conditional probability
that the system has made a particular transition multiplied by the probability that
it then emitted the visible symbol in our target sequence. All these are captured in
our paramters aij and bkj , and thus Eq. 91 can be evaluated directly. Alas, this is an
O(cTT ) calculation, which is quite prohibitive in practice. For instance, if c = 10 and
T = 20, we must perform on the order of 1021 calculations.
A computationaly simpler algorithm for the same goal is as follows. We can
calculate P (VT ) recursively, since each term P (v(t)|ω(t))P (ω(t)|ω(t − 1)) involves
only v(t), ω(t) and ω(t− 1). We do this by defining
αi(t) =
0 t = 0 and i �= initial state
1 t = 0 and i = initial state∑
j α(t− 1)aijbjkv(t) otherwise,
(92)
where the notation bjkv(t) means the transition probability bjk selected by the visible
state emitted at time t. thus the only non-zero contribution to the sum is for the
index k which matches the visible state v(t). Thus αi(t) represents the probability
that our HMM is in hidden state ωi at step t having generated the first t elements of
VT . This calculation is implemented in the Forward algorithm in the following way:
Algorithm 2 (HMM Forward)
1 initialize ω(1), t = 0, aij , bjk, visible sequence VT , α(0) = 1
2 for t← t + 1
3 αj(t)←
c∑
i=1
αi(t− 1)aijbjk
4 until t = T
5 return P (VT )← α0(T )
6 end
where in line 5, α0 denotes the probability of the associated sequence ending to the
known final state. The Forward algorithm has, thus, a computational complexity of
O(c2T ) — far more efficient than the complexity associated with exhaustive enumer-
ation of paths of Eq. 91 (Fig. 3.11). For the illustration of c = 10, T = 20 above, we
would need only on the order of 2000 calculations — more than 17 orders of magnitude
faster than that to examine each path individually.
We shall have cause to use the Backward algorithm, which is the time-reversed
version of the Forward algorithm.
Algorithm 3 (HMM Backward)
3.10. *HIDDEN MARKOV MODELS 47
ω1 ω1 ω1 ω1 ω1
ω2 ω2 ω2 ω2 ω2
ω3 ω3 ω3 ω3 ω3
ωc ωc ωc ωc ωc
t = 1 2 3 T-1 T
α1(2)
α2(2)
α3(2)
αc(2)
ac2
a32
a22
a12
b2k
vk
Figure 3.11: The computation of probabilities by the Forward algorithm can be vi-
sualized by means of a trellis — a sort of “unfolding” of the HMM through time.
Suppose we seek the probability that the HMM was in state ω2 at t = 3 and gener-
ated the observed visible up through that step (including the observed visible symbol
vk). The probability the HMM was in state ωj(t = 2) and generated the observed
sequence through t = 2 is αj(2) for j = 1, 2, ..., c. To find α2(3) we must sum these
and multiply the probability that state ω2 emitted the observed symbol vk. Formally,
for this particular illustration we have α2(3) = b2k
c∑
j=1
αj(2)aj2.
1 initialize ω(T ), t = T, aij , bjk, visible sequence V T
2 for t← t− 1;
4 βj(t)←
c∑
i=1
βi(t + 1)aijbjkv(t + 1)
5 until t = 1
7 return P (V T )← βi(0) for the known initial state
8 end
Example 4: Hidden Markov Model
To clarify the evaluation problem, consider an HMM such as shown in Fig. 3.10,
but with an explicit absorber state and unique null visible symbol V0 with the following
transition probabilities (where the matrix indexes begin at 0):
aij =
1 0 0 0
0.2 0.3 0.1 0.4
0.2 0.5 0.2 0.1
0.8 0.1 0.0 0.1
and
bjk =
1 0 0 0 0
0 0.3 0.4 0.1 0.2
0 0.1 0.1 0.7 0.1
0 0.5 0.2 0.1 0.2
.
48 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
What is the probability it generates the particular sequence V5 = {v3, v1, v3, v2, v0}?
Suppose we know the initial hidden state at t = 0 to be ω1. The visible symbol at
each step is shown above, and the αi(t) in each unit. The circles show the value for
αi(t) as we progress left to right. The product aijbjk is shown along each transition
link for the step t = 1 to t = 2. The final probability, P (VT |θ) is hence 0.0011.
ω2
ω0
ω3ω1
V1
V4V1 V4
V0
V1
V4
V3 V1 V3 V2 V0
ω0
ω1
ω2
ω3
t = 0 1 2 3 4
.0011
.0052
.0077
.0057
0 0 0 0
0
0
0
0
0
1 .09
.01
.2
.0024
.0002
.0007
0.2
x
0
0.3 x 0.3
0.1 x 0.1
0.4 x 0.5
The HMM (above) consists of four hidden states (one of which is an absorber state,
ω0), each emitting one of five visible states; only the allowable transitions to visible
states are shown. The trellis for this HMM is shown below. In each node is αi(t) — the
probability the model generated the observed visible sequence up to t. For instance,
we know that the system was in hidden state ω1 at t = 1, and thus α1(0) = 1 and
αi(0) = 0 for i �= 1. The arrows show the calculation of αi(1). for instance, since
visible state v1 was emitted at t = 1, we have α0(1) = α1(0)a10b01 = 1[0.2 × 0] = 0.
as shown by the top arrow. Likewise the nest highest arrow corresponds to the
calculation α1(1) = α1(0)a11b11 = 1[0.3×0.3] = 0.09. In this example, the calculation
of αi(1) is particularly simple, since only transitions from the known initial hidden
state need be considered; all other transitions have zero contribution to αi(1). For
subsequent times, however, the caculation requires a sum over all hidden states at
the previous time, as given by line 3 in the Forward algorithm. The probability
shown in the final (absorbing) state gives the probability of the full sequence observed,
P (VT |θ) = 0.0011.
If we denote our model — the a’s and b’s — by θ, we have by Bayes’ formula that
the probability of the model given the observed sequence is:
P (θ|VT ) = P (V
T |θ)P (θ)
P (VT )
(93)
3.10. *HIDDEN MARKOV MODELS 49
In HMM pattern recognition we would have a number of HMMs, one for each category
and classify a test sequence according to the model with the highest probability. Thus
in HMM speech recognition we could have a model for “cat” and another one for
“dog” and for a test utterance determine which model has the highest probability. In
practice, nearly all HMMs for speech are left-to-right models (Fig. 3.12). left-to-
right
model
ω1 ω2 ω3 ω4 ω5 ω0
Figure 3.12: A left-to-right HMM commonly used in speech recognition. For instance,
such a model could describe the utterance “viterbi,” where ω1 represents the phoneme
/v/, ω2 represents /i/, ..., and ω0 a final silent state. Such a left-to-right model is
more restrictive than the general HMM in Fig. 3.10, and precludes transitions “back”
in time.
The Forward algorithm gives us P (V T |θ). The prior probability of the model,
P (θ), is given by some external source, such as a language model in the case of speech.
This prior probability might depend upon the semantic context, or the previous words,
or yet other information. In the absence of such information, it is traditional to assume
a uniform density on P (θ), and hence ignore it in any classification problem. (This
is an example of a “non-informative” prior.)
3.10.5 Decoding
Given a sequence of visible states VT , the decoding problem is to find the most
probable sequence of hidden states. While we might consider enumerating every
possible path and calculating the probability of the visible sequence observed, this is
an O(cTT ) calculation and prohibitive. Instead, we use perhaps the simplest decoding
algorithm:
Algorithm 4 (HMM decoding)
1 begin initialize Path = {}, t = 0
2 for t← t + 1
4 k = 0, α0 = 0
5 for k ← k + 1
7 αk(t)← bjkv(t)
c∑
i=1
αi(t− 1)aij
8 until k = c
10 j′ ← arg max
j
αj(t)
11 AppendTo Path ωj′
12 until t = T
13 return Path
14 end
A closely related algorithm uses logarithms of the probabilities and calculates total
probabilities by addition of such logarithms; this method has complexity O(c2T )
(Problem 48).
50 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
ω1 ω1 ω1 ω1 ω1
ω2 ω2 ω2 ω2 ω2
ω3 ω3 ω3 ω3 ω3
ωc ωc ωc ωc ωc
t = 1 3 4 T-1 T
αmax(2)
ω1
ω2
ω3
ωc
2
αmax(1)
αmax(3) αmax(T-1)
ω0 ω0 ω0 ω0 ω0ω0
αmax(T)
Figure 3.13: The decoding algorithm finds at each time step t the state that has the
highest probability of having come from the previous step and generated the observed
visible state vk. The full path is the sequence of such states. Because this is a
local optimization (dependent only upon the single previous time step, not the full
sequence), the algorithm does not guarantee that the path is indeed allowable. For
instance, it might be possible that the maximum at t = 5 is ω1 and at t = 6 is ω2, and
thus these would appear in the path. This can even occur if a12 = P (ω2(t+1)|ω1(t)) =
0, precluding that transition.
The red line in Fig. 3.13 corresponds to Path, and connects the hidden states with
the highest value of αi at each step t. There is a difficulty, however. Note that there
is no guarantee that the path is in fact a valid one — it might not be consistent with
the underlying models. For instance, it is possible that the path actually implies a
transition that is forbidden by the model, as illustrated in Example 5.
Example 5: HMM decoding
We find the path for the data of Example 4 for the sequence {ω1, ω3, ω2, ω1, ω0}.
Note especially that the transition from ω3 to ω2 is not allowed according to the tran-
sition probabilities aij given in Example 4. The path locally optimizes the probability
through the trellis.
3.10. *HIDDEN MARKOV MODELS 51
V3 V1 V3 V2 V0
ω0
ω1
ω2
ω3
t = 1 2 3 4 5
.0011
.0052
.0077
.0057
0 0 0 0
0
0
0
0
0
1 .09
.01
.2
.0024
.0002
.0007
The locally optimal path through the HMM trellis of Example 4.
HMMs address the problem of rate invariance in the following two ways. The first
is that the transition probabilities themselves incorporate probabilistic structure of
the durations. Moreover, using postprocessing, we can delete repeated states and just
get the sequence somewhat independent of variations in rate. Thus in post-processing
we can convert the sequence {ω1, ω1, ω3, ω2, ω2, ω2} to {ω1, ω3, ω2}, which would be
appropriate for speech recognition, where the fundamental phonetic units are not
repeated in natural speech.
3.10.6 Learning
The goal in HMM learning is to determine model parameters — the transition prob-
abilities aij and bjk — from an ensemble of training samples. There is no known
method for obtaining the optimal or most likely set of parameters from the data, but
we can nearly always determine a good solution by a straightforward technique.
The Forward-backward Algorithm
The Forward-backward algorithm is an instance of a generalized Expectation-Maximization
algorithm. The general approach will be to iteratively update the weights in order to
better explain the observed training sequences.
Above, we defined αi(t) as the probability that the model is in state ωi(t) and has
generated the target sequence up to step t. We can analogously define βi(t) to be
the probability that the model is in state ωi(t) and will generate the remainder of the
given target sequence, i.e., from t + 1→ T . We express βi(t) as:
βi(t) =
0 ωi(t) �= sequence’s final state and t = T
1 ωi(t) = sequence’s final state and t = T∑
j
aijbjkv(t + 1)βj(t + 1) otherwise,
(94)
To understand Eq. 94, imagine we knew αi(t) up to step T − 1, and we wanted to
calculate the probability that the model would generate the remaining single visible
52 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
symbol. This probability, βi(T ), is just the probability we make a transition to state
ωi(T ) multiplied by the probability that this hidden state emitted the correct final visi-
ble symbol. By the definition of βi(T ) in Eq. 94, this will be either 0 (if ωi(T ) is not the
final hidden state) or 1 (if it is). Thus it is clear that βi(T −1) =
∑
j aijbijv(T )βi(T ).
Now that we have determined βi(T − 1), we can repeat the process, to determine
βi(T − 2), and so on, backward through the trellis of Fig. ??.
But the αi(t) and βi(t) we determined are merely estimates of their true values,
since we don’t know the actual value of the transition probabilities aij and bij in
Eq. 94. We can calculate an improved value by first defining γij(t) — the probability
of transition between ωi(t−1) and ωj(t), given the model generated the entire training
sequence VT by any path. We do this by defining γij(t), as follows:
γij(t) =
αi(t− 1)aijbijβi(t)
P (V T |θ)
, (95)
where P (VT |θ) is the probability that the model generated sequence VT by any path.
Thus γij(t) is the probability of a transition from state ωi(t − 1) to ωj(t) given that
the model generated the complete visible sequence V T .
We can now calculate an improved estimate for aij . The expected number of
transitions between state ωi(t − 1) and ωj(t) at any time in the sequence is simply∑T
t=1 γij(t), whereas at step t it is
∑T
t=1
∑
k γik(t). Thus âij (the estimate of the
probability of a transition from ωi(t − 1) to ωj(t)) can be found by taking the ratio
between the expected number of transitions from ωi to ωj and the total expected
number of any transitions from ωi. That is:
âij =
T∑
t=1
γij(t)
T∑
t=1
∑
k
γik(t)
. (96)
In the same way, we can obtain an improved estimate b̂ij by calculating the ratio
between the frequency that any particular symbol vk is emitted and that for any
symbol. Thus we have
b̂jk =
∑
γjk(t)
T∑
t=1
γjk(t)
. (97)
In short, then, we start with rough or arbitrary estimates of aij and bjk, calculate
improved estimates by Eqs. 96 & 97, and repeat until some convergence criterion
is met (e.g., sufficiently small change in the estimated values of the parameters on
subsequent iterations). This is the Baum-Welch or Forward-backward algorithm —
an example of a Generalized Expectation-Maximumization algorithm (Sec. 3.8):
Algorithm 5 (Forward-backward)
1 begin initialize aij , bjk, training sequence V T , convergence criterion θ
2 do z ← z + 1
3 Compute â(z) from a(z − 1) and b(z − 1) by Eq. 96
4 Compute b̂(z) from a(z − 1) and b(z − 1) by Eq. 97
5 aij(z)← âij(z − 1)
6 bjk(z)← b̂jk(z − 1)
3.10. SUMMARY 53
7 until max
i,j,k
[aij(z)− aij(z − 1), bjk(z)− bjk(z − 1)] < θ; convergence achievedln : ForBackstop
8 return aij ← aij(z); bjk ← bjk(z)
9 end
The stopping or convergence criterion in line ?? halts learning when no estimated
transition probability changes more than a predetermined amount, θ. In typical
speech recognition applications, convergence requires several presentations of each
training sequence (fewer than five is common). Other popular stopping criteria are
based on overall probability that the learned model could have generated the full
training data.
Summary
If we know a parametric form of the class-conditional probability densities, we can
reduce our learning task from one of finding the distribution itself, to that of find-
ing the parameters (represented by a vector θi for each category ωi), and use the
resulting distributions for classification. The maximum likelihood method seeks to
find the parameter value that is best supported by the training data, i.e., maximizes
the probability of obtaining the samples actually observed. (In practice, for com-
putational simplicity one typically uses log-likelihood.) In Bayesian estimation the
parameters are considered random variables having a known a priori density; the
training data convert this to an a posteriori density. The recursive Bayes method
updates the Bayesian parameter estimate incrementally, i.e., as each training point
is sampled. While Bayesian estimation is, in principle, to be preferred, maximum
likelihood methods are generally easier to implement and in the limit of large training
sets give classifiers nearly as accurate.
A sufficient statistic s for θ is a function of the samples that contains all infor-
mation needed to determine θ. Once we know the sufficient statistic for models of a
given form (e.g., exponential family), we need only estimate their value from data to
create our classifier — no other functions of the data are relevant.
Expectation-Maximization is an iterative scheme to maximize model parameters,
even when some data are missing. Each iteration employs two steps: the expectation
or E step which requires marginalizing over the missing variables given the current
model, and the maximization or M step, in which the optimum parameters of a new
model are chosen. Generalized Expectation-Maximization algorithms demand merely
that parameters be improved — not optimized — on each iteration and have been
applied to the training of a large range of models.
Bayesian belief nets allow the designer to specify, by means of connection topology,
the functional dependences and independencies among model variables. When any
subset of variables is clamped to some known values, each node comes to a proba-
bility of its value through a Bayesian inference calculation. Parameters representing
conditional dependences can be set by an expert.
Hidden Markov models consist of nodes representing hidden states, interconnected
by links describing the conditional probabilities of a transition between the states.
Each hidden state also has an associated set of probabilities of emiting a particular
visible states. HMMs can be useful in modelling sequences, particularly context depen-
dent ones, such as phonemes in speech. All the transition probabilities can be learned
(estimated) iteratively from sample sequences by means of the Forward-backward or
Baum-Welch algorithm, an example of a generalized EM algorithm. Classification
54 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
proceeds by finding the single model among candidates that is most likely to have
produced a given observed sequence.
Bibliographical and Historical Remarks
Maximum likelihood and Bayes estimation have a long history. The Bayesian ap-
proach to learning in pattern recognition began by the suggestion that the proper
way to use samples when the conditional densities are unknown is the calculation
of P (ωi|x,D), [6]. Bayes himself appreciated the role of non-informative priors. An
analysis of different priors from statistics appears in [21, 15] and [4] has an extensive
list of references.
The origins of Bayesian belief nets traced back to [33], and a thorough literature
review can be found in [8]; excellent modern books such as [24, 16] and tutorials [7]
can be recommended. An important dissertation on the theory of belief nets, with
an application to medical diagnosis is [14], and a summary of work on diagnosis of
machine faults is [13]. While we have focussed on directed acyclic graphs, belief nets
are of broader use, and even allow loops or arbitrary topologies — a topic that would
lead us far afield here, but which is treated in [16].
The Expectation-Maximization algorithm is due to Dempster et al.[11] and a thor-
ough overview and history appears in [23]. On-line or incremental versions of EM are
described in [17, 31]. The definitive compendium of work on missing data, including
much beyond our discussion here, is [27].
Markov developed what later became called the Markov framework [22] in order
to analyze the the text of his fellow Russian Pushkin’s masterpiece Eugene Onegin.
Hidden Markov models were introduced by Baum and collaborators [2, 3], and have
had their greatest applications in the speech recognition [25, 26], and to a lesser extent
statistical language learning [9], and sequence identification, such as in DNA sequences
[20, 1]. Hidden Markov methods have been extended to two-dimensions and applied
to recognizing characters in optical document images [19]. The decoding algorithm is
related to pioneering work of Viterbi and followers [32, 12]. The relationship between
hidden Markov models and graphical models such as Bayesian belief nets is explored
in [29].
Knuth’s classic [18] was the earliest compendium of the central results on com-
putational complexity, the majority due to himself. The standard books [10], which
inspired several homework problems below, are a bit more accessible for those with-
out deep backgrounds in computer science. Finally, several other pattern recognition
textbooks, such as [28, 5, 30] which take a somewhat different approach to the field
can be recommended.
Problems
⊕
Section 3.2
1. Let x have an exponential density
p(x|θ) =
{
θe−θx x ≥ 0
0 otherwise.
(a) Plot p(x|θ) versus x for θ = 1. Plot p(x|θ) versus θ, (0 ≤ θ ≤ 5), for x = 2.
3.10. PROBLEMS 55
(b) Suppose that n samples x1, ..., xn are drawn independently according to p(x|θ).
Show that the maximum likelihood estimate for θ is given by
θ̂ =
1
1
n
n∑
k=1
xk
.
(c) On your graph generated with θ = 1 in part (a), mark the maximum likelihood
estimate θ̂ for large n.
2. Let x have a uniform density
p(x|θ) ∼ U(0, θ) =
{
1/θ 0 ≤ x ≤ θ
0 otherwise.
(a) Suppose that n samples D = {x1, ..., xn} are drawn independently according to
p(x|θ). Show that the maximum likelihood estimate for θ is max[D], i.e., the
value of the maximum element in D.
(b) Suppose that n = 5 points are drawn from the distribution and the maximum
value of which happens to be max
k
xk = 0.6. Plot the likelihood p(D|θ) in the
range 0 ≤ θ ≤ 1. Explain in words why you do not need to know the values of
the other four points.
3. Maximum likelihood methods apply to estimates of prior probabilities as well.
Let samples be drawn by successive, independent selections of a state of nature ωi
with unknown probability P (ωi). Let zik = 1 if the state of nature for the kth sample
is ωi and zik = 0 otherwise.
(a) Show that
P (zi1, . . . , zin|P (ωi)) =
n∏
k=1
P (ωi)
zik(1− P (ωi))1−zik .
(b) Show that the maximum likelihood estimate for P (ωi) is
P̂ (ωi) =
1
n
n∑
k=1
zik.
Interpret your result in words.
4. Let x be a d-dimensional binary (0 or 1) vector with a multivariate Bernoulli
distribution
P (x|θ) =
d∏
i=1
θxii (1− θi)
1−xi ,
where θ = (θ1, ..., θd)t is an unknown parameter vector, θi being the probability that
xi = 1. Show that the maximum likelihood estimate for θ is
θ̂ =
1
n
n∑
k=1
xk.
56 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
5. Let each component xi of x be binary valued (0 or 1) in a two-category problem
with P (ω1) = P (ω2) = 0.5. Suppose that the probability of obtaining a 1 in any
component is
pi1 = p
pi2 = 1− p,
and we assume for definiteness p > 1/2. The probability of error is known to approach
zero as the dimensionality d approaches infinity. This problem asks you to explore the
behavior as we increase the number of features in a single sample — a complementary
situation.
(a) Suppose that a single sample x = (x1, …, xd)t is drawn from category ω1. Show
that the maximum likelihood estimate for p is given by
p̂ =
1
d
d∑
i=1
xi.
(b) Describe the behavior of p̂ as d approaches infinity. Indicate why such behavior
means that by letting the number of features increase without limit we can
obtain an error-free classifier even though we have only one sample from each
class.
(c) Let T = 1/d
d∑
j=1
xj represent the proportion of 1’s in a single sample. Plot
P (T |ωi) vs. T for the case P = 0.6, for small d and for large d (e.g., d = 11 and
d = 111, respectively). Explain your answer in words.
6. Derive Eqs. 18 & 19 for the maximum likelihood estimation of the mean and
covariance of a multidimensional Gaussian. State clearly any assumptions you need
to invoke.
7. Show that if our model is poor, the maximum likelihood classifier we derive
is not the best — even among our (poor) model set — by exploring the following
example. Suppose we have two equally probable categories (i.e., P (ω1) = P (ω2) =
0.5). Further, we know that p(x|ω1) ∼ N(0, 1) but assume that p(x|ω2) ∼ N(µ, 1).
(That is, the parameter θ we seek by maximum likelihood techniques is the mean of
the second distribution.) Imagine however that the true underlying distribution is
p(x|ω2) ∼ N(1, 106).
(a) What is the value of our maximum likelihood estimate µ̂ in our poor model,
given a large amount of data?
(b) What is the decision boundary arising from this maximum likelihood estimate
in the poor model?
(c) Ignore for the moment the maximum likelihood approach, and use the methods
from Chap. ?? to derive the Bayes optimal decision boundary given the true
underlying distributions — p(x|ω1) ∼ N(0, 1) and p(x|ω2) ∼ N(1, 106). Be
careful to include all portions of the decision boundary.
(d) Now consider again classifiers based on the (poor) model assumption of p(x|ω2) ∼ N(µ, 1).
Using your result immediately above, find a new value of µ that will give lower
error than the maximum likelihood classifier.
3.10. PROBLEMS 57
(e) Discuss these results, with particular attention to the role of knowledge of the
underlying model.
8. Consider an extreme case of the general issue discussd in Problem 7, one in
which it is possible that the maximum likelihood solution leads to the worst possible
classifier, i.e., one with an error that approaches 100% (in probability). Suppose our
data in fact comes from two one-dimensional distributions of the forms
p(x|ω1) ∼ [(1− k)δ(x− 1) + kδ(x + X)] and
p(x|ω2) ∼ [(1− k)δ(x + 1) + kδ(x−X)],
where X is positive, 0 ≤ k < 0.5 represents the portion of the total probability mass concentrated at the point ±X, and δ(·) is the Dirac delta function. Suppose our poor models are of the form p(x|ω1, µ1) ∼ N(µ1, σ21) and p(x|ω2, µ2) ∼ N(µ2, σ22) and we form a maximum likelihood classifier. (a) Consider the symmetries in the problem and show that in the infinite data case the decision boundary will always be at x = 0, regardless of k and X. (b) Recall that the maximum likelihood estimate of either mean, µ̂i, is the mean of its distribution. For a fixed k, find the value of X such that the maximum likelihood estimates of the means “switch,” i.e., where µ̂1 ≥ µ̂2. (c) Plot the true distributions and the Gaussian estimates for the particular case k = .2 and X = 5. What is the classification error in this case? (d) Find a dependence X(k) which will guarantee that the estimated mean µ̂1 of p(x|ω1) is less than zero. (By symmetry, this will also insure µ̂2 > 0.)
(e) Given your X(k) just derived, state the classification error in terms of k.
(f) Suppose we constrained our model space such that σ21 = σ
2
2 = 1 (or indeed any
other constant). Would that change the above results?
(g) Discuss how if our model is wrong (here, does not include the delta functions),
the error can approaches 100% (in probability). Does this surprising answer
arise because we have found some local minimum in parameter space?
9. Prove the invariance property of maximum likelihood estimators, i.e., that if θ̂ is
the maximum likelihood estimate of θ, then for any differentiable function τ(·), the
maximum likelihood estimate of τ(θ) is τ(θ̂).
10. Suppose we employ a novel method for estimating the mean of a data set
D = {x1,x2, …,xn}: we assign the mean to be the value of the first point in the set,
i.e., x1.
(a) Show that this method is unbiased.
(b) State why this method is nevertheless highly undesirable.
11. One measure of the difference between two distributions in the same space is the
Kullback-Leibler divergence of Kullback-Leibler “distance”:
DKL(p1(x), p2(x)) =
∫
p1(x)ln
p1(x)
p2(x)
dx.
58 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
(This “distance,” does not obey the requisite symmetry and triangle inequalities for a
metric.) Suppose we seek to approximate an arbitrary distribution p2(x) by a normal
p1(x) ∼ N(µ,Σ). Show that the values that lead to the smallest Kullback-Leibler
divergence are the obvious ones:
µ = E2[x]
Σ = E2[(x− µ)(x− µ)t],
where the expectation taken is over the density p2(x).⊕
Section 3.3
12. Justify all the statements in the text leading from Eq. 25 to Eq. 26.⊕
Section 3.4
13. Let p(x|Σ) ∼ N(µ,Σ) where µ is known and Σ is unknown. Show that the
maximum likelihood estimate for Σ is given by
Σ̂ =
1
n
n∑
k=1
(xk − µ)(xk − µ)t
by carrying out the following argument:
(a) Prove the matrix identity atAa = tr[Aaat], where the trace, tr[A], is the sum
of the diagonal elements of A.
(b) Show that the likelihood function can be written in the form
p(x1, …,xn|Σ) =
1
(2π)nd/2
|Σ−1|n/2exp
[
−1
2
tr
[
Σ−1
n∑
k=1
(xk − µ)(xk − µ)t
]]
.
(c) Let A = Σ−1Σ̂ and λ1, …, λn be the eigenvalues of A; show that your result
above leads to
p(x1, …,xn|Σ) =
1
(2π)nd/2|Σ̂|n/2
(λ1 · · ·λd)n/2exp
[
− n
2
(λ1 + · · ·+ λd)
]
.
(d) Complete the proof by showing that the likelihood is maximized by the choice
λ1 = · · · = λd = 1. Explain your reasoning.
14. Suppose that p(x|µi,Σ, ωi) ∼ N(µi,Σ), where Σ is a common covariance matrix
for all c classes. Let n samples x1, …,xn be drawn as usual, and let l1, …, ln be their
labels, so that lk = i if the state of nature for xk was ωi.
(a) Show that
p(x1, …,xn, l1, …, ln|µ1, …,µc,Σ) =
n∏
k=1
P (ωlk)
(2π)nd/2|Σ|n/2
exp
[
− 1
2
n∑
k=1
(xk − µlk)
tΣ−1(xk − µlk)
]
.
3.10. PROBLEMS 59
(b) Using the results for samples drawn from a single normal population, show that
the maximum likelihood estimates for µi and Σ are given by
µ̂ =
∑
lk=i
xk
∑
lk=1
1
and
Σ̂ =
1
n
n∑
k=1
(xk − µ̂lk)(xk − µ̂lk)
t.
Interpret your answer in words.
15. Consider the problem of learning the mean of a univariate normal distribution.
Let n0 = σ2/σ20 be the dogmatism, and imagine that µ0 is formed by averaging n0
fictitious samples xk, k = −n0 + 1,−n0 + 2, …, 0.
(a) Show that Eqs. 32 & 33 for µn and σ2n yield
µn =
1
n + n0
n∑
k=−n0+1
xk
and
σ2n =
σ2
n + n0
.
(b) Use this result to give an interpretation of the a priori density p(µ) ∼ N(µ0, σ20).
16. Suppose that A and B are nonsingular matrices of the same order.
(a) Prove the matrix identity
(A−1 + B−1)−1 = A(A + B)−1B = B(A + B)−1A.
(b) Must these matrixes be square for this identity to hold?
(c) Use this result in showing that Eqs. 46 & 47 do indeed follow from Eqs. 42 &
43.
⊕
Section 3.5
17. The purpose of this problem is to derive the Bayesian classifier for the d-
dimensional multivariate Bernoulli case. As usual, work with each class separately,
interpreting P (x|D) to mean P (x|Di, ωi). Let the conditional probability for a given
category be given by
P (x|θ) =
d∏
i=1
θxii (1− θi)
1−xi ,
and let D = {x1, …,xn} be a set of n samples independently drawn according to this
probability density.
60 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
(a) If s = (s1, …, sd)t is the sum of the n samples, show that
P (D|θ) =
d∏
i=1
θsii (1− θi)
n−si .
(b) Assuming a uniform a priori distribution for θ and using the identity
1∫
0
θm(1− θ)n dθ = m!n!
(m + n + 1)!
,
show that
p(θ|D) =
d∏
i=1
(n + 1)!
si!(n− si)!
θsii (1− θi)
n−si .
(c) Plot this density for the case d = 1, n = 1, and for the two resulting possibilities
for s1.
(d) Integrate the product P (x|θ)p(θ|D) over θ to obtain the desired conditional
probability
P (x|D) =
d∏
i=1
(si + 1
n + 2
)xi(
1− si + 1
n + 2
)1−xi
.
(e) If we think of obtaining P (x|D) by substituting an estimate θ̂ for θ in P (x|θ),
what is the effective Bayesian estimate for θ?
18. Consider how knowledge of an invariance can guide our creation of a prior in the
following case. Suppose we have a binary (0 or 1) variable x, chosen independently
with a probability p(θ) = p(x = 1). Imagine we have observed Dn = {x1, x2, …, xn},
and now wish to evaluate the probability that xn+1 = 1, which we express as a ratio:
P (xn+1 = 1|Dn)
P (xn+1 = 0|Dn)
.
(a) Define s = x1 + · · · + xn and p(t) = P (x1 + · · · + xn+1 = t). Assume now
invariance of exchangeability, i.e., that the samples in any set Dn could have
been selected in an arbitrary order and it would not affect any probabilities.
Show how this assumption of exchangeability implies the ratio in question can
be written
p(s + 1)/
(
n+1
s+1
)
p(s)/
(
n+1
s
) ,
where
(
n+1
s
)
= (n+1)!
s!(n+1−s)! is the binomial coefficient.
(b) Evaluate this ratio given the assumption p(s)
p(s+ 1), when n and n− s and
s are not too small. Interpret your answer in words.
3.10. PROBLEMS 61
(c) In the binomial framework, we now seek a prior p(θ) such that p(s) does not
depend upon s, where
p(s) =
1∫
0
(
n
s
)
θs(1− θ)n−sp(θ) dθ.
Show that this requirement is satisfied if p(θ) is uniform, i.e., p(θ) ∼ U(0, 1).
19. Assume we have training data from a Gaussian distribution of known covari-
ance Σ but unknown mean µ. Suppose further that this mean itself is random, and
characterized by a Gaussian density having mean m0 and covariance Σ0.
(a) What is the MAP estimator for µ?
(b) Suppose we transform our coordinates by a linear transform x′ = Ax, for non-
singular matrix A, and accordingly for other terms. Determine whether your
MAP estimator gives the appropriate estimate for the transformed mean µ′.
Explain.
20. Suppose for a given class with parameter s the density can be written as:
p(x|α) = 1
α
f
(x
α
)
.
In such a case we say that α is a scale parameter. For instance, the standard deviation
σ is a scale parameter for a one-dimensional Gaussian.
(a) Imagine that we measure x′ = αx instead of x, for some constant α. Show that
the density now can be written as
p(x′|α′) = 1
α′
f
(
x′
α′
)
.
Find α′.
(b) Find the non-informative prior for α′, written as p′(α′). You will need to note
that for any interval ∆ ∈ (0,∞) the following equation should hold:
∫
∆
p(α)dα =
∫
∆
p′(α′) dα′.
21. State the conditions on p(x|θ), on p(θ), and on Dn that insure that the estimate
p(θ|Dn) in Eq. 54 converges in the limit n→∞.
⊕
Section 3.6
22. Employ the notation of the chapter and suppose s is a sufficient statistic statis-
tics for which p(θ|s,D) = p(θ|s). Assume p(θ|s) �= 0 and prove that p(D|s,θ) is
independent of θ.
62 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
23. Using the results given in Table 3.1, show that the maximum likelihood estimate
for the parameter θ of a Rayleigh distribution is given by
θ̂ =
1
1
n
n∑
k=1
x2k
.
24. Using the results given in Table 3.1, show that the maximum likelihood estimate
for the parameter θ of a Maxwell distribution is given by
θ̂ =
3/2
1
n
n∑
k=1
x2k
.
25. Using the results given in Table 3.1, show that the maximum likelihood estimate
for the parameter θ of a multinomial distribution is given by
θ̂i =
si
d∑
j=1
sj
.
where the vector s = (s1, …, sd)t is the average of the n samples x1, …,xn.
26. Demonstrate that sufficiency is an integral concept, i.e., that if s is sufficient for
θ, then corresponding components of s and θ need not be sufficient. Do this for the
case of a univariate Gaussian p(x) ∼ N(µ, σ2) where θ =
(
µ
σ2
)
is the full vector of
parameters.
(a) Verify that the statistic
s =
(
s1
s2
)
=
1
n
n∑
k=1
xk
1
n
n∑
k=1
x2k
is indeed sufficient for θ, as given in Table 3.1.
(b) Show that s1 taken alone is not sufficient for µ. Does your answer depend upon
whether σ2 is known?
(c) Show that s2 taken alone is not sufficient for σ2. Does your answer depend upon
whether µ is known?
27. Suppose s is a statistic for which p(θ|x,D) = p(θ|s).
(a) Assume p(θ|s) �= 0, and prove that p(D|s,θ) is independent of θ.
(b) Create an example to show that the inequality p(θ|s) �= 0 is required for your
proof above.
28. Consider the Cauchy distribution,
p(x) =
1
πb
· 1
1 +
(
x−a
b
)2 ,
for b > 0 and arbitrary real a.
3.10. PROBLEMS 63
(a) Confirm that the distribution is indeed normalized.
(b) For a fixed a and b, try to calculate the mean and the standard deviation of the
distribution. Explain your results.
(c) Prove that this distribution has no sufficient statistics for the mean and standard
deviation.
⊕
Section 3.7
29. In the following, suppose a and b are constants and n a variable parameter.
(a) Is an+1 = O(an)?
(b) Is abn = O(an)?
(c) Is an+b = O(an)?
(d) Prove f(n) = O(f(n)).
30. Consider the evaluation of a polynomial function f(x) =
n−1∑
i=0
aix
i, where the n
coefficients ai are given.
(a) Write pseudocode for a simple Θ(n2)-time algorithm for evaluating f(x).
(b) Show that such a polynomial can be rewritten as:
f(x) =
n−1∑
i=0
aix
i = (· · · (an−1x + an−2)x + · · ·+ a1)x + a0,
and so forth — a method known as Horner’s rule. Use the rule to write pseu-
docode for a Θ(n)-time algorithm for evaluating f(x).
31. For each of the short procedures, state the computational complexity in terms
of the variables N , M , P , and K, as appropriate. Assume that all data structures
are defined, that those without indexes are scalars and that those with indexes have
the number of dimensions shown.
Algorithm 6
1 begin for i← i + 1
2 s← s + i3
3 until i = N
4 return s
5 end
Algorithm 7
1 begin for i← i + 1
2 s← s + xi × xi
3 until i = N
4 return
√
s
5 end
64 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
Algorithm 8
1 begin for j ← j + 1
2 for i← i + 1
3 sj ← sj + wijxi
4 until i = I
5 until j = J
6 for k ← k + 1
7 for j ← j + 1
8 rk ← rk + wjksj
9 until j = J
10 until k = K
11 end
32. Consider a computer having a uniprocessor that can perform one operation per
nanosecond (10−9 sec). The left column of the table shows the functional dependence
of such operations in different hypothetical algorithms. For each such function, fill in
the number of operations n that can be performed in the total time listed along the
top.
f(n) 1 sec 1 hour 1 day 1 year
log2n√
n
n
nlog2n
n2
n3
2n
en
n!
33. Show that the estimator of Eq. 21 is indeed unbiased for:
(a) Normal distributions.
(b) Cauchy distributions.
(c) Binomial distributions.
(d) Prove that the estimator of Eq. 20 is asymptotically unbiased.
34. Let the sample mean µ̂n and the sample covariance matrix Cn for a set of n
samples x1, …,xn (each of which is d-dimensional) be defined by
µ̂n =
1
n
n∑
k=1
xk
and
Cn =
1
n− 1
n∑
k=1
(xk − µ̂n)(xk − µ̂n)t.
We call these the “non-recursive” formulas.
3.10. PROBLEMS 65
(a) What is the computational complexity of calculating µ̂n and Cn by these for-
mulas?
(b) Show that alternative, “recursive” techniques for calculating µ̂n and Cn based
on the successive addition of new samples xn+1 can be derived using the recur-
sion relations
µ̂n+1 = µ̂n +
1
n + 1
(xn+1 − µ̂n)
and
Cn+1 =
n− 1
n
Cn +
1
n + 1
(xn+1 − µ̂n)(xn+1 − µ̂n)t.
(c) What is the computational complexity of finding µ̂n and Cn by these recursive
methods?
(d) Describe situations where you might prefer to use the recursive method for com-
puting µ̂n and Cn, and ones where you might prefer the non-recursive method.
35. In pattern classification, one is often interested in the inverse of the covariance
matrix, for instance when designing a Bayes classifier for Gaussian distributions. Note
that the non-recursive calculation of C−1n (the inverse of the covariance matrix based
on n samples, cf., Problem 34) might require the O(n3) inversion of Cn by standard
matrix methods. We now explore an alternative, “recursive” method for computing
C−1n .
(a) Prove the so-called Sherman-Morrison-Woodbury matrix identity
(A + xyt)−1 = A−1 − A
−1xytA−1
1 + ytA−1x
.
(b) Use this and the results of Problem 34 to show that
C−1n+1 =
n
n− 1
[
C−1n −
C−1n (xn+1 − µ̂n)(xn+1 − µ̂n)tC−1n
n2−1
n
+ (xn+1 − µ̂n)tC−1n (xn+1 − µ̂n)
]
.
(c) What is the computational complexity of this calculation?
(d) Describe situations where you would use the recursive method, and ones where
you would use instead the non-recursive method.
36. Suppose we wish to simplify (or regularize) a Gaussian classifier for two categories
by means of shrinkage. Suppose that the estimated distributions are N(µ1,Σ1) and
N(µ2,Σ2). In order to employ shrinkage of an assumed common covariance toward
the identity matrix as given in Eq. 77, show that one must first normalize the data
to have unit variance.⊕
Section 3.8
37. Consider the convergence of the Expectation-Maximization algorithm, i.e., that
if l(θ,Dg) = lnp(Dg; θ) is not already optimum, then the EM algorithm increases it.
Prove this as follows:
66 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
(a) First note that
l(θ; Dg) = lnp(Dg,Db; θ)− lnp(Db|Dg; θ).
Let E ′[·] denote the expectation with respect to the distribution p(Db|Dg; θ′).
Take such an expectation of l(θ; Dg), and express your answer in terms of
Q(θ,θ′) of Eq. 78.
(b) Define φ(Db) = p(Db|Dg; θ)/p(Db|Dg; θ′) to be the ratio of expectations as-
suming the two distributions. Show that E ′[lnφ(Db)] ≤ E ′[φ(Db)]− 1 = 0.
(c) Use this result to show that if Q(θt+1,θt) > Q(θt,θt), achieved by the M step
in Algorithm ??, then l(θt+1; Dg) > l(θt; Dg).
38. Suppose we seek to estimate θ describing a multidimensional distribution from
data D, some of whose points are missing features. Consider an iterative algorithm in
which the maximum likelihood value of the missing values is calculated, then assumed
to be correct for the purposes of restinating θ and iterated.
(a) Is this always equivalent to an Expectation-Maximization algorithm, or just a
generalized Expectation-Maximization algorithm?
(b) If it is an Expectation-Maximization algorithm, what is Q(θ,θt), as described
by Eq. 78?
39. Consider data D =
{(
2
3
)
,
(
3
1
)
,
(
5
4
)
,
(
4
∗
)
,
(∗
6
)}
, sampled from a two-dimensional
uniform distribution
p(x) ∼ U(xl,xu) =
1
|xu1−xl1||xu2−xl2|
if xl1 ≤ x1 ≤ xu1
and xl2 ≤ x2 ≤ xu2
0 otherwise,
where ∗ represents missing feature values.
(a) Start with an initial estimate
θ0 =
(
xl
xu
)
=
0
0
10
10
,
and analytically calculate Q(θ,θ0) — the E step in the EM algorithm.
(b) Find the θ that maximizes your Q(θ,θ0) — the M step.
(c) Plot your data and the bounding rectangle.
(d) Without having to iterate further, state the estimate of θ that would result after
convergence of the EM algorithm.
40. Consider data D =
{(
1
1
)
,
(
3
3
)
,
(
2
∗
)}
, sampled from a two-dimensional (separable)
distribution p(x1, x2) = p(x1)p(x2), with
3.10. PROBLEMS 67
p(x1) ∼
{
1
θ1
e−θ1×1 if x1 ≥ 0
0 otherwise,
and
p(x2) ∼ U(0, θ2) =
{
1
θ2
if 0 ≤ x2 ≤ θ
0 otherwise.
As usual, ∗ represents a missing feature value.
(a) Start with an initial estimate θ0 =
(
2
4
)
and analytically calculate Q(θ,θ0) —
the E step in the EM algorithm. Be sure to consider the normalization of your
distribution.
(b) Find the θ that maximizes your Q(θ,θ0) — the M step.
(c) Plot your data on a two-dimensional graph and indicate the new parameter
estimates.
41. Repeat Problem 40 but with data D =
{(
1
1
)
,
(
3
3
)
,
(∗
2
)}
.⊕
Section 3.9
42. Use the conditional probability matrices in Example 3 to answer the following
separate problems.
(a) Suppose it is December 20 — the end of autumn and the beginning of winter
— and thus let P (a1) = P (a4) = 0.5. Furthermore, it is known that the fish
was caught in the north Atlantic, i.e., P (b1) = 1. Suppose the lightness has not
been measured but it is known that the fish is thin, i.e., P (d2) = 1. Classify the
fish as salmon or sea bass. What is the expected error rate?
(b) Suppse all we know is that a fish is thin and medium lightness. What season is
it now, most likely? What is your probability of being correct?
(c) Suppose we know a fish is thin and medium lightness and that it was caught in
the north Atlantic. What season is it, most likely? What is the probability of
being correct?
43. One of the simplest assumptions is that of the naive Bayes rule or idiot Bayes
rule expressed in Eq. 85. Draw the belief net for a three-category problem with five
features xi, i = 1, 2, …5.
44. Consider a Bayesian belief net with several nodes having unspecified values.
Suppose that one such node is selected at random, the probabilities of its nodes
computed by the formulas described in the text. Next another such node is chosen at
random (possibly even a node already visited), and the probabilities similarly updated.
Prove that this procedure will converge to the desired probabilities throughout the
full network.⊕
Section 3.10
45. Consider training an HMM by the Forward-backward algorithm, for a single
sequence of length T where each symbol could be one of c values. What is the
computational complexity of a single revision of all values âij and b̂jk?
46. The standard method for calculating the probability of a sequence in a given
HMM is to use the forward probabilities αi(t).
68 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
(a) Show by a simple substitution that a symmetric method can be derived using
the backward probabilities βi(t).
(b) Prove that one can get the probability by combining the forward and the back-
ward probabilities at any place in the middle of the sequence. That is, show
that
P (ωT
′
) =
T ′∑
i=1
αi(t)βi(t),
where ωT
′
is a particular sequence of length T ′ < T .
(c) Show that your formula reduces to the known values at the beginning and end
of the sequence.
47. Suppose we have a large number of symbol sequences emitted from an HMM
that has a particular transition probability ai′j′ = 0 for some single value of i′ and
j′. We use such sequences to train a new HMM, one that happens also to start
with its ai′j′ = 0. Prove that this parameter will remain 0 throughout training by
the Forward-backward algorithm. In other words, if the topology of the trained model
(pattern of non-zero connections) matches that of the generating HMM, it will remain
so after training.
48. Consider the decoding algorithm (Algorithm 4) in the text.
(a) Take logarithms of HMM model parameters and write pseudocode for an equiv-
alent algorithm.
(b) Explain why taking logarithms is an O(n) calculation, and thus the complexity
of your algorithm in (a) is O(c2T ).
49. Explore the close relationship between Bayesian belief nets and hidden Markov
models as follows.
(a) Prove that the forward and the backward equations for hidden Markov models
are special cases of Eq. 84.
(b) Use your answer to explain the relationship between these two general classes
of models.
Computer exercises
Several exercises will make use of the following three-dimensional data sampled from
three categories, denoted ωi.
3.10. COMPUTER EXERCISES 69
ω1 ω2 ω3
point x1 x2 x3 x1 x2 x3 x1 x2 x3
1 0.42 -0.087 0.58 -0.4 0.58 0.089 0.83 1.6 -0.014
2 -0.2 -3.3 -3.4 -0.31 0.27 -0.04 1.1 1.6 0.48
3 1.3 -0.32 1.7 0.38 0.055 -0.035 -0.44 -0.41 0.32
4 0.39 0.71 0.23 -0.15 0.53 0.011 0.047 -0.45 1.4
5 -1.6 -5.3 -0.15 -0.35 0.47 0.034 0.28 0.35 3.1
6 -0.029 0.89 -4.7 0.17 0.69 0.1 -0.39 -0.48 0.11
7 -0.23 1.9 2.2 -0.011 0.55 -0.18 0.34 -0.079 0.14
8 0.27 -0.3 -0.87 -0.27 0.61 0.12 -0.3 -0.22 2.2
9 -1.9 0.76 -2.1 -0.065 0.49 0.0012 1.1 1.2 -0.46
10 0.87 -1.0 -2.6 -0.12 0.054 -0.063 0.18 -0.11 -0.49
⊕
Section 3.2
1. Consider Gaussian density models in different dimensions.
(a) Write a program to find the maximum likelihood values µ̂ and σ̂2. Apply your
program individually to each of the three features xi of category ω1 in the table
above.
(b) Modify your program to apply to two-dimensional Gaussian data p(x) ∼ N(µ,Σ).
Apply your data to each of the three possible pairings of two features for ω1.
(c) Modify your program to apply to three-dimensional Gaussian data. Apply your
data to the full three-dimensional data for ω1.
(d) Assume your three-dimensional model is separable, so that Σ = diag(σ21 , σ
2
2 , σ
2
3).
Write a program to estimate the mean and the diagonal components of Σ. Apply
your program to the data in ω2.
(e) Compare your results for the mean of each feature µi calculated in the above
ways. Explain why they are the same or different.
(f) Compare your results for the variance of each feature σ2i calculated in the above
ways. Explain why they are the same or different.
⊕
Section 3.3
2. Consider a one-dimensional model of a triangular density governed by two scalar
parameters:
p(x|θ) ≡ T (µ, δ) =
{
(δ − |x− µ|)/δ2 for |x− µ| < δ
0 otherwise,
where θ =
(
µ
δ
)
. Write a program to calculate a density p(x|D) via Bayesian methods
(Eq. 26) and apply it to the x2 feature of category ω2. Plot your resulting posterior
density p(x|D).⊕
Section 3.4
3. Consider Bayesian estimation of the mean of a one-dimensional Gaussian. Suppose
you are given the prior for the mean is p(µ) ∼ N(µ0, σ0).
70 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
(a) Write a program that plots the density p(x|D) given µ0, σ0, σ and training set
D = {x1, x2, ..., xn}.
(b) Estimate σ for the x2 component of ω3 in the table above. Now assume µ0 = −1
and plot your estimated densities p(x|D) for each of the following values of the
dogmatism, σ2/σ20 : 0.1, 1.0, 10, 100.
⊕
Section 3.5
4. Suppose we have reason to believe that our data is sampled from a two-dimensional
uniform density
p(x|θ) ∼ U(xl,xu) =
1
|xu1−xl1||xu2−xl2| for xl1 ≤ x1 ≤ xu1 and xl2 ≤ x2 ≤ xu2
0 otherwise,
where xl1 is the x1 component of the “lower” bounding point xl, and analogously for
the x2 component and for the upper point. Suppose we have reliable prior information
that the density is zero outside the box defined by xl =
(−6
−6
)
and xu =
(
+6
+6
)
. Write
a program that calculates p(x|D) via recursive Bayesian estimation and apply it to
the x1−x2 components of ω1, in sequence, from the table above. For each expanding
data set Dn (2 ≤ n ≤ 10) plot your posterior density.
⊕
Section 3.6
5. Write a single program to calculate sufficient statistics for any members of the
exponential family (Eq. 69). Assume that the x3 data from ω3 in the table come from
an exponential density, and use your program to calculate the sufficient statistics for
each of the following exponential forms: Gaussian, Rayleigh and Maxwell.
⊕
Section 3.7
6. Consider error rates in different dimensions.
(a) Use maximum likelihood to train a dichotomizer using the three-dimensional
data for categories ω1 and ω2 in the Table above. Numerically integrate to
estimate the classification error rate.
(b) Now consider the data projected into a two-dimensional subspace. For each of
the three subspaces — defined by x1 = 0 or x2 = 0 or x3 = 0 — train a Gaussian
dichotomizer. Numerically integrate to estimate the error rate.
(c) Now consider the data projected onto one-dimensional subspaces, defined by
each of the three axes. Train a Gaussian classifier, and numerically integrate to
estimate the error rate.
(d) Discuss the rank order of the error rates you find.
(e) Assuming that you resestimate the distribution in the different dimensions, log-
ically must the Bayes error be higher in the projected spaces.
3.10. COMPUTER EXERCISES 71
7. Repeat the steps in Exercise 6 but for categories ω1 and ω3.
8. Consider the classification of Gaussian data employing shrinkage of covariance
matrixes to a common one.
(a) Generate 20 training points from each of three equally probable three-dimensional
Gaussian distributions N(µi,Σi) with the following parameters:
µ1 = (0, 0, 0)
t, Σ1 = diag[3, 5, 2]
µ2 = (1, 5,−3)t, Σ2 =
1 0 00 4 1
0 1 6
µ3 = (0, 0, 0)
t, Σ3 = 10I.
(b) Write a program to estimate the means and covariances of your data.
(c) Write a program that takes α and shrinks these estimated covariance matrixes
according to Eq. 76.
(d) Plot the training error as a function of α, where 0 < α < 1.
(e) Use your program from part (a) to generate 50 test points from each category.
Plot the test error as a function of α.
⊕
Section 3.8
9. Suppose we know that the ten data points in category ω1 in the table above come
from a three-dimensional Gaussian. Suppose, however, that we do not have access to
the x3 components for the even-numbered data points.
(a) Write an EM program to estimate the mean and covariance of the distribution.
Start your estimate with µ0 = 0 and Σ0 = I, the three-dimensional identity
matrix.
(b) Compare your final esimate with that for the case when there is no missing data.
10. Suppose we know that the ten data points in category ω2 in the table above
come from a three-dimensional uniform distribution p(x|ω2) ∼ U(xl,xu). Suppose,
however, that we do not have access to the x3 components for the even-numbered
data points.
(a) Write an EM program to estimate the six scalars comprising xl and xu of the dis-
tribution. Start your estimate with xl = (−2,−2,−2)t and xu = (+2,+2,+2)t.
(b) Compare your final esimate with that for the case when there is no missing data.
⊕
Section 3.9
Write a program to evaluate the Bayesian belief net for fish in Example 3, including
the information in P (xi|aj),P(xi|bj), P (ci|xj), and P (di|xj). Test your program on
the calculation given in the Example. Apply your program to the following cases, and
state any assumptions you need to make.
72 CHAPTER 3. MAXIMUM LIKELIHOOD AND BAYESIAN ESTIMATION
(a) A dark, thin fish is caught in the north Atlantic in summer. What is the
probability it is a salmon?
(b) A thin, medium fish is caught in the north Atlantic. What is the probability it
is winter? spring? summer? autumn?
(c) A light, wide fish is caught in the autumn. What is the probability it came from
the north Atlantic?
⊕
Section 3.10
11. Consider the use of hidden Markov models for classifying sequences of four visible
states, A-D. Train two hidden Markov models, each consisting of three hidden states
(plus a null initial state and a null final state), fully connected, with the following
data. Assume that each sequence starts with a null symbol and ends with an end null
symbol (not listed).
sample ω1 ω2
1 AABBCCDD DDCCBBAA
2 ABBCBBDD DDABCBA
3 ACBCBCD CDCDCBABA
4 AD DDBBA
5 ACBCBABCDD DADACBBAA
6 BABAADDD CDDCCBA
7 BABCDCC BDDBCAAAA
8 ABDBBCCDD BBABBDDDCD
9 ABAAACDCCD DDADDBCAA
10 ABD DDCAAA
(a) Print out the full transition matrices for each of the models.
(b) Assume equal prior probabilities for the two models and classify each of the
following sequences: ABBBCDDD, DADBCBAA, CDCBABA, and ADBBBCD.
(c) As above, classify the test pattern BADBDCBA. Find the prior probabilities for your
two trained models that would lead to equal posteriors for your two categories
when applied to this pattern.
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Index
O(·), see big oh
δ(·), see Dirac delta (δ(·))
Θ(·), see big theta
θ, see vector, parameter
Baum-Welch Algorithm, see Forward-
backward Algorithm
Bayes
maximum likelihood comparison,
see maximum likelihood, Bayes
comparison
Bayes error
dependence on number of features,
27
Bayes estimation
maximum likelihood comparison,
19
Bayes’ formula, 10
density estimation, 16
Bayesian
learning, see learning, Bayesian
Bayesian belief networks, see Belief net-
works
Bayesian estimation, see learning, Bayesian
Gaussian
multidimensional, 16
Bayesian learning, see learning, Bayesian
BEL(·), see belief, function
belief
function, 37
belief net
node, 37
Belief networks, 36
Bernoulli, see distribution, Bernoulli
Beta, see distribution, Beta
bias-variance
tradeoff, 19
big oh
non-uniqueness, 28
notation, 28
big theta, 29
Binomial, see distribution, Binomial
cardinality, 38
Cauchy distribution, see distribution,
Cauchy
causal network, see belief network
child (belief net), 37
class
independence, 4, 10
classifier
Bayes, 10
complexity
computational, 28
maximum likelihood classifier, 29
exponential, 30
polynomial, 30
space, 30
time, 30
computational complexity, see complex-
ity, computational, 28–32
of estimation, 19
conjugate prior, see prior, conjugate
covariance
matrix
sample, 9
of sum distribution, 15
DAG, see directed acyclic graph
data
training, 3
density
class-conditional, 11
estimation, 3
estimation, 11
sequence, 17
Gaussian, 3
joint
estimate, 11
design sample, see sample, design
determinant
complexity, 29
76
INDEX 77
Dirac delta, 13, 17
directed acyclic graph (DAG), 36
discriminant
regularized, 32
distance
Mahalanobis, 27
distribution
Bernoulli, 26
Beta, 26
Binomial, 26
Cauchy, 62
exponential, 26
Gamma, 26
Gaussian, 26
identifiable, 18
Maxwell, 26
multinomial, 26
normal, see distribution, Gaussian,
26
Poisson, 26
Rayleigh, 26
uniform, 17
dogmatism, 59
EM algorithm, see Expectation-Maximization
error
Bayes, 20
Gaussian, 27
dependence on number of features,
27
estimation, 20
indistinguisability, see error,Bayes
model, 9, 20
estimation
complexity, 19
estimation error, see error, estimation
estimator
absolutely unbiased, 9
asymptotically unbiased, 9
unbiased, 8
exchangeability
invariance, 60
Expectation-Maximization, 32–36
Algorithm, 33
Example, 33
exponential, see distribution, exponen-
tial
Factorization Theorem, 22
feature
independent, 27
related to error, 27
Forward-backward Algorithm, 52
function
Dirac delta, 13
Gamma, see distribution, Gamma
Gaussian, see distribution, Gaussian
Gaussian elimination, 29
GEM algorithm, see Expectation-Maximization,
generalized
generalized expectation maximization,
see Expectation-Maximization,
generalized
generalized Expectation-Maximization,
51
gradient
operator, 6
hidden Markov model
Example, 47
causal, 44
computation, 44
decoding, 49–51
ergodic, 44
evaluation, 45–49
learning, 51–53
Forward-backward Algorithm, 52
state
absorber, 44
final, 44
HMM
decoding
Example, 50
left-to-right, 49
Horner’s rule, 63
i.i.d., 4
identifiable, 18
idiot Bayes rule, see naive Bayes rule
improper prior, see prior, improper
independence
class, 10
independent features, 27
indistinguisability error, see error, Bayes
invariance
exchangeability, 60
scale, 20, 61
translation, 20
knowledge
78 INDEX
prior, 10
Kullback-Leibler divergence
Gaussian, 57
learning
Bayesian, 4, 17
pattern classification, 9
degenerate, 13
incremental
recursive Bayes, 17
supervised, 4
unsupervised, 4
likelihood, 5
extremum, 6
in belief net, 39
smoothness assumption, 11
log-likelihood, 5
function, 6
MAP, see maximum a posteriori
matrix
covariance
complexity, 29
estimates, 9
inversion
complexity, 29
sweep methods, 29
trace, 58
maximum a posteriori (MAP), 6
estimator, 6
maximum likelihood, 3, 5
Bayes comparison, 4, 19
Gaussian
mean, 7
mean and covariance, 7
solution non-uniqueness, 18
Maxwell distribution, see distribution,
Maxwell
mean
sample, see sample mean
mode
MAP estimation, 6
model error, see error, model
Monte-Carlo, 11
multinomial, see distribution, multino-
mial
naive Bayes rule, 42, 67
node
belief net, see belief net, node
node (belief net), 38
child, 38
parent, 37
normal, see distribution, normal
on-line learning, see learning, incremen-
tal
order of a function, 28
overdetermined solution, 31
parameter
estimation, 3
space, 11
parameter estimation
Bayesian
Gaussian case, 15
recursive Bayes, 17
parent (belief net), 37
Poisson distribution, see distribution,
Poisson
posterior
convergence, 18
delta function, 18
prior
conjugate, 12
determination, 10
estimation, 3
improper, 20
probability
density
estimation, 3
Rayleigh distribution, see distribution,
Rayleigh
recursive Bayes, 17
Example, 17
true, 17
regression
ridge, 32
ridge regression, see regression, ridge
sample
design, 3
mean, 7
Sherman-Morrison-Woodbury formula,
65
shrinkage, 32, 65
state of nature (ω), 4
stopping criterion, 53
sufficient statistic, 17
sufficient statistics, 21–27
INDEX 79
integral nature, 23, 62
temporality, 42
trace, 58
training data, see data, training
transition probability
Markov model, 43
uniform distribution, see distribution,
uniform
variance
addition, 14
bias, 8
vector
parameter (θ), 4, 11
true, 11
Contents
4 Nonparametric techniques 3
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4.2 Density estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4.3 Parzen Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.3.1 Convergence of the Mean . . . . . . . . . . . . . . . . . . . . . 9
4.3.2 Convergence of the Variance . . . . . . . . . . . . . . . . . . . 9
4.3.3 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.3.4 Classification example . . . . . . . . . . . . . . . . . . . . . . . 14
4.3.5 Probabilistic Neural Networks (PNNs) . . . . . . . . . . . . . . 15
Algorithm 1: PNN training . . . . . . . . . . . . . . . . . . . . . . . . 16
Algorithm 2: PNN classification . . . . . . . . . . . . . . . . . . . . . . 17
4.3.6 Choosing the window function . . . . . . . . . . . . . . . . . . 18
4.4 kn–Nearest-Neighbor Estimation . . . . . . . . . . . . . . . . . . . . . 18
4.4.1 Estimation of a posteriori probabilities . . . . . . . . . . . . . . 21
4.5 The Nearest-Neighbor Rule . . . . . . . . . . . . . . . . . . . . . . . . 21
4.5.1 Convergence of the Nearest Neighbor . . . . . . . . . . . . . . . 23
4.5.2 Error Rate for the Nearest-Neighbor Rule . . . . . . . . . . . . 23
4.5.3 Error Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.5.4 The k-Nearest-Neighbor Rule . . . . . . . . . . . . . . . . . . . 26
4.5.5 Computational Complexity of the k–Nearest-Neighbor Rule . . 28
Algorithm 3: Nearest-neighbor editing . . . . . . . . . . . . . . . . . . 30
4.6 *Metrics and Nearest-Neighbor Classification . . . . . . . . . . . . . . 31
4.6.1 Tangent distance . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.7 *Fuzzy Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.7.1 Are Fuzzy Category Memberships just Probabilities? . . . . . . 37
4.8 *Relaxation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Algorithm 4: RCE training . . . . . . . . . . . . . . . . . . . . . . . . 40
Algorithm 5: RCE classification . . . . . . . . . . . . . . . . . . . . . . 40
4.9 *Approximations by Series Expansions . . . . . . . . . . . . . . . . . . 42
4.10 *Fisher Linear Discriminant . . . . . . . . . . . . . . . . . . . . . . . . 44
4.11 *Multiple Discriminant Analysis . . . . . . . . . . . . . . . . . . . . . 47
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Bibliographical and Historical Remarks . . . . . . . . . . . . . . . . . . . . 51
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Computer exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
1
2 CONTENTS
Chapter 4
Nonparametric techniques
4.1 Introduction
I n Chap. ?? we treated supervised learning under the assumption that the formsof the underlying density functions were known. Alas, in most pattern recognition
applications this assumption is suspect; the common parametric forms rarely fit the
densities actually encountered in practice. In particular, all of the classical parametric
densities are unimodal (have a single local maximum), whereas many practical prob-
lems involve multimodal densities. Further, our hopes are rarely fulfilled that a high-
dimensional density might be simply represented as the product of one-dimensional
functions. In this chapter we shall examine nonparametric procedures that can be
used with arbitrary distributions and without the assumption that the forms of the
underlying densities are known.
There are several types of nonparametric methods of interest in pattern recogni-
tion. One consists of procedures for estimating the density functions p(x|ωj) from
sample patterns. If these estimates are satisfactory, they can be substituted for the
true densities when designing the classifier. Another consists of procedures for directly
estimating the a posteriori probabilities P (ωj |x). This is closely related to nonpara-
metric design procedures such as the nearest-neighbor rule, which bypass probability
estimation and go directly to decision functions. Finally, there are nonparametric
procedures for transforming the feature space in the hope that it may be possible to
employ parametric methods in the transformed space. These discriminant analysis
methods include the Fisher linear discriminant, which provides an important link be-
tween the parametric techniques of Chap. ?? and the adaptive techniques of Chaps. ??
& ??.
4.2 Density estimation
The basic ideas behind many of the methods of estimating an unknown probability
density function are very simple, although rigorous demonstrations that the estimates
converge require considerable care. The most fundamental techniques rely on the fact
that the probability P that a vector x will fall in a region R is given by
3
4 CHAPTER 4. NONPARAMETRIC TECHNIQUES
P =
∫
R
p(x′) dx′. (1)
Thus P is a smoothed or averaged version of the density function p(x), and we can
estimate this smoothed value of p by estimating the probability P . Suppose that n
samples x1, ...,xn are drawn independently and identically distributed (i.i.d.) accord-
ing to the probability law p(x). Clearly, the probability that k of these n fall in R is
given by the binomial law
Pk =
(
n
k
)
P k(1− P )n−k, (2)
and the expected value for k is
E [k] = nP. (3)
P = .7 1
k/n
2
4
6
8
Pk
0
100
50
20
Figure 4.1: The probability Pk of finding k patterns in a volume where the space
averaged probability is P as a function of k/n. Each curve is labelled by the total
number of patterns n. For large n, such binomial distributions peak strongly at
k/n = P (here chosen to be 0.7).
Moreover, this binomial distribution for k peaks very sharply about the mean, so that
we expect that the ratio k/n will be a very good estimate for the probability P , and
hence for the smoothed density function. This estimate is especially accurate when n
is very large (Fig. 4.1). If we now assume that p(x) is continuous and that the region
R is so small that p does not vary appreciably within it, we can write
∫
R
p(x′) dx′ � p(x)V, (4)
where x is a point within R and V is the volume enclosed by R. Combining Eqs. 1,
3 & 4, we arrive at the following obvious estimate for p(x):
4.2. DENSITY ESTIMATION 5
p(x) � k/n
V
. (5)
There are several problems that remain — some practical and some theoretical.
If we fix the volume V and take more and more training samples, the ratio k/n will
converge (in probability) as desired, but we have only obtained an estimate of the
space-averaged value of p(x),
P
V
=
∫
R
p(x′) dx′
∫
R
dx′
. (6)
If we want to obtain p(x) rather than just an averaged version of it, we must be
prepared to let V approach zero. However, if we fix the number n of samples and let
V approach zero, the region will eventually become so small that it will enclose no
samples, and our estimate p(x) � 0 will be useless. Or if by chance one or more of
the training samples coincide at x, the estimate diverges to infinity, which is equally
useless.
From a practical standpoint, we note that the number of samples is always limited.
Thus, the volume V can not be allowed to become arbitrarily small. If this kind of
estimate is to be used, one will have to accept a certain amount of variance in the
ratio k/n and a certain amount of averaging of the density p(x).
From a theoretical standpoint, it is interesting to ask how these limitations can
be circumvented if an unlimited number of samples is available. Suppose we use the
following procedure. To estimate the density at x, we form a sequence of regions
R1,R2, ..., containing x — the first region to be used with one sample, the second
with two, and so on. Let Vn be the volume of Rn, kn be the number of samples falling
in Rn, and pn(x) be the nth estimate for p(x):
pn(x) =
kn/n
Vn
. (7)
If pn(x) is to converge to p(x), three conditions appear to be required:
• lim
n→∞
Vn = 0
• lim
n→∞
kn =∞
• lim
n→∞
kn/n = 0.
The first condition assures us that the space averaged P/V will converge to p(x),
provided that the regions shrink uniformly and that p(·) is continuous at x. The
second condition, which only makes sense if p(x) = 0, assures us that the frequency
ratio will converge (in probability) to the probability P . The third condition is clearly
necessary if pn(x) given by Eq. 7 is to converge at all. It also says that although a
huge number of samples will eventually fall within the small region Rn, they will form
a negligibly small fraction of the total number of samples.
There are two common ways of obtaining sequences of regions that satisfy these
conditions (Fig. 4.2). One is to shrink an initial region by specifying the volume Vn
as some function of n, such as Vn = 1/
√
n. It then must be shown that the random
variables kn and kn/n behave properly, or more to the point, that pn(x) converges to
6 CHAPTER 4. NONPARAMETRIC TECHNIQUES
p(x). This is basically the Parzen-window method that will be examined in Sect. 4.3.
The second method is to specify kn as some function of n, such as kn =
√
n. Here
the volume Vn is grown until it encloses kn neighbors of x. This is the kn-nearest-
neighbor estimation method. Both of these methods do in fact converge, although it
is difficult to make meaningful statements about their finite-sample behavior.
n=1 2 3 10
Figure 4.2: Two methods for estimating the density at a point x (at the center of
each square) are to xxx.
4.3 Parzen Windows
The Parzen-window approach to estimating densities can be introduced by temporar-
ily assuming that the region Rn is a d-dimensional hypercube. If hn is the length of
an edge of that hypercube, then its volume is given by
Vn = h
d
n. (8)
We can obtain an analytic expression for kn, the number of samples falling in the
hypercube, by defining the following window function:window
function
ϕ(u) =
{
1 |uj | ≤ 1/2 j = 1, ..., d
0 otherwise.
(9)
Thus, ϕ(u) defines a unit hypercube centered at the origin. It follows that ϕ((x− xi)/hn)
is equal to unity if xi falls within the hypercube of volume Vn centered at x, and is
zero otherwise. The number of samples in this hypercube is therefore given by
kn =
n∑
i=1
ϕ
(
x− xi
hn
)
, (10)
and when we substitute this into Eq. 7 we obtain the estimate
pn(x) =
1
n
n∑
i=1
1
Vn
ϕ
(
x− xi
hn
)
. (11)
This equation suggests a more general approach to estimating density functions.
Rather than limiting ourselves to the hypercube window function of Eq. 9, suppose
we allow a more general class of window functions. In such a case, Eq. 11 expresses
our estimate for p(x) as an average of functions of x and the samples xi. In essence,
4.3. PARZEN WINDOWS 7
the window function is being used for interpolation — each sample contributing to
the estimate in accordance with its distance from x.
It is natural to ask that the estimate pn(x) be a legitimate density function, i.e.,
that it be nonnegative and integrate to one. This can be assured by requiring the
window function itself be a density function. To be more precise, if we require that
ϕ(x) ≥ 0 (12)
and
∫
ϕ(u) du = 1, (13)
and if we maintain the relation Vn = hdn, then it follows at once that pn(x) also
satisfies these conditions.
Let us examine the effect that the window width hn has on pn(x). If we define the
function δn(x) by
δn(x) =
1
Vn
ϕ
(
x
hn
)
, (14)
then we can write pn(x) as the average
pn(x) =
1
n
n∑
i=1
δn(x− xi). (15)
Since Vn = hdn, hn clearly affects both the amplitude and the width of δn(x) (Fig. 4.3).
If hn is very large, the amplitude of δn is small, and x must be far from xi before
δn(x − xi) changes much from δn(0). In this case, pn(x) is the superposition of n
broad, slowly changing functions and is a very smooth “out-of-focus” estimate of
p(x). On the other hand, if hn is very small, the peak value of δn(x−xi) is large and
occurs near x = xi. In this case p(x) is the superposition of n sharp pulses centered
at the samples — an erratic, “noisy” estimate (Fig. 4.4). For any value of hn, the
distribution is normalized, i.e.,
∫
δn(x− xi) dx =
∫
1
Vn
ϕ
(
x− xi
hn
)
dx =
∫
ϕ(u) du = 1. (16)
Thus, as hn approaches zero, δn(x−xi) approaches a Dirac delta function centered at
xi, and pn(x) approaches a superposition of delta functions centered at the samples.
Clearly, the choice of hn (or Vn) has an important effect on pn(x). If Vn is too
large, the estimate will suffer from too little resolution; if Vn is too small, the estimate
will suffer from too much statistical variability. With a limited number of samples, the
best we can do is to seek some acceptable compromise. However, with an unlimited
number of samples, it is possible to let Vn slowly approach zero as n increases and
have pn(x) converge to the unknown density p(x).
In discussing convergence, we must recognize that we are talking about the con-
vergence of a sequence of random variables, since for any fixed x the value of pn(x)
depends on the random samples x1, ...,xn. Thus, pn(x) has some mean p̄n(x) and
variance σ2n(x). We shall say that the estimate pn(x) converges to p(x) if
lim
n→∞
p̄n(x) = p(x) (17)
8 CHAPTER 4. NONPARAMETRIC TECHNIQUES
-2
-1
0
1
2 -2
-1
0
1
2
0
0.05
0.1
0.15
-2
-1
0
1
2
h = 1
δ(x)
-2
-1
0
1
2 -2
-1
0
1
2
0
0.2
0.4
0.6
-2
-1
0
1
2
h = .5
δ(x)
-2
-1
0
1
2 -2
-1
0
1
2
0
1
2
3
4
-2
-1
0
1
2
h = .2
δ(x)
Figure 4.3: Examples of two-dimensional circularly symmetric normal Parzen windows
ϕ(x/h) for three different values of h. Note that because the δk(·) are normalized,
different vertical scales must be used to show their structure.
p(x)
p(x) p(x)
Figure 4.4: Three Parzen-window density estimates based on the same set of five
samples, using the window functions in Fig. 4.3. As before, the vertical axes have
been scaled to show the structure of each function.
and
lim
n→∞
σ2n(x) = 0. (18)
To prove convergence we must place conditions on the unknown density p(x), on
the window function ϕ(u), and on the window width hn. In general, continuity of
p(·) at x is required, and the conditions imposed by Eqs. 12 & 13 are customarily
invoked. With care, it can be shown that the following additional conditions assure
convergence (Problem 1):
sup
u
ϕ(u) <∞ (19)
lim
‖u‖→∞
ϕ(u)
d∏
i=1
ui = 0 (20)
4.3. PARZEN WINDOWS 9
lim
n→∞
Vn = 0 (21)
and
lim
n→∞
nVn =∞. (22)
Equations 19 & 20 keep ϕ(·) well behaved, and are satisfied by most density functions
that one might think of using for window functions. Equations 21 & 22 state that the
volume Vn must approach zero, but at a rate slower than 1/n. We shall now see why
these are the basic conditions for convergence.
4.3.1 Convergence of the Mean
Consider first p̄n(x), the mean of pn(x). Since the samples xi are i.i.d. according to
the (unknown) density p(x), we have
p̄n(x) = E [pn(x)]
=
1
n
n∑
i=1
E
[ 1
Vn
ϕ
(x− xi
hn
)]
=
∫
1
Vn
ϕ
(x− v
hn
)
p(v) dv
=
∫
δn(x− v)p(v) dv. (23)
This equation shows that the expected value of the estimate is an averaged value
of the unknown density — a convolution of the unknown density and the window convolution
function (Appendix ??). Thus, p̄n(x) is a blurred version of p(x) as seen through the
averaging window. But as Vn approaches zero, δn(x−v) approaches a delta function
centered at x. Thus, if p is continuous at x, Eq. 21 ensures that p̄n(x) will approach
p(x) as n approaches infinity.
4.3.2 Convergence of the Variance
Equation 23 shows that there is no need for an infinite number of samples to make
p̄n(x) approach p(x); one can achieve this for any n merely by letting Vn approach
zero. Of course, for a particular set of n samples the resulting “spiky” estimate is
useless; this fact highlights the need for us to consider the variance of the estimate.
Since pn(x) is the sum of functions of statistically independent random variables, its
variance is the sum of the variances of the separate terms, and hence
σ2n(x) =
n∑
i=1
E
[(
1
nVn
ϕ
(
x− xi
hn
− 1
n
p̄n(x)
))2]
= n E
[ 1
n2V 2n
ϕ2
(x− xi
hn
)]
− 1
n
p̄2n(x)
=
1
nVn
∫
1
Vn
ϕ2
(x− v
hn
)
p(v) dv − 1
n
p̄2n(x). (24)
10 CHAPTER 4. NONPARAMETRIC TECHNIQUES
By dropping the second term, bounding ϕ(·) and using Eq. 21, we obtain
σ2n(x) ≤
sup(ϕ(·)) p̄n(x)
nVn
. (25)
Clearly, to obtain a small variance we want a large value for Vn, not a small one —
a large Vn smooths out the local variations in density. However, since the numerator
stays finite as n approaches infinity, we can let Vn approach zero and still obtain zero
variance, provided that nVn approaches infinity. For example, we can let Vn = V1/
√
n
or V1/ln n or any other function satisfying Eqs. 21 & 22.
This is the principal theoretical result. Unfortunately, it does not tell us how to
choose ϕ(·) and Vn to obtain good results in the finite sample case. Indeed, unless we
have more knowledge about p(x) than the mere fact that it is continuous, we have no
direct basis for optimizing finite sample results.
4.3.3 Illustrations
It is interesting to see how the Parzen window method behaves on some simple ex-
amples, and particularly the effect of the window function. Consider first the case
where p(x) is a zero-mean, unit-variance, univariate normal density. Let the window
function be of the same form:
ϕ(u) =
1√
2π
e−u
2/2. (26)
Finally, let hn = h1/
√
n, where h1 is a parameter at our disposal. Thus pn(x) is an
average of normal densities centered at the samples:
pn(x) =
1
n
n∑
i=1
1
hn
ϕ
(x− xi
hn
)
. (27)
While it is not hard to evaluate Eqs. 23 & 24 to find the mean and variance of
pn(x), it is even more interesting to see numerical results. When a particular set of
normally distributed random samples was generated and used to compute pn(x), the
results shown in Fig. 4.5 were obtained. These results depend both on n and h1. For
n = 1, pn(x) is merely a single Gaussian centered about the first sample, which of
course has neither the mean nor the variance of the true distribution. For n = 10
and h1 = 0.1 the contributions of the individual samples are clearly discernible; this
is not the case for h1 = 1 and h1 = 5. As n gets larger, the ability of pn(x) to resolve
variations in p(x) increases. Concomitantly, pn(x) appears to be more sensitive to
local sampling irregularities when n is large, although we are assured that pn(x) will
converge to the smooth normal curve as n goes to infinity. While one should not judge
on visual appearance alone, it is clear that many samples are required to obtain an
accurate estimate. Figure 4.6 shows analogous results in two dimensions.
As a second one-dimensional example, we let ϕ(x) and hn be the same as in
Fig. 4.5, but let the unknown density be a mixture of two uniform densities:
p(x) =
1 −2.5 < x < −2
1/4 0 < x < 2
0 otherwise.
(28)
Figure 4.7 shows the behavior of Parzen-window estimates for this density. As before,
the case n = 1 tells more about the window function than it tells about the unknown
4.3. PARZEN WINDOWS 11
h = 1 h = .5 h = .1
n = 1
n = 10
n = 100
n = ∞
Figure 4.5: Parzen-window estimates of a univariate normal density using different
window widths and numbers of samples. The vertical axes have been scaled to best
show the structure in each graph. Note particularly that the n =∞ estimates are the
same (and match the true generating function), regardless of window width h.
12 CHAPTER 4. NONPARAMETRIC TECHNIQUES
-2
-1
0
1
2 -2
-1
0
1
2
0.01
0.02
0.03
0.04
-2
-1
0
1
2
n=1
h=2
-2
-1
0
1
2 -2
-1
0
1
2
0
.05
.1
.15
-2
-1
0
1
2
n = 1
h = 1
-2
-1
0
1
2 -2
-1
0
1
2
0
.2
.4
.6
-2
-1
0
1
2
n = 1
h = .5
-2
-1
0
1
2 -2
-1
0
1
2
0
.05
.1
.15
-2
-1
0
1
2
n = 10
h = 2
-2
-1
0
1
2 -2
-1
0
1
2
0
.1
.2
.3
.4
-2
-1
0
1
2
n = 10
h = 1
-2
-1
0
1
2 -2
-1
0
1
2
0
.25
.5
.75
-2
-1
0
1
2
n = 10
h = .5
-2
-1
0
1
2 -2
-1
0
1
2
0
.1
.2
-2
-1
0
1
2
n = 100
h = 2
-2
-1
0
1
2 -2
-1
0
1
2
0
.2
.4
-2
-1
0
1
2
n = 100
h1 = 1
-2
-1
0
1
2 -2
-1
0
1
2
0
0.5
1
-2
-1
0
1
2
n = 100
h1 = .5
-2
-1
0
1
2 -2
-1
0
1
2
0
.05
.1
.15
-2
-1
0
1
2
n = ∞
h = 2
-2
-1
0
1
2 -2
-1
0
1
2
0
.05
.1
.15
-2
-1
0
1
2
n = ∞
h = 1
-2
-1
0
1
2 -2
-1
0
1
2
0
.05
.1
.15
-2
-1
0
1
2
n = ∞
h = .5
Figure 4.6: Parzen-window estimates of a bivariate normal density using different
window widths and numbers of samples. The vertical axes have been scaled to best
show the structure in each graph. Note particularly that the n =∞ estimates are the
same (and match the true generating distribution), regardless of window width h.
4.3. PARZEN WINDOWS 13
density. For n = 16, none of the estimates is particularly good, but results for n = 256
and h1 = 1 are beginning to appear acceptable.
14 CHAPTER 4. NONPARAMETRIC TECHNIQUES
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
n = 1
n = 16
n = 256
n = ∞
h1 = 1 h1 = .5 h1 = .2
Figure 4.7: Parzen-window estimates of a bimodal distribution using different window
widths and numbers of samples. Note particularly that the n =∞ estimates are the
same (and match the true generating distribution), regardless of window width h.
4.3.4 Classification example
In classifiers based on Parzen-window estimation, we estimate the densities for each
category and classify a test point by the label corresponding to the maximum poste-
rior. If there are multiple categories with unequal priors we can easily include these
too (Problem 4). The decision regions for a Parzen-window classifier depend upon
the choice of window function, of course, as illustrated in Fig. 4.8. In general, the
training error — the empirical error on the training points themselves — can be made
arbitrarily low by making the window width sufficiently small.∗ However, the goal of
creating a classifier is to classify novel patterns, and alas a low training error does
not guarantee a small test error, as we shall explore in Chap. ??. Although a generic
Gaussian window shape can be justified by considerations of noise, statistical inde-
pendence and uncertainty, in the absense of other information about the underlying
distributions there is little theoretical justification of one window width over another.
These density estimation and classification examples illustrate some of the power
and some of the limitations of nonparametric methods. Their power resides in their
generality. Exactly the same procedure was used for the unimodal normal case and
the bimodal mixture case and we did not need to make any assumptions about the
∗ We ignore cases in which the same feature vector has been assigned to multiple categories.
4.3. PARZEN WINDOWS 15
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Figure 4.8: The decision boundaries in a two-dimensional Parzen-window di-
chotomizer depend on the window width h. At the left a small h leads to boundaries
that are more complicated than for large h on same data set, shown at the right.
Apparently, for this data a small h would be appropriate for the upper region, while
a large h for the lower region; no single window width is ideal overall.
distributions ahead of time. With enough samples, we are essentially assured of
convergence to an arbitrarily complicated target density. On the other hand, the
number of samples needed may be very large indeed — much greater than would be
required if we knew the form of the unknown density. Little or nothing in the way of
data reduction is provided, which leads to severe requirements for computation time
and storage. Moreover, the demand for a large number of samples grows exponentially
with the dimensionality of the feature space. This limitation is related to the “curse of
dimensionality,” and severely restricts the practical application of such nonparametric
procedures (Problem 11). The fundamental reason for the curse of dimensionality is
that high-dimensional functions have the potential to be much more complicated than
low-dimensional ones, and that those complications are harder to discern. The only
way to beat the curse is to incorporate knowledge about the data that is correct.
4.3.5 Probabilistic Neural Networks (PNNs)
A hardware implementation of the Parzen windows approach is found in Probabilistic
Neural Networks (Fig. 4.9). Suppose we wish to form a Parzen estimate based on n
patterns, each of which is d-dimensional, randomly sampled from c classes. The PNN
for this case consists of d input units comprising the input layer, each unit is connect input unit
to each of the n pattern units; each pattern unit is, in turn, connected to one and
pattern
unit
only one of the c category units. The connections from the input to pattern units
category
unit
represent modifiable weights, which will be trained. (While these weights are merely
weight
parameters and could be represented by a vector θ, in keeping with the established
terminology in neural networks we shall use the symbol w.) Each link from a pattern
unit to its associated category unit is of a single constant magnitude.
The PNN is trained in the following way. First, each pattern x of the training set is
16 CHAPTER 4. NONPARAMETRIC TECHNIQUES
-1 1
-1
1
-1 1
-1
1
-1 1
-1
1
-1 1
-1
1
1 2 3 n
1 2 c
x1 x2 xd
input
pattern
category
Figure 4.9: A probabilistic neural network (PNN) consists of d input units, n pattern
units and c category units. Each pattern unit forms the inner product of its weight
vector and the normalized pattern vector x to form z = wtx, and then emits exp[(z−
1)/σ2]. Each category unit sums such contributions from the pattern unit connected
to it. This insures that the activity in each of the category units represents the Parzen-
window density estimate using a circularly symmetric Gaussian window of covariance
σ2I, where I is the d× d identity matrix.
normalized to have unit length, i.e., is scaled so that
d∑
i=1
x2i = 1.
∗ The first normalized
training pattern is placed on the input units. The modifiable weights linking the input
units and the first pattern unit are set such that w1 = x1. (Note that because of the
normalization of x1, w1 is normalized too.) Then, a single connection from the first
pattern unit is made to the category unit corresponding to the known class of that
pattern. The process is repeated with each of the remaining training patterns, setting
the weights to the successive pattern units such that wk = xk for k = 1, 2, ..., n.
After such training we have a network that is fully connected between input and
pattern units, and sparsely connected from pattern to category units. If we denote
the components of the jth pattern as xjk and the weights to the jth pattern unit wjk,
for j = 1, 2, ..., n and k = 1, 2, ..., d, then our algorithm is:
Algorithm 1 (PNN training)
1 begin initialize j = 0, n = #patterns
2 do j ← j + 1
3 normalize : xjk ← xjk/
(
d∑
i
x2ji
)1/2
4 train : wjk ← xjk
5 if x ∈ ωi then aic ← 1
6 until j = n
∗ Such normalization collapses two vectors having the same direction but different magnitude. In
order to avoid this, we can augment the pattern with a feature of magnitude 1.0, making it (d+1)-
dimensional, and then normalize.
4.3. PARZEN WINDOWS 17
7 end
The trained network is then used for classification in the following way. A nor-
malized test pattern x is placed at the input units. Each pattern unit computes the
inner product
zk = w
t
kx, (29)
and emits a nonlinear function of zk; each output unit sums the contributions from
all pattern units connected to it. The nonlinear function is e(zk−1)/σ
2
, where σ is a
parameter set by the user and is equal to
√
2 times the width of the effective Gaussian
window. To understand this choice of nonlinearity, consider an (unnormalized) Gaus-
sian window centered on the position of one of the training patterns wk. We work
backwards from the desired Gaussian window function to infer the nonlinear transfer
function that should be employed by the pattern units. That is, if we let our effective
width hn be a constant, the window function is
ϕ
(
xk −wk
hn
)
∝
desired Gaussian︷ ︸︸ ︷
e−(x−wk)
t(x−wk)/2σ2
= e−(x
tx+wtkwk−2x
twk)/2σ
2
= e(zk−1)/σ
2
︸ ︷︷ ︸
transfer
function
, (30)
where we have used our normalization conditions xtx = wtkwk = 1. Thus each pattern
unit contributes to its associated category unit a signal equal to the probability the
test point was generated by a Gaussian centered on the associated training point.
The sum of these local estimates (computed at the corresponding category unit) gives
the discriminant function gi(x) — the Parzen window estimate of the underlying
distribution. The max
i
gi(x) operation gives the desired category for the test point
(Algorithm 2).
Algorithm 2 (PNN classification)
1 begin initialize k = 0,x = test pattern
2 do k ← k + 1
3 zk ← wtkx
4 if akc = 1 then gc ← gc + exp[(zk − 1)/σ2]
5 until k = n
6 return class← arg max
i
gi(x)
7 end
One of the benefits of PNNs is their speed of learning, since the learning rule
(i.e., setting wk = xk) is simple and requires only a single pass through the training
data. The space complexity (amount of memory) for the PNN is easy to determine by
counting the number of wires in Fig. 4.9 — O((n+ 1)d). This can be quite severe for
instance in a hardware application, since both n and d can be quite large. The time
complexity for classification by the parallel implementation of Fig. 4.9 is O(1), since
the n inner products of Eq. 29 can be done in parallel. Thus this PNN architecture
18 CHAPTER 4. NONPARAMETRIC TECHNIQUES
could find uses where recognition speed is important and storage is not a severe
limitation. Another benefit is that new training patterns can be incorporated into
a previously trained classifier quite easily; this might be important for a particular
on-line application.
4.3.6 Choosing the window function
As we have seen, one of the problems encountered in the Parzen-window/PNN ap-
proach concerns the choice of the sequence of cell volumes sizes V1, V2, ... or overall
window size (or indeed other window parameters, such as shape or orientation). For
example, if we take Vn = V1/
√
n, the results for any finite n will be very sensitive to
the choice for the initial volume V1. If V1 is too small, most of the volumes will be
empty, and the estimate pn(x) will be very erratic (Fig. 4.7). On the other hand, if
V1 is too large, important spatial variations in p(x) may be lost due to averaging over
the cell volume. Furthermore, it may well be the case that a cell volume appropriate
for one region of the feature space might be entirely unsuitable in a different region
(Fig. 4.8). In Chap. ?? we shall consider general methods, including cross-validation,
which are often used in conjunction with Parzen windows. Now, though, we turn to an
important alternative method that is both useful and has solvable analytic properties.
4.4 kn–Nearest-Neighbor Estimation
A potential remedy for the problem of the unknown “best” window function is to
let the cell volume be a function of the training data, rather than some arbitrary
function of the overall number of samples. For example, to estimate p(x) from n
training samples or prototypes we can center a cell about x and let it grow until itprototypes
captures kn samples, where kn is some specified function of n. These samples are
the kn nearest-neighbors of x. It the density is high near x, the cell will be relatively
small, which leads to good resolution. If the density is low, it is true that the cell will
grow large, but it will stop soon after it enters regions of higher density. In either
case, if we take
pn(x) =
kn/n
Vn
(31)
we want kn to go to infinity as n goes to infinity, since this assures us that kn/n
will be a good estimate of the probability that a point will fall in the cell of volume
Vn. However, we also want kn to grow sufficiently slowly that the size of the cell
needed to capture kn training samples will shrink to zero. Thus, it is clear from
Eq. 31 that the ratio kn/n must go to zero. Although we shall not supply a proof,
it can be shown that the conditions lim
n→∞
kn = ∞ and lim
n→∞
kn/n = 0 are necessary
and sufficient for pn(x) to converge to p(x) in probability at all points where p(x) is
continuous (Problem 5). If we take kn =
√
n and assume that pn(x) is a reasonably
good approximation to p(x) we then see from Eq. 31 that Vn � 1/(
√
np(x)). Thus,
Vn again has the form V1/
√
n, but the initial volume V1 is determined by the nature
of the data rather than by some arbitrary choice on our part. Note that there are
nearly always discontinuities in the slopes of these estimates, and these lie away from
the prototypes themselves (Figs. 4.10 & 4.11).
It is instructive to compare the performance of this method with that of the
Parzen-window/PNN method on the data used in the previous examples. With n = 1
4.4. KN–NEAREST-NEIGHBOR ESTIMATION 19
x
p(x)
3
5
Figure 4.10: Eight points in one dimension and the k-nearest-neighbor density esti-
mates, for k = 3 and 5. Note especially that the discontinuities in the slopes in the
estimates generally occur away fom the positions of the points themselves.
0
1
2
3
Figure 4.11: The k-nearest-neighbor estimate of a two-dimensional density for k = 5.
Notice how such a finite n estimate can be quite “jagged,” and that discontinuities in
the slopes generally occur along lines away from the positions of the points themselves.
20 CHAPTER 4. NONPARAMETRIC TECHNIQUES
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
0 1 2 3 4
1
n = 1
kn = 1
n = 16
kn = 4
n = 256
kn = 16
n = ∞
kn = ∞
Figure 4.12: Several k-nearest-neighbor estimates of two unidimensional densities: a
Gaussian and a bimodal distribution. Notice how the finite n estimates can be quite
“spiky.”
and kn =
√
n = 1, the estimate becomes
pn(x) =
1
2|x− x1|
. (32)
This is clearly a poor estimate of p(x), with its integral embarrassing us by diverging
to infinity. As shown in Fig. 4.12, the estimate becomes considerably better as n gets
larger, even though the integral of the estimate remains infinite. This unfortunate fact
is compensated by the fact that pn(x) never plunges to zero just because no samples
fall within some arbitrary cell or window. While this might seem to be a meager
compensation, it can be of considerable value in higher-dimensional spaces.
As with the Parzen-window approach, we could obtain a family of estimates by
taking kn = k1
√
n and choosing different values for k1. However, in the absense of
any additional information, one choice is as good as another, and we can be confident
only that the results will be correct in the infinite data case. For classification, one
popular method is to adjust the window width until the classifier has the lowest error
on a separate set of samples, also drawn from the target distributions, a technique we
shall explore in Chap. ??.
4.5. THE NEAREST-NEIGHBOR RULE 21
4.4.1 Estimation of a posteriori probabilities
The techniques discussed in the previous sections can be used to estimate the a pos-
teriori probabilities P (ωi|x) from a set of n labelled samples by using the samples
to estimate the densities involved. Suppose that we place a cell of volume V around
x and capture k samples, ki of which turn out to be labelled ωi. Then the obvious
estimate for the joint probability p(x, ωi) is
pn(x, ωi) =
ki/n
V
, (33)
and thus a reasonable estimate for P (ωi|x) is
Pn(ωi|x) =
pn(x, ωi)
c∑
j=1
pn(x, ωj)
=
ki
k
. (34)
That is, the estimate of the a posteriori probability that ωi is the state of nature is
merely the fraction of the samples within the cell that are labelled ωi. Consequently,
for minimum error rate we select the category most frequently represented within the
cell. If there are enough samples and if the cell is sufficiently small, it can be shown
that this will yield performance approaching the best possible.
When it comes to choosing the size of the cell, it is clear that we can use either
the Parzen-window approach or the kn-nearest-neighbor approach. In the first case,
Vn would be some specified function of n, such as Vn = 1/
√
n. In the second case,
Vn would be expanded until some specified number of samples were captured, such
as k =
√
n. In either case, as n goes to infinity an infinite number of samples will fall
within the infinitely small cell. The fact that the cell volume could become arbitrarily
small and yet contain an arbitrarily large number of samples would allow us to learn
the unknown probabilities with virtual certainty and thus eventually obtain optimum
performance. Interestingly enough, we shall now see that we can obtain comparable
performance if we base our decison solely on the label of the single nearest neighbor
of x.
4.5 The Nearest-Neighbor Rule
While the k-nearest-neighbor algorithm was first proposed for arbitrary k, the crucial
matter of determining the error bound was first solved for k = 1. This nearest-
neighbor algorithm has conceptual and computational simplicity. We begin by letting
Dn = {x1, ...,xn} denote a set of n labelled prototypes, and x′ ∈ Dn be the prototype
nearest to a test point x. Then the nearest-neighbor rule for classifying x is to assign
it the label associated with x′. The nearest-neighbor rule is a sub-optimal procedure;
its use will usually lead to an error rate greater than the minimum possible, the Bayes
rate. We shall see, however, that with an unlimited number of prototypes the error
rate is never worse than twice the Bayes rate.
Before we get immersed in details, let us try to gain a heuristic understanding of
why the nearest-neighbor rule should work so well. To begin with, note that the label
θ′ associated with the nearest neighbor is a random variable, and the probability
that θ′ = ωi is merely the a posteriori probability P (ωi|x′). When the number of
samples is very large, it is reasonable to assume that x′ is sufficiently close to x that
P (ω|x′) � P (ωi|x). Since this is exactly the probability that nature will be in state
ωi, the nearest-neighbor rule is effectively matching probabilities with nature.
22 CHAPTER 4. NONPARAMETRIC TECHNIQUES
If we define ωm(x) by
P (ωm|x) = max
i
P (ωi|x), (35)
then the Bayes decision rule always selects ωm. This rule allows us to partition the
feature space into cells consisting of all points closer to a given training point x′ than
to any other training points. All points in such a cell are thus labelled by the category
of the training point — a so-called Voronoi tesselation of the space (Fig. 4.13).Voronoi
tesselation
Figure 4.13: In two dimensions, the nearest-neighbor algorithm leads to a partitioning
of the input space into Voronoi cells, each labelled by the category of the training point
it contains. In three dimensions, the cells are three-dimensional, and the decision
boundary resembles the surface of a crystal.
When P (ωm|x) is close to unity, the nearest-neighbor selection is almost always
the same as the Bayes selection. That is, when the minimum probability of error
is small, the nearest-neighbor probability of error is also small. When P (ωm|x) is
close to 1/c, so that all classes are essentially equally likely, the selections made by
the nearest-neighbor rule and the Bayes decision rule are rarely the same, but the
probability of error is approximately 1 − 1/c for both. While more careful analysis
is clearly necessary, these observations should make the good performance of the
nearest-neighbor rule less surprising.
Our analysis of the behavior of the nearest-neighbor rule will be directed at ob-
taining the infinite-sample conditional average probability of error P (e|x), where the
averaging is with respect to the training samples. The unconditional average proba-
bility of error will then be found by averaging P (e|x) over all x:
P (e) =
∫
P (e|x)p(x) dx. (36)
In passing we should recall that the Bayes decision rule minimizes P (e) by minimizing
P (e|x) for every x. Recall from Chap. ?? that if we let P ∗(e|x) be the minimum
possible value of P (e|x), and P ∗ be the minimum possible value of P (e), then
P ∗(e|x) = 1− P (ωm|x) (37)
4.5. THE NEAREST-NEIGHBOR RULE 23
and
P ∗ =
∫
P ∗(e|x)p(x) dx. (38)
4.5.1 Convergence of the Nearest Neighbor
We now wish to evaluate the average probability of error for the nearest-neighbor
rule. In particular, if Pn(e) is the n-sample error rate, and if
P = lim
n→∞
Pn(e), (39)
then we want to show that
P ∗ ≤ P ≤ P ∗
(
2− c
c− 1
P ∗
)
. (40)
We begin by observing that when the nearest-neighbor rule is used with a par-
ticular set of n samples, the resulting error rate will depend on the accidental char-
acteristics of the samples. In particular, if different sets of n samples are used to
classify x, different vectors x′ will be obtained for the nearest-neighbor of x. Since
the decision rule depends on this nearest-neighbor, we have a conditional probability
of error P (e|x,x′) that depends on both x and x′. By averaging over x′, we obtain
P (e|x) =
∫
P (e|x,x′)p(x′|x) dx′. (41)
where we understand that there is an implicit dependence upon the number n of
training points.
It is usually very difficult to obtain an exact expression for the conditional density
p(x′|x). However, since x′ is by definition the nearest-neighbor of x, we expect this
density to be very peaked in the immediate vicinity of x, and very small elsewhere.
Furthermore, as n goes to infinity we expect p(x′|x) to approach a delta function
centered at x, making the evaluation of Eq. 41 trivial. To show that this is indeed the
case, we must assume that at the given x, p(·) is continuous and not equal to zero.
Under these conditions, the probability that any sample falls within a hypersphere S
centered about x is some positive number Ps:
Ps =
∫
x′∈S
p(x′) dx′. (42)
Thus, the probability that all n of the independently drawn samples fall outside
this hypersphere is (1 − Ps)n, which approaches zero as n goes to infinity. Thus x′
converges to x in probability, and p(x′|x) approaches a delta function, as expected. In
fact, by using measure theoretic methods one can make even stronger (as well as more
rigorous) statements about the convergence of x′ to x, but this result is sufficient for
our purposes.
4.5.2 Error Rate for the Nearest-Neighbor Rule
We now turn to the calculation of the conditional probability of error Pn(e|x,x′).
To avoid a potential source of confusion, we must state the problem with somewhat
24 CHAPTER 4. NONPARAMETRIC TECHNIQUES
greater care than has been exercised so far. When we say that we have n inde-
pendently drawn labelled samples, we are talking about n pairs of random variables
(x1, θ1), (x2, θ2), ..., (xn, θn), where θj may be any of the c states of nature ω1, ..., ωc.
We assume that these pairs were generated by selecting a state of nature ωj for θj with
probability P (ωj) and then selecting an xj according to the probability law p(x|ωj),
with each pair being selected independently. Suppose that during classification nature
selects a pair (x, θ), and that x′j , labelled θ
′
j , is the training sample nearest x. Since
the state of nature when x′j was drawn is independent of the state of nature when x
is drawn, we have
P (θ, θ′j |x,x′j) = P (θ|x)P (θ′j |x′j). (43)
Now if we use the nearest-neighbor decision rule, we commit an error whenever θ = θ′j .
Thus, the conditional probability of error Pn(e|x,x′j) is given by
Pn(e|x,x′j) = 1−
c∑
i=1
P (θ = ωi, θ
′ = ωi|x,x′j)
= 1−
c∑
i=1
P (ωi|x)P (ωi|x′j). (44)
To obtain Pn(e) we must substitute this expression into Eq. 41 for Pn(e|x) and
then average the result over x. This is very difficult, in general, but as we remarked
earlier the integration called for in Eq. 41 becomes trivial as n goes to infinity and
p(x′|x) approaches a delta function. If P (ωi|x) is continuous at x, we thus obtain
lim
n→∞
Pn(e|x) =
∫ [
1−
c∑
i=1
P (ωi|x)P (ωi|x′)
]
δ(x′ − x) dx′
= 1−
c∑
i=1
P 2(ωi|x). (45)
Therefore, provided we can exchange some limits and integrals, the asymptotic nearest-
neighbor error rate is given by
P = lim
n→∞
Pn(e)
= lim
n→∞
∫
Pn(e|x)p(x) dx
=
∫ [
1−
c∑
i=1
P 2(ωi|x)
]
p(x) dx. (46)
4.5.3 Error Bounds
While Eq. 46 presents an exact result, it is more illuminating to obtain bounds on P in
terms of the Bayes rate P ∗. An obvious lower bound on P is P ∗ itself. Furthermore,
it can be shown that for any P ∗ there is a set of conditional and prior probabilities
for which the bound is achieved, so in this sense it is a tight lower bound.
4.5. THE NEAREST-NEIGHBOR RULE 25
The problem of establishing a tight upper bound is more interesting. The basis
for hoping for a low upper bound comes from observing that if the Bayes rate is
low, P (ωi|x) is near 1.0 for some i, say i = m. Thus the integrand in Eq. 46 is
approximately 1− P 2(ωm|x) � 2(1− P (ωm|x)), and since
P ∗(e|x) = 1− P (ωm|x), (47)
integration over x might yield about twice the Bayes rate, which is still low and
acceptable for some applications. To obtain an exact upper bound, we must find out
how large the nearest-neighbor error rate P can become for a given Bayes rate P ∗.
Thus, Eq. 46 leads us to ask how small
c∑
i=1
P 2(ωi|x) can be for a given P (ωm|x). First
we write
c∑
i=1
P 2(ωi|x) = P 2(ωm|x) +
∑
i
=m
P 2(ωi|x), (48)
and then seek to bound this sum by minimizing the second term subject to the
following constraints:
• P (ωi|x) ≥ 0
•
∑
i
=m
P (ωi|x) = 1− P (ωm|x) = P ∗(e|x).
With a little thought we see that
c∑
i=1
P 2(ωi|x) is minimized if all of the a posteriori
probabilities except the mth are equal. The second constraint yields
P (ωi|x) =
{
P∗(e|x)
c−a i = m
1− P ∗(e|x) i = m.
(49)
Thus we have the inequalities
c∑
i=1
P 2(ωi|x) ≥ (1− P ∗(e|x))2 +
P ∗2(e|x)
c− 1
(50)
and
1−
c∑
i=1
P 2(ωi|x) ≤ 2P ∗(e|x)−
c
c− 1
P ∗2(e|x). (51)
This immediately shows that P ≤ 2P ∗, since we can substitute this result in
Eq. 46 and merely drop the second term. However, a tighter bound can be obtained
by observing that the variance is:
Var[P ∗(e|x)] =
∫
[P ∗(e|x)− P ∗]2p(x) dx
=
∫
P ∗2(e|x)p(x) dx− P ∗2 ≥ 0,
so that
26 CHAPTER 4. NONPARAMETRIC TECHNIQUES
∫
P ∗2(e|x)p(x) dx ≥ P ∗2, (52)
with equality holding if and only if the variance of P ∗(e|x) is zero. Using this result
and substituting Eq. 51 into Eq. 46, we obtain the desired bounds on the nearest-
neighbor error P in the case of an infinite number of samples:
P ∗ ≤ P ≤ P ∗
(
2− c
c− 1
P ∗
)
. (53)
It is easy to show that this upper bound is achieved in the so-called zero-information
case in which the densities p(x|ωi) are identical, so that P (ωi|x) = P (ωi) and further-
more P ∗(e|x) is independent of x (Problem 17). Thus the bounds given by Eq. 53 are
as tight as possible, in the sense that for any P ∗ there exist conditional and a priori
probabilities for which the bounds are achieved. In particular, the Bayes rate P ∗ can
be anywhere between 0 and (c− 1)/c and the bounds meet at the two extreme values
for the probabilities. When the Bayes rate is small, the upper bound is approximately
twice the Bayes rate (Fig. 4.14).
___c-1
c
___c-1
c
P*
P
Figure 4.14: Bounds on the nearest-neighbor error rate P in a c-category problem
given infinite training data, where P ∗ is the Bayes error (Eq. 53). At low error rates,
the nearest-neighbor error rate is bounded above by twice the Bayes rate.
Since P is always less than or equal to 2P ∗, if one had an infinite collection of data
and used an arbitrarily complicated decision rule, one could at most cut the error rate
in half. In this sense, at least half of the classification information in an infinite data
set resides in the nearest neighbor.
It is natural to ask how well the nearest-neighbor rule works in the finite-sample
case, and how rapidly the performance converges to the asymptotic value. Unfor-
tunately, despite prolonged effort on such problems, the only statements that can
be made in the general case are negative. It can be shown that convergence can
be arbitrarily slow, and the error rate Pn(e) need not even decrease monotonically
with n. As with other nonparametric methods, it is difficult to obtain anything other
than asymptotic results without making further assumptions about the underlying
probability structure (Problems 13 & 14).
4.5.4 The k-Nearest-Neighbor Rule
An obvious extension of the nearest-neighbor rule is the k-nearest-neighbor rule. As
one would expect from the name, this rule classifies x by assigning it the label most
frequently represented among the k nearest samples; in other words, a decision is made
4.5. THE NEAREST-NEIGHBOR RULE 27
by examining the labels on the k nearest neighbors and taking a vote (Fig. 4.15). We
shall not go into a thorough analysis of the k-nearest-neighbor rule. However, by
considering the two-class case with k odd (to avoid ties), we can gain some additional
insight into these procedures.
x
x1
x2
Figure 4.15: The k-nearest-neighbor query starts at the test point and grows a spher-
ical region until it encloses k training samples, and labels the test point by a majority
vote of these samples. In this k = 5 case, the test point x would be labelled the
category of the black points.
The basic motivation for considering the k-nearest-neighbor rule rests on our ear-
lier observation about matching probabilities with nature. We notice first that if
k is fixed and the number n of samples is allowed to approach infinity, then all of
the k nearest neighbors will converge to x. Hence, as in the single-nearest-neighbor
cases, the labels on each of the k-nearest-neighbors are random variables, which in-
dependently assume the values ωi with probabilities P (ωi|x), i = 1, 2. If P (ωm|x)
is the larger a posteriori probability, then the Bayes decision rule always selects ωm.
The single-nearest-neighbor rule selects ωm with probability P (ωm|x). The k-nearest-
neighbor rule selects ωm if a majority of the k nearest neighbors are labeled ωm, an
event of probability
k∑
i=(k+1)/2
(
k
i
)
P (ωm|x)i[1− P (ωm|x)]k−i. (54)
In general, the larger the value of k, the greater the probability that ωm will be
selected.
We could analyze the k-nearest-neighbor rule in much the same way that we
analyzed the single-nearest-neighbor rule. However, since the arguments become more
involved and supply little additional insight, we shall content ourselves with stating
the results. It can be shown that if k is odd, the large-sample two-class error rate for
the k-nearest-neighbor rule is bounded above by the function Ck(P ∗), where Ck(P ∗)
is defined to be the smallest concave function of P ∗ greater than
(k−1)/2∑
i=0
(
k
i
) [
(P ∗)i+1(1− P ∗)k−i + (P ∗)k−i(1− P ∗)i+1
]
. (55)
28 CHAPTER 4. NONPARAMETRIC TECHNIQUES
Here the summation over the first bracketed term represents the probability of error
due to i points coming from the category having the minimum probability and k−i > i
points from the other category. The summation over the second term in the brackets
is the probability that k − i points are from the minimum-probability category and
i + 1 < k − i from the higher probability category. Both of these cases constitute
errors under the k-nearest-neighbor decision rule, and thus we must add them to find
the full probability of error (Problem 18).
Figure 4.16 shows the bounds on the k-nearest-neighbor error rates for several
values of k. As k increases, the upper bounds get progressively closer to the lower
bound — the Bayes rate. In the limit as k goes to infinity, the two bounds meet and
the k-nearest-neighbor rule becomes optimal.
0 0.1 0.2 0.3 0.4
0.1
0.2
0.3
0.4
Ba
yes
Ra
te
P*
P
1
3
5
9
15
0.5
Figure 4.16: The error-rate for the k-nearest-neighbor rule for a two-category problem
is bounded by Ck(P ∗) in Eq. 55. Each curve is labelled by k; when k = ∞, the
estimated probabilities match the true probabilities and thus the error rate is equal
to the Bayes rate, i.e., P = P ∗.
At the risk of sounding repetitive, we conclude by commenting once again on the
finite-sample situation encountered in practice. The k-nearest-neighbor rule can be
viewed as another attempt to estimate the a posteriori probabilities P (ωi|x) from
samples. We want to use a large value of k to obtain a reliable estimate. On the
other hand, we want all of the k nearest neighbors x′ to be very near x to be sure
that P (ωi|x′) is approximately the same as P (ωi|x). This forces us to choose a
compromise k that is a small fraction of the number of samples. It is only in the limit
as n goes to infinity that we can be assured of the nearly optimal behavior of the
k-nearest-neighbor rule.
4.5.5 Computational Complexity of the k–Nearest-Neighbor
Rule
The computational complexity of the nearest-neighbor algorithm — both in space
(storage of prototypes) and time (search) — has received a great deal of analy-
sis. There are a number of elegant theorems from computational geometry on the
construction of Voronoi tesselations and nearest-neighbor searches in one- and two-
dimensional spaces. However, because the greatest use of nearest-neighbor techniques
is for problems with many features, we concentrate on the more general d-dimensional
case.
Suppose we have n labelled training samples in d dimensions, and seek to find
the closest to a test point x (k = 1). In the most naive approach we inspect each
stored point in turn, calculate its Euclidean distance to x, retaining the identity only
of the current closest one. Each distance calculation is O(d), and thus this search
4.5. THE NEAREST-NEIGHBOR RULE 29
OR
ω1
1 2 d
. . .
1 2 d
. . .
1 2 d
. . .
AND
. . .
1 2 d
. . .
1 2 d
. . .
1 2 d
. . .
AND
. . .
1 2 d
. . .
1 2 d
. . .
1 2 d
. . .
AND
. . .
A
B
C
D
E
ω1
Figure 4.17: A parallel nearest-neighbor circuit can perform search in constant —
i.e., O(1) — time. The d-dimensional test pattern x is presented to each box, which
calculates which side of a cell’s face x lies on. If it is on the “close” side of every face
of a cell, it lies in the Voronoi cell of the stored pattern, and receives its label.
is O(dn2). An alternative but straightforward parallel implementation is shown in
Fig. 4.17, which is O(1) in time and O(n) in space.
There are three general algorithmic techniques for reducing the computational
burden in nearest-neighbor searches: computing partial distances, prestructuring, and
editing the stored prototypes. In partial distance, we calculate the distance using some partial
distancesubset r of the full d dimensions, and if this partial distance is too great we do not
compute further. The partial distance based on r selected dimensions is
Dr(a,b) =
(
r∑
k=1
(ak − bk)2
)1/2
(56)
where r < d. Intuitively speaking, partial distance methods assume that what we
know about the distance in a subspace is indicative of the full space. Of course, the
partial distance is strictly non-decreasing as we add the contributions from more and
more dimensions. Consequently, we can confidently terminate a distance calculation
to any prototype once its partial distance is greater than the full r = d Euclidean
distance to the current closest prototype.
In presturcturing we create some form of search tree in which prototypes are selec- search
treetively linked. During classification, we compute the distance of the test point to one
or a few stored “entry” or “root” prototypes and then consider only the prototypes
linked to it. Of these, we find the one that is closest to the test point, and recursively
consider only subsequent linked prototypes. If the tree is properly structured, we will
reduce the total number of prototypes that need to be searched.
Consider a trivial illustration of prestructuring in which we store a large number
of prototypes that happen to be distributed uniformly in the unit square, i.e., p(x) ∼
30 CHAPTER 4. NONPARAMETRIC TECHNIQUES
U
((
0
0
)
,
(
1
1
))
. Imagine we prestructure this set using four entry or root prototypes —
at
(
1/4
1/4
)
,
(
1/4
3/4
)
,
(
3/4
1/4
)
and
(
3/4
3/4
)
— each fully linked only to points in its corresponding
quadrant. When a test pattern x is presented, the closest of these four prototypes
is determined, and then the search is limited to the prototypes in the corresponding
quadrant. In this way, 3/4 of the prototypes need never be queried.
Note that in this method we are no longer guaranteed to find the closest prototype.
For instance, suppose the test point is near a boundary of the quadrants, e.g., x =(
0.499
0.499
)
. In this particular case only prototypes in the first quadrant will be searched.
Note however that the closest prototype might actually be in one of the other three
quadrants, somewhere near
(
0.5
0.5
)
. This illustrates a very general property in pattern
recognition: the tradeoff of search complexity against accuracy.
More sophisticated search trees will have each stored prototype linked to a small
number of others, and a full analysis of these methods would take us far afield. Nev-
ertheless, here too, so long as we do not query all training prototypes, we are not
guaranteed that the nearest prototype will be found.
The third method for reducing the complexity of nearest-neighbor search is to
eliminate “useless” prototypes during training, a technique known variously as editing,editing
pruning or condensing. A simple method to reduce the O(n) space complexity is to
eliminate prototypes that are surrounded by training points of the same category
label. This leaves the decision boundaries — and hence the error — unchanged, while
reducing recall times. A simple editing algorithm is as follows.
Algorithm 3 (Nearest-neighbor editing)
1 begin initialize j = 0,D = data set, n = #prototypes
2 construct the full Voronoi diagram of D
3 do j ← j + 1; for each prototype x′j
4 Find the Voronoi neighbors of x′j
5 if any neighbor is not from the same class as x′j then mark x
′
j
6 until j = n
7 Discard all points that are not marked
8 Construct the Voronoi diagram of the remaining (marked) prototypes
9 end
The complexity of this editing algorithm is O(d3n�d/2
lnn), where here the “floor”
operation (�·�) implies �d/2� = k if d is even, and 2k − 1 if d is odd (Problem 10).
According to Algorithm 3, if a prototype contributes to a decision boundary (i.e.,
at least one of its neighbors is from a different category), then it remains in the set;
otherwise it is edited away (Problem 15). This algorithm does not guarantee that the
minimal set of points is found (Problem 16), nevertheless, it is one of the examples in
pattern recognition in which the computational complexity can be reduced — some-
times significantly — without affecting the accuracy. One drawback of such pruned
nearest neighbor systems is that one generally cannot add training data later, since
the pruning step requires knowledge of all the training data ahead of time (Computer
exercise ??). We conclude this section by noting the obvious, i.e., that we can com-
bine these three complexity reduction methods. We might first edit the prototypes,
then form a search tree during training, and finally compute partial distances during
classification.
4.6. *METRICS AND NEAREST-NEIGHBOR CLASSIFICATION 31
4.6 Metrics and Nearest-Neighbor Classification
The nearest-neighbor classifier relies on a metric or “distance” function between pat-
terns. While so far we have assumed the Euclidean metric in d dimensions, the notion
of a metric is far more general, and we now turn to the use alternate measures of
distance to address key problems in classification. First let us review the properties of
a metric. A metric D(·, ·) is merely a function that gives a generalized scalar distance
between two argument patterns. A metric must have four properties: for all vectors
a, b and c
non-negativity: D(a,b) ≥ 0
reflexivity: D(a,b) = 0 if and only if a = b
symmetry: D(a,b) = D(b,a)
triangle inequality: D(a,b) +D(b, c) ≥ D(a, c).
It is easy to verify that if the Euclidean formula for distance in d dimensions,
D(a,b) =
(
d∑
k=1
(ak − bk)2
)1/2
, (57)
obeys the properties of metric. Moreover, if each coordinate is multiplied by an
arbitrary constant, the resulting space also obeys a metric (Problem 19), though it
can lead to problems in nearest-neighbor classifiers (Fig. 4.18).
x1
x2 x2
αx1
x x
Figure 4.18: Even if each coordinate is scaled by some constant, the resulting space
still obeys the properties of a metric. However, a nearest-neighbor classifier would
have different results depending upon such rescaling. Consider the test point x and
its nearest neighbor. In the original space (left), the black prototype is closest. In
the figure at the right, the x1 axis has been rescaled by a factor 1/3; now the nearest
prototype is the red one. If there is a large disparity in the ranges of the full data in
each dimension, a common procedure is to rescale all the data to equalize such ranges,
and this is equivalent to changing the metric in the original space.
One general class of metrics for d-dimensional patterns is the Minkowski metric Minkowsi
metric
Lk(a,b) =
(
d∑
i=1
|ai − bi|k
)1/k
, (58)
also referred to as the Lk norm (Problem 20); thus, the Euclidean distance is the L2
norm. The L1 norm is sometimes called the Manhattan or city block distance, the Manhattan
distanceshortest path between a and b, each segment of which is parallel to a coordinate axis.
32 CHAPTER 4. NONPARAMETRIC TECHNIQUES
(The name derives from the fact that the streets of Manhattan run north-south and
east-west.) Suppose we compute the distances between the projections of a and b
onto each of the d coordinate axes. The L∞ distance between a and b corresponds
to the maximum of these projected distances (Fig. 4.19).
1
4
2
∞
0,0,0
1,0,0
0,1,0
1,1,1
Figure 4.19: Each colored surface consists of points a distance 1.0 from the origin,
measured using different values for k in the Minkowski metric (k is printed in red).
Thus the white surfaces correspond to the L1 norm (Manhattan distance), light gray
the L2 norm (Euclidean distance), dark gray the L4 norm, and red the L∞ norm.
The Tanimoto metric finds most use in taxonomy, where the distance between twoTanimoto
metric sets is defined as
DTanimoto(S1,S2) =
n1 + n2 − 2n12
n1 + n2 − n12
, (59)
where n1 and n2 are the number of elements in sets S1 and S2, respectively, and n12 is
the number that is in both sets. The Tanimoto metric finds greatest use for problems
in which two patterns or features — the elements in the set — are either the same or
different, and there is no natural notion of graded similarity (Problem 27).
The selection among these or other metrics is generally dictated by computational
concerns, and it is hard to base a choice on prior knowledge about the distributions.
One exception is when there is great difference in the range of the data along different
axes in a multidmensional data. Here, we should scale the data — or equivalently
alter the metric — as suggested in Fig. 4.18.
4.6.1 Tangent distance
There may be drawbacks inherent in the uncritical use of a particular metric in
nearest-neighbor classifiers, and these drawbacks can be overcome by the careful use
of more general measures of distance. On crucial such problem is that of invariance.
Consider a 100-dimensional pattern x′ representing a 10× 10 pixel grayscale image of
a handwritten 5. Consider too the Euclidean distance from x′ to the pattern repre-
senting an image that is shifted horizontally but otherwise identical (Fig. 4.20). Even
if the relative shift is a mere three pixels, the Euclidean distance grows very large —
4.6. *METRICS AND NEAREST-NEIGHBOR CLASSIFICATION 33
much greater than the distance to an unshifted 8. Clearly the Euclidean metric is of
little use in a nearest-neighbor classifier that must be insensitive to such translations.
Likewise, other transformations, such as overall rotation or scale of the image,
would not be well accommodated by Euclidean distance in this manner. Such draw-
backs are especially pronounced if we demand that our classifier be simultaneously
invariant to several transformations, such as horizontal translation, vertical transla-
tion, overall scale, rotation, line thickness, shear, and so on (Computer exercise 7 &
8). While we could preprocess the images by shifting their centers to coalign, then
have the same bounding box, and so forth, such an approach has its own difficulties,
such as sensitivity to outlying pixels or to noise. We explore here alternatives to such
preprocessing.
1 2 3 4 5
2.58
x8 x' x'(s=3)
D(x,x(s))
D(x',x8)
s
Figure 4.20: The uncritical use of Euclidean metric cannot address the problem of
translation invariance. Pattern x′ represents a handwritten 5, and x′(s = 3) the same
shape but shifted three pixels to the right. The Euclidean distance D(x′,x′(s = 3)) is
much larger than D(x′,x8), where x8 represents the handwritten 8. Nearest-neighbor
classification based on the Euclidean distance in this way leads to very large errors.
Instead, we seek a distance measure that would be insensitive to such translations, or
indeed other known invariances, such as scale or rotation.
Ideally, during classification we would like to first transform the patterns to be
as similar to one another and only then compute their similarity, for instance by
the Euclidean distance. Alas, the computational complexity of such transformations
make this ideal unattainable. Merely rotating a k× k image by a known amount and
interpolating to a new grid is O(k2). But of course we do not know the proper rotation
angle ahead of time and must search through several values, each value requiring a
distance calculation to test the whether the optimal setting has been found. If we must
search for the optimal set of parameters for several transformations for each stored
prototype during classification, the computational burden is prohibitive (Problem 25).
The general approach in tangent distance classifiers is to use a novel measure of
distance and a linear approximation to the arbitrary transforms. Suppose we believe
there are r transformations applicable to our problem, such as horizontal translation,
vertical translation, shear, rotation, scale, and line thinning. During construction of
34 CHAPTER 4. NONPARAMETRIC TECHNIQUES
the classifier we take each stored prototype x′ and perform each of the transformations
Fi(x′; αi) on it. Thus Fi(x′; αi) could represent the image described by x′, rotated
by a small angle αi. We then construct a tangent vector TVi for each transformation:tangent
vector
TVi = Fi(x′; αi)− x′. (60)
While such a transformation may be compute intensive — as, for instance, the line
thinning transform — it need be done only once, during training when computational
constraints are lax. In this way we construct for each prototype x′ an r × d matrix
T, consisting of the tangent vectors at x′. (Such vectors can be orthonormalized, but
we need assume here only that they are linearly independent.) It should be clear, too
that this method will not work with binary images, since they lack a proper notion
of derivative. If the data are binary, then, it is traditional to blur the images before
creating a tangent distance based classifier.
Each point in the subspace spanned by the r tangent vectors passing through
x′ represents the linearized approximation to the full combination of transforms, as
shown in Fig. 4.21. During classification we search for the point in the tangent space
that is closest to a test point x — the linear approximation to our ideal. As we shall
see, this search can be quite fast.
Now we turn to computing the tangent distance from a test point x to a particular
stored prototype x′. Formally, given a matrix T consisting of the r tangent vectors
at x′, the tangent distance from x′ to x is
Dtan(x
′,x) = min
a
[‖(x′ + Ta)− x‖], (61)
i.e., the Euclidean distance from x to the tangent space of x′. Equation 61 describes
the so-called “one-sided” tangent distance, because only one pattern, x′, is trans-
formed. The two-sided tangent distance allows both x and x′ to be transformed but
improves the accuracy only slightly at a large added computational burden (Prob-
lem 23); for this reason we shall concentrate on the one-sided version.
During classification of x we will find its tangent distance to x′ by finding the
optimizing value of a required by Eq. 61. This minimization is actually quite simple,
since the argument is a paraboloid as a function of a, as shown in pink in Fig. 4.22.
We find the optimal a via iterative gradient descent. For gradient descent we need
the derivative of the (squared) Euclidean distance. The Euclidean distance in Eq. 61
obeys
D2(x′ + Ta,x) = ‖(x′ + Ta)− x‖2, (62)
and we compute the gradient with respect to the vector of parameters a — the pro-
jections onto the tangent vectors — as
∇aD2(x′ + Ta,x) = 2Tt(x′ + Ta− x). (63)
Thus we can start with an arbitrary a and take a step in the direction of the negative
gradient, updating our parameter vector as
a(t+ 1) = a(t)− ηTt(Ta(t) + x′ − x), (64)
where η is the scalar step size controlling the rate of convergence. So long as the step
is not too large, we will reduce the squared Euclidean distance. When the minimum
of such Euclidean distance is found, we have our tangent distance (Eq. 61). The
4.6. *METRICS AND NEAREST-NEIGHBOR CLASSIFICATION 35
0 281 0 694
641 660 924 1283
0 973 1535 1856
1628 1767 2122 2373
TV2
(thinning)
TV1
(rotation)
0 0.5 1.5 a1
a2
0
0.5
1.5
prototype
Figure 4.21: The pixel image of the handwritten 5 prototype at the lower left was
subjected to two transformations, rotation, and line thinning, to obtain the tangent
vectors TV1 and TV2; images corresponding to these tangent vectors are shown out-
side the axes. Each of the 16 images within the axes represents the prototype plus
linear combination of the two tangent vectors with coefficients a1 and a2. The small
red number in each image is the Euclidean distance between the tangent approxi-
mation and the image generated by the unapproximated transformations. Of course,
this Euclidean distance is 0 for the prototype and for the cases a1 = 1, a2 = 0 and
a1 = 0, a2 = 1. (The patterns generated with a1 + a2 > 1 have a gray background
because of automatic grayscale conversion of images with negative pixel values.)
optimal a can also be found by standard matrix methods, but these generally have
higher computational complexities, as is explored in Problems 21 & 22. We note that
the methods for editing and prestructuring data sets described in Sec. 4.5.5 can be
applied to tangent distance classifers too.
Nearest-neighbor classifiers using tangent distance have been shown to be highly
accurate, but they require the designer to know which invariances and to be able to
perform them on each prototype. Some of the insights from tangent approach can
also be used for learning which invariances underly the training data — a topic we
shall revisit in Chap. ??.
36 CHAPTER 4. NONPARAMETRIC TECHNIQUES
x1
x2
x3
x ‘
TV1
TV
2 tange
nt spa
ce
x 2
x 1
Ta
D
tan (x’, x 2 )
Figure 4.22: A stored prototype x′, if transformed by combinations of two basic
transformations, would fall somewhere on a complicated curved surface in the full
d-dimensional space (gray). The tangent space at x′ is an r-dimensional Euclidean
space, spanned by the tangent vectors (here TV1 and TV2). The tangent distance
Dtan(x′,x) is the smallest Euclidean distance from x to the tangent space of x′, shown
in the solid red lines for two points, x1 and x2. Thus although the Euclidean distance
from x′ to x1 is less than to x2, for the tangent distance the situation is reversed. The
Euclidean distance from x2 to the tangent space of x′ is a quadratic function of the
parameter vector a, as shown by the pink paraboloid. Thus simple gradient descent
methods can find the optimal vector a and hence the tangent distance Dtan(x′,x2).
4.7 Fuzzy Classification
Occassionally we may have informal knowledge about a problem domain where we
seek to build a classifier. For instance, we might feel, generally speaking, that an
adult salmon is oblong and light in color, while a sea bass is stouter and dark. The
approach taken in fuzzy classification is to create so-called “fuzzy category member-
ships functions,” which convert an objectively measurable parameter into a subjective
“category memberships,” which are then used for classification. We must stress im-
mediately that the term “categories” used by fuzzy practitioners refers not to the final
class as we have been discussing, but instead just overlapping ranges of feature values.
For instance, if we consider the feature value of lightness, fuzzy practitioners might
split this into five “categories” — dark, medium-dark, medium, medium-light and
light. In order to avoid misunderstandings, we shall use quotations when discussing
such “categories.”
For example we might have the lightness and shape of a fish be judged as in
Fig. 4.23. Next we need a way to convert an objective measurement in several features
into a category decision about the fish, and for this we need a merging or conjunction
rule — a way to take the “category memberships” (e.g., lightness and shape) andconjunction
rule yield a number to be used for making the final decision. Here fuzzy practitioners have
4.7. *FUZZY CLASSIFICATION 37
1
x
Figure 4.23: “Category membership” functions, derived from the designer’s prior
knowledge, together with a lead to discriminants. In this figure x might represent an
objectively measureable value such as the reflectivity of a fish’s skin. The designer
believes there are four relevant ranges, which might be called dark, medium-dark,
medium-light and light. Note, the memberships are not in true categories we wish to
classify, but instead merely ranges of feature values.
at their disposal a large number of possible functions. Indeed, most functions can be
used and there are few principled criteria to preference one over another. One guiding
principle that is often invoked is that that in the extreme cases the membership
functions have value 0 or 1, the conjunction reduces to standard predicate logic;
likewise, symmetry in the arguments is virtually always assumed. Nevertheless, there
are no strong principled reasons to impose these conditions, nor are they sufficient to
determine the “categories.”
Suppose the designer feels that the final category based on lightness ahd shape can
be described as medium-light and oblong. While the heuristic category membership
function (µ(·)) converts the objective measurements to two “category memberships,”
we now need a conjunction rule to transform the component “membership values” conjunction
ruleinto a discriminant function. There are many ways to do this, but the most popular
is
1−Min[µx(x), µy(y)]. (65)
and the obvious extension if there are more then two features.
It must be emphasized that fuzzy techniques are completely and thoroughly sub-
sumed by the general notion of discriminant function discussed in Chap. ?? (Prob-
lem 29).
4.7.1 Are Fuzzy Category Memberships just Probabilities?
Even before the introduction of fuzzy methods and category membership functions,
the statistics, pattern recognition and even mathematical philosophy communities ar-
gued a great deal over the fundamental nature of probability. Some questioned the
applicability of the concept to single, non-repeatable events, feeling that statements
about a single event — what was the probability of rain on Tuesday? — were mean-
ingless. Such discussion made it quite clear that “probability” need not apply only
to repeatable events. Instead, since the first half of the 20th century, probability has
been used as the logic of reasonable inference — work that highlighted the notion of
subjective probability. Moreover, pattern recognition practitioners had happily used
38 CHAPTER 4. NONPARAMETRIC TECHNIQUES
1
x1
x
2
1 x1
x2
Figure 4.24: “Category membership” functions and a conjunction rule based on the
designer’s prior knowledge lead to discriminant functions. Here x1 and x2 are objec-
tively measurable feature values. The designer believes that a particular class can be
described as the conjunction of two “category memberships,” here shown bold. Here
the conjunction rule of Eq. 65 is used to give the discriminant function. The resulting
discriminant function for the final category is indicated by the grayscale in the middle:
the greater the discriminant, the darker. The designer constructs discriminant func-
tions for other categories in a similar way (possibly also using disjunctions). During
classification, the maximum discriminant function is chosen.
discriminant functions without concern over whether they represented probabilities,
subjective probabilities, approximations to frequencies, or other fundamental entities.
While a full analysis of these topics would lead us away from our development of
pattern recognition techniques, it pays to consider the claims of fuzzy logic proponents,
since in order to be a good pattern recognition practitioner, we must understand what
is or is not afforded by any technique. Proponents of fuzzy logic are adamant that
category membership functions do not represent probabilities — subjective or not.
Fuzzy practitioners point to examples such as when a half teaspoon of sugar is placed
in a cup of tea, and conclude that the “membership” in the category sweet is 0.5, and
that it would be incorrect to state that the probability the tea was sweet was 50%.
But this situation be viewed simply as some sweetness feature value is 0.5, and there
is some discriminant function, whose arguments include this feature value. One need
not entertain xxx
Rather than debate the fundamental nature of probability, we should really be
concerned with the nature of inference, i.e., how we take measurements and infer a
category. Cox’s axioms — sometimes called CoxJaynesaxioms−−− are
1. If P (a|d) > P (b|d) and P (b|d) > P (c|d) then P (a|d) > P (c|d). That is, degrees
of belief have a natural ordering, given by real numbers.
2. P (not a|d) = F1[P (a|d)]. That is, the degree of belief that a proposition is not
the case is some function of the degree of belief that it is the case. Note that
4.8. *RELAXATION METHODS 39
such degrees of belief are graded values.
3. P (a, b|d) = F2[P (a|d), P (b|a, d)]
The first axiom states merely that the probability of not having proposition b
given a, is some function F1 of the probability of b given a. The second, though not
as evident, is
From these two, along with classical inference, we get the laws of probability. Any
consistent inference method is formally equivalent to standard probabilistic inference.
In spite of the arguments on such foundational issues, many practitioners are happy
to use fuzzy logic feeling that “whatever works” should be part of their repertoire. It
is important, therefore, to understand the methodological strengths and limitations
of the method. The limitations are formidable:
• Fuzzy methods are of very limited use in high dimensions or on complex prob-
lems. Pure fuzzy methods contribute little or nothing to problems with dozens
or hundreds of features, and where there is training data.
• The amount of information the designer can be expected to bring to a problem
is quite limited — the number, positions and widths of “category memberships.”
• Because of their lack of normalization, pure fuzzy methods are poorly suited to
problems in which there is a changing cost matrix λij (Computer exercise 9).
• Pure fuzzy methods do not make use of training data. When such pure fuzzy
methods (as outlined above) have unacceptable performance, it has been tradi-
tional to try to graft on adaptive (e.g., “neuro-fuzzy”) methods.
If there is a contribution of fuzzy approaches to pattern recognition, it would lie
in giving the steps by which one takes knowledge in a linguistic form and casts it
into discriminant functions. It is unlikely that the verbal knowledge could extend to
problems with dozens — much less hundreds — of features, the domain of the majority
of real-world pattern recognition problems. A severe limitation of pure fuzzy methods
is they do not rely on data, and when unsatisfactory results on problems of moderate
size, it has been traditional to try to use neural or other adaptive techniques to
compensate. At best, these are equivalent to maximum likelihood methods.
4.8 Relaxation methods
We have seen how the Parzen-window method uses a fixed window throughout the
feature space, and that this could lead to difficulties: in some regions a small window
width was appropriate while elsewhere a large one would be best. The k-nearest-
neighbor method addressed this problem by adjusting the region based on the density
of the points. Informally speaking, an approach that is intermediate between these
two is to adjust the size of the window during training according to the distance to the
nearest point of a different category. This is the method of some relaxation techniques.
(The term “relaxation” refers to the underlying mathematical techniques for setting
the parameters; we will consider only such relaxation issues, and concentrate instead
on their effects.)
The simplest method is that of potential functions — which merely consists of an potential
functioninterpolation function. The difference with Parzen windows is that the magnitude
40 CHAPTER 4. NONPARAMETRIC TECHNIQUES
of each is adjusted so as to properly classify the training data. One representative
method — called the reduced coulomb energy or RCE network — has the form shown reduced
coulomb
energy
in Fig. 4.25, which has the same topology as a Probabilistic neural network (Fig. 4.9).
1 2 3 n
1 2 c
x1 x2 xd
input
pattern
category
λ1 λ2 λ3 λn
Figure 4.25: An RCE network is topologically equivalent to the PNN of Fig. 4.9. Dur-
ing training the wghts are adjusted to have the same values as the pattern presented,
just as in a PNN. However, pattern units in an RCE network also have a modifiable
“radius” parameter λ. During training, each λ is adjusted so that the region is as
large as possible without containing training patterns from a different category.
The primary difference is that in an RCE network each pattern unit has an ad-
justable parameter that corresponds to the radius of the d-dimensional sphere. During
training, each radius is adjusted so that each pattern unit covers a region as large as
possible without containing a training point from another category.
Algorithm 4 (RCE training)
1 begin initialize j = 0, n = #patterns, 1 = small param, λm = max radius
2 do j ← j + 1
3 train weight: wjk ← xk
4 find nearest pt not in ωi: x̂← arg min
x/∈ωi
D(x,x′)
5 set radius: λj ←Min[D(x̂,x′)− 1, λm]
6 if x ∈ ωi then aic ← 1
7 until j = n
8 end
There are several subtleties that we need not consider right here. For instance, if
the radius of a pattern unit becomes too small (i.e., less than some threshold λmin),
then it indicates that different categories are highly overlapping. In that case, the
pattern unit is called a “probabilistic” unit, and so marked.
During classification, a test point is classified by the label of any point is by
presenting the unit, getting activation. If probabilistic units overlap, Any region that
is overlapped is considered ambiguous. Such ambiguous regions can be useful, since
the teacher can be queried as to the identity of points in that region. If we continue
to let λj be the radius around stored prototype x′j and now let Dt be the set of stored
prototypes in whose hypershperes test point x lies, then our classification algorithm
is written as:
Algorithm 5 (RCE classification)
4.8. *RELAXATION METHODS 41
1 begin initialize j = 0, k = 0,x = test pattern,Dt = {}
2 do j ← j + 1
3 if D(x,x′j) < λj then Dt ← Dt ∪ x′j
4 until j = n
5 if cat of all x′j ∈ Dt is the same then return label of all xk ∈ Dt
6 else return “ambiguous” label
7 end
λm
301510
4 5 6
1 2 3
Figure 4.26: During training, each pattern has a parameter — equivalent to a radius
in the d-dimensional space — that is adjusted to be as large as possible, without
enclosing any points from a different category. As new patterns are presented, each
such radius is decreased accordingly (and can never increase). In this way, each
pattern unit can enclose several prototypes, but only those having the same category
label. The number of points is shown in each component figure. The figure at the
bottom shows the final complicated decision regions, colored by category.
42 CHAPTER 4. NONPARAMETRIC TECHNIQUES
4.9 Approximations by Series Expansions
The nonparametric methods described thus far suffer from the requirement that in
general all of the samples must be stored or that the designer have extensive knowledge
of the problem. Since a large number of samples is needed to obtain good estimates,
the memory requirements can be severe. In addition, considerable computation time
may be required each time one of the methods is used to estimate p(x) or classify a
new x.
In certain circumstances the Parzen-window procedure can be modified to reduce
these problems considerably. The basic idea is to approximate the window function
by a finite series expansion that is acceptably accurate in the region of interest. If
we are fortunate and can find two sets of functions ψj(x) and χj(x) that allow the
expansion
ϕ
(x− xi
hn
)
=
m∑
j=1
ajψj(x)χj(xi), (66)
then we can split the dependence upon x and xi as
n∑
i=1
ϕ
(x− xi
hn
)
=
m∑
j=1
ajψj(x)
n∑
i=1
χj(xi). (67)
Then from Eq. 11 we have
pn(x) =
m∑
j=1
bjψj(x), (68)
where
bj =
aj
nVn
n∑
i=1
χj(xi). (69)
If a sufficiently accurate expansion can be obtained with a reasonable value for
m, this approach has some obvious advantages. The information in the n samples
is reduced to the m coefficients bj . If additional samples are obtained, Eq. 69 for
bj can be updated easily, and the number of coefficients remains unchanged. If the
functions ψj(·) and χj(·) are polynomial functions of the components of x and xi,
the expression for the estimate pn(x) is also a polynomial, which can be computed
relatively efficiently. Furthermore, use of this estimate p(x|ωi)P (ωi) leads to a simple
way of obtaining polynomial discriminant functions.polynomial
discriminant Before becoming too enthusiastic, however, we should note one of the problems
with this approach. A key property of a useful window function is its tendency
to peak at the origin and fade away elsewhere. Thus ϕ((x − xi)/hn) should peak
sharply at x = xi, and contribute little to the approximation of pn(x) for x far from
xi. Unfortunately, polynomials have the annoying property of becoming unbounded.
Thus, in a polynomial expansion we might find the terms associated with an xi far
from x contributing most (rather than least) to the expansion. It is quite important,
therefore, to be sure that the expansion of each windown function is in fact accurate
in the region of interest, and this may well require a large number of terms.
There are many types of series expansions one might consider. Readers familiar
with integral equations will naturally interpret Eq. 66 as an expansion of the kernel
4.9. *APPROXIMATIONS BY SERIES EXPANSIONS 43
ϕ(x,xi) in a series of eigenfunctions. (In analogy with eigenvectors and eigenvalues,eigen-
function eigenfunctions are solutions to certain differential equations with fixed real-number
coefficients.) Rather than computing eigenfunctions, one might choose any reasonable
set of functions orthogonal over the region of interest and obtain a least-squares fit
to the window function. We shall take an even more straightforward approach and
expand the window function in a Taylor series. For simplicity, we confine our attention
to a one-dimensional example using a Gaussian window function:
√
π ϕ(u) = e−u
2
�
m−1∑
j=0
(−1)j u
2j
j!
.
This expansion is most accurate near u = 0, and is in error by less than u2m/m!. If
we substitute u = (x− xi)/h, we obtain a polynomial of degree 2(m− 1) in x and xi.
For example, if m = 2 the window function can be approximated as
√
πϕ
(x− xi
h
)
� 1−
(x− xi
h
)2
= 1 +
2
h2
x xi −
1
h2
x2 − 1
h2
x2i ,
and thus
√
πpn(x) =
1
nh
n∑
i=1
√
πϕ
(x− xi
h
)
� b0 + b1x+ b2x2, (70)
where the coefficients are
b0 =
1
h
− 1
h3
1
n
n∑
i=1
x2i
b1 =
2
h3
1
n
n∑
i=1
xi
b2 = −
1
h3
.
This simple expansion condenses the information in n samples into the values,
b0, b1, and b2. It is accurate if the largest value of |x − xi| is not greater than h.
Unfortunately, this restricts us to a very wide window that is not capable of much
resolution. By taking more terms we can use a narrower window. If we let r be the
largest value of |x − xi| and use the fact that the error is the m-term expansion of√
π ϕ((x− xi)/h) is less than (r/h)2mm!, then using Stirling’s approximation for m!
we find that the error in approximating pn(x) is less than
1√
πh
(
r/h
)2m
m!
� 1√
πh
√
2πm
[( e
m
) ( r
h
)2]m
. (71)
Thus, the error becomes small only when m > e(r/h)2. This implies the need for
many terms if the window size h is small relative to the distance r from x to the most
44 CHAPTER 4. NONPARAMETRIC TECHNIQUES
distant sample. Although this example is rudimentary, similar considerations arise
in the multidimensional case even when more sophisticated expansions are used, and
the procedure is most attractive when the window size is relatively large.
4.10 Fisher Linear Discriminant
One of the recurring problems encountered in applying statistical techniques to pat-
tern recognition problems has been called the “curse of dimensionality.” Procedures
that are analytically or computationally manageable in low-dimensional spaces can be-
come completely impractical in a space of 50 or 100 dimensions. Pure fuzzy methods
are particularly ill-suited to such high-dimensional problems since it is implausible
that the designer’s linguistic intuition extends to such spaces. Thus, various tech-
niques have been developed for reducing the dimensionality of the feature space in
the hope of obtaining a more manageable problem.
We can reduce the dimensionality from d dimensions to one dimension if we merely
project the d-dimensional data onto a line. Of course, even if the samples formed
well-separated, compact clusters in d-space, projection onto an arbitrary line will
usually produce a confused mixture of samples from all of the classes, and thus poor
recognition performance. However, by moving the line around, we might be able to
find an orientation for which the projected samples are well separated. This is exactly
the goal of classical discriminant analysis.
Suppose that we have a set of n d-dimensional samples x1, …,xn, n1 in the subset
D1 labelled ω1 and n2 in the subset D2 labelled ω2. If we form a linear combination
of the components of x, we obtain the scalar dot product
y = wtx (72)
and a corresponding set of n samples y1, …, yn divided into the subsets Y1 and Y2.
Geometrically, if ‖w‖ = 1, each yi is the projection of the corresponding xi onto a
line in the direction of w. Actually, the magnitude of w is of no real significance,
since it merely scales y. The direction of w is important, however. If we imagine
that the samples labelled ω1 fall more or less into one cluster while those labelled ω2
fall in another, we want the projections falling onto the line to be well separated, not
thoroughly intermingled. Figure 4.27 illustrates the effect of choosing two different
values for w for a two-dimensional example. It should be abundantly clear that if the
original distributions are multimodal and highly overlapping, even the “best” w is
unlikely to provide adequate seaparation, and thus this method will be of little use.
We now turn to the matter of finding the best such direction w, one we hope will
enable accurate classification. A measure of the separation between the projected
points is the difference of the sample means. If mi is the d-dimensional sample mean
given by
mi =
1
ni
∑
x∈Di
x, (73)
then the sample mean for the projected points is given by
m̃i =
1
ni
∑
y∈Yi
y
4.10. *FISHER LINEAR DISCRIMINANT 45
0.5 1 1.5
0.5
1
1.5
2
0.5 1 1.5
x1
-0.5
0.5
1
1.5
2
x2
w
w
x1
x2
Figure 4.27: Projection of samples onto two different lines. The figure on the right
shows greater separation between the red and black projected points.
=
1
ni
∑
y∈Yi
wtx = wtmi. (74)
and is simply the projection of mi.
It follows that the distance between the projected means is
|m̃1 − m̃2| = |wt(m1 −m2)|, (75)
and that we can make this difference as large as we wish merely by scaling w. Of
course, to obtain good separation of the projected data we really want the difference
between the means to be large relative to some measure of the standard deviations for
each class. Rather than forming sample variances, we define the scatter for projected scatter
samples labelled ωi by
s̃2i =
∑
y∈Yi
(y − m̃i)2. (76)
Thus, (1/n)(s̃21 + s̃
2
2) is an estimate of the variance of the pooled data, and s̃
2
1 + s̃
2
2
is called the total within-class scatter of the projected samples. The Fisher linear within-
class
scatter
discriminant employs that linear function wtx for which the criterion function
J(w) =
|m̃1 − m̃2|2
s̃21 + s̃
2
2
(77)
is maximum (and independent of ‖w‖). While the w maximizing J(·) leads to the
best separation between the two projected sets (in the sense just described), we will
also need a threshold criterion before we have a true classifier. We first consider how
to find the optimal w, and later turn to the issue of thresholds.
To obtain J(·) as an explicit function of w, we define the scatter matrices Si and scatter
matricesSW by
Si =
∑
x∈Di
(x−mi)(x−mi)t (78)
and
46 CHAPTER 4. NONPARAMETRIC TECHNIQUES
SW = S1 + S2. (79)
Then we can write
s̃2i =
∑
x∈Di
(wtx−wtmi)2
=
∑
x∈Di
wt(x−mi)(x−mi)tw
= wtSiw; (80)
therefore the sum of these scatters can be written
s̃21 + s̃
2
2 = w
tSWw. (81)
Similarly, the separations of the projected means obeys
(m̃1 − m̃2)2 = (wtm1 −wtm2)2
= wt(m1 −m2)(m1 −m2)tw
= wtSBw, (82)
where
SB = (m1 −m2)(m1 −m2)t. (83)
We call SW the within-class scatter matrix. It is proportional to the sample co-within-
class
scatter
variance matrix for the pooled d-dimensional data. It is symmetric and positive
semidefinite, and is usually nonsingular if n > d. Likewise, SB is called the between-
class scatter matrix. It is also symmetric and positive semidefinite, but because it is
between-
class
scatter
the outer product of two vectors, its rank is at most one. In particular, for any w,
SBw is in the direction of m1 −m2, and SB is quite singular.
In terms of SB and SW , the criterion function J(·) can be written as
J(w) =
wtSBw
wtSWw
. (84)
This expression is well known in mathematical physics as the generalized Rayleigh
quotient. It is easy to show that a vector w that maximizes J(·) must satisfy
SBw = λSWw, (85)
for some constant λ, which is a generalized eigenvalue problem (Problem 36). This
can also be seen informally by noting that at an extremum of J(w) a small change in
w in Eq. 84 should leave unchanged the ratio of the numerator to the denominator.
If SW is nonsingular we can obtain a conventional eigenvalue problem by writing
S−1W SBw = λw. (86)
In our particular case, it is unnecessary to solve for the eigenvalues and eigenvectors
of S−1W SB due to the fact that SBw is always in the direction of m1 −m2. Since the
4.11. *MULTIPLE DISCRIMINANT ANALYSIS 47
scale factor for w is immaterial, we can immediately write the solution for the w that
optimizes J(·):
w = S−1W (m1 −m2). (87)
Thus, we have obtained w for Fisher’s linear discriminant — the linear function
yielding the maximum ratio of between-class scatter to within-class scatter. (The
solution w given by Eq. 87 is sometimes called the canonical variate.) Thus the
classification has been converted from a d-dimensional problem to a hopefully more
manageable one-dimensional one. This mapping is many-to-one, and in theory can not
possibly reduce the minimum achievable error rate if we have a very large training set.
In general, one is willing to sacrifice some of the theoretically attainable performance
for the advantages of working in one dimension. All that remains is to find the
threshold, i.e., the point along the one-dimensional subspace separating the projected
points.
When the conditional densities p(x|ωi) are multivariate normal with equal co-
variance matrices Σ, we can calculate the threshold directly. In that case we recall
(Chap. ??, Sect. ??) that the optimal decision boundary has the equation
wtx + w0 = 0 (88)
where
w = Σ−1(µ1 − µ2), (89)
and where w0 is a constant involving w and the prior probabilities. If we use sample
means and the sample covariance matrix to estimate µi and Σ, we obtain a vector
in the same direction as the w of Eq. 89 that maximized J(·). Thus, for the normal,
equal-covariance case, the optimal decision rule is merely to decide ω1 if Fisher’s linear
discriminant exceed some threshold, and to decide ω2 otherwise. More generally, if
we smooth the projected data, or fit it with a univariate Gaussian, we then should
choose w0 where the posteriors in the one dimensional distributions are equal.
The computational complexity of finding the optimal w for the Fisher linear dis-
criminant (Eq. 87) is dominated by the calculation of the within-category total scatter
and its inverse, an O(d2n) calculation.
4.11 Multiple Discriminant Analysis
For the c-class problem, the natural generalization of Fisher’s linear discriminant
involves c − 1 discriminant functions. Thus, the projection is from a d-dimensional
space to a (c − 1)-dimensional space, and it is tacitly assumed that d ≥ c. The
generalization for the within-class scatter matrix is obvious:
SW =
c∑
i=1
Si (90)
where, as before,
Si =
∑
x∈Di
(x−mi)(x−mi)t (91)
and
48 CHAPTER 4. NONPARAMETRIC TECHNIQUES
mi =
1
ni
∑
x∈Di
x. (92)
The proper generalization for SB is not quite so obvious. Suppose that we define
a total mean vector m and a total scatter matrix ST bytotal
mean
vector
total
scatter
matrix
m =
1
n
∑
x
x =
1
n
c∑
i=1
nimi (93)
and
ST =
∑
x
(x−m)(x−m)t. (94)
Then it follows that
ST =
c∑
i=1
∑
x∈Di
(x−mi + mi −m)(x−mi + mi −m)t
=
c∑
i=1
∑
x∈Di
(x−mi)(x−mi)t +
c∑
i=1
∑
x∈Di
(mi −m)(mi −m)t
= SW +
c∑
i=1
ni(mi −m)(mi −m)t. (95)
It is natural to define this second term as a general between-class scatter matrix,
so that the total scatter is the sum of the within-class scatter and the between-class
scatter:
SB =
c∑
i=1
ni(mi −m)(mi −m)t (96)
and
ST = SW + SB . (97)
If we check the two-class case, we find that the resulting between-class scatter matrix
is n1n2/n times our previous definition.∗
The projection from a d-dimensional space to a (c − 1)-dimensional space is ac-
complished by c− 1 discriminant functions
yi = w
t
ix i = 1, …, c− 1. (98)
If the yi are viewed as components of a vector y and the weight vectors wi are viewed
as the columns of a d-by-(c − 1) matrix W, then the projection can be written as a
single matrix equation
y = Wtx. (99)
∗ We could redefine SB for the two-class case to obtain complete consistency, but there should be
no misunderstanding of our usage.
4.11. *MULTIPLE DISCRIMINANT ANALYSIS 49
The samples x1, …,xn project to a corresponding set of samples y1, …,yn, which
can be described by their own mean vectors and scatter matrices. Thus, if we define
m̃i =
1
ni
∑
y∈Yi
y (100)
m̃ =
1
n
c∑
i=1
nim̃i (101)
S̃W =
c∑
i=1
∑
y∈Yi
(y − m̃i)(y − m̃i)t (102)
and
S̃B =
c∑
i=1
ni(m̃i − m̃)(m̃i − m̃)t, (103)
it is a straightforward matter to show that
S̃W = W
tSWW (104)
and
S̃B = W
tSBW. (105)
These equations show how the within-class and between-class scatter matrices are
transformed by the projection to the lower dimensional space (Fig. 4.28). What we
seek is a transformation matrix W that in some sense maximizes the ratio of the
between-class scatter to the within-class scatter. A simple scalar measure of scatter
is the determinant of the scatter matrix. The determinant is the product of the
eigenvalues, and hence is the product of the “variances” in the principal directions,
thereby measuring the square of the hyperellipsoidal scattering volume. Using this
measure, we obtain the criterion function
J(W) =
|S̃B |
|S̃W |
=
|WtSBW|
|WtSWW|
. (106)
The problem of finding a rectangular matrix W that maximizes J(·) is tricky,
though fortunately it turns out that the solution is relatively simple. The columns of
an optimal W are the generalized eigenvectors that correspond to the largest eigen-
values in
SBwi = λiSWwi. (107)
A few observations about this solution are in order. First, if SW is non-singular,
this can be converted to a conventional eigenvalue problem as before. However, this
is actually undesirable, since it requires an unnecessary computation of the inverse of
SW . Instead, one can find the eigenvalues as the roots of the characteristic polynomial
|SB − λiSW | = 0 (108)
and then solve
50 CHAPTER 4. NONPARAMETRIC TECHNIQUES
W1
W2
Figure 4.28: Three three-dimensional distributions are projected onto two-dimensional
subspaces, described by a normal vectors w1 and w2. Informally, multiple discrimi-
nant methods seek the optimum such subspace, i.e., the one with the greatest sepa-
ration of the projected distributions for a given total within-scatter matrix, here as
associated with w1.
(SB − λiSW )wi = 0 (109)
directly for the eigenvectors wi. Because SB is the sum of c matrices of rank one or
less, and because only c−1 of these are independent, SB is of rank c−1 or less. Thus,
no more than c − 1 of the eigenvalues are nonzero, and the desired weight vectors
correspond to these nonzero eigenvalues. If the within-class scatter is isotropic, the
eigenvectors are merely the eigenvectors of SB , and the eigenvectors with nonzero
eigenvalues span the space spanned by the vectors mi −m. In this special case the
columns of W can be found simply by applying the Gram-Schmidt orthonormalization
procedure to the c − 1 vectors mi −m, i = 1, …, c − 1. Finally, we observe that in
general the solution for W is not unique. The allowable transformations include
rotating and scaling the axes in various ways. These are all linear transformations
from a (c− 1)-dimensional space to a (c− 1)-dimensional space, however, and do not
change things in any significant way; in particular, they leave the criterion function
J(W) invariant and the classifier unchanged.
If we have very little data, we would tend to project to a subspace of low dimen-
sion, while if there is more data, we can use a higher dimension, as we shall explore
in Chap. ??. Once we have projected the distributions onto the optimal subspace
(defined as above), we can use the methods of Chapt. ?? to create our full classifier.
As in the two-class case, multiple discriminant analysis primarily provides a reason-
able way of reducing the dimensionality of the problem. Parametric or nonparametric
techniques that might not have been feasible in the original space may work well in
the lower-dimensional space. In particular, it may be possible to estimate separate
covariance matrices for each class and use the general multivariate normal assump-
tion after the transformation where this could not be done with the original data. In
general, if the transformation causes some unnecessary overlapping of the data and
increases the theoretically achievable error rate, then the problem of classifying the
data still remains. However, there are other ways to reduce the dimensionality of
4.11. SUMMARY 51
data, and we shall encounter this subject again in Chap. ??. We note that there are
also alternate methods of discriminant analysis — such as the selection of features
based on statistical sigificance — some of which are given in the references for this
chapter. Of these, Fisher’s method remains a fundamental and widely used technique.
Summary
There are two overarching approaches to non-parametric estimation for pattern clas-
sification: in one the densities are estimated (and then used for classification), in the
other the category is chosen directly. The former approach is exemplified by Parzen
windows and their hardware implementation, Probabilistic neural networks. The lat-
ter is exemplified by k-nearest-neighbor and several forms of relaxation networks. In
the limit of infinite training data, the nearest-neighbor error rate is bounded from
above by twice the Bayes error rate. The extemely high space complexity of the
nominal nearest-neighbor method can be reduced by editing (e.g., removing those
prototypes that are surrounded by prototypes of the same category), prestructuring
the data set for efficient search, or partial distance calculations. Novel distance mea-
sures, such as the tangent distance, can be used in the nearest-neighbor algorithm for
incorporating known tranformation invariances.
Fuzzy classification methods employ heuristic choices of “category membership”
and heuristic conjunction rules to obtain discriminant functions. Any benefit of such
techniques is limited to cases where there is very little (or no) training data, small
numbers of features, and when the knowledge can be gleaned from the designer’s prior
knowledge.
Relaxation methods such as potential functions create “basins of attraction” sur-
rounding training prototypes; when a test pattern lies in such a basin, the corre-
sponding prototype can be easily identified along with its category label. Reduced
coloumb energy networks are one in the class of such relaxation networks, the basins
are adjusted to be as large as possible yet not include prototypes from other categories.
The Fisher linear discriminant finds a good subspace in which categories are best
separated; other techniques can then be applied in the subspace. Fisher’s method
can be extended to cases with multiple categories projected onto subspaces of higher
dimension than a line.
Bibliographical and Historical Remarks
Parzen introduced his window method for estimating density functions [32], and its
use in regression was pioneered by Ndaraya and Watson [?, ?]. Its natural application
to classification problems stems from the work of Specht [39], including its PNN
hardware implementation [40].
Nearest-neighbor methods were first introduced by [16, 17], but it was over fifteen
years later that computer power had increased, thereby making it practical and re-
newing interest in its theoretical foundations. Cover and Hart’s foundational work
on asymptotic bounds [10] were expanded somewhat through the analysis of Devroye
[14]. The first pruning or editing work in [23] was followed by a number of related al-
gorithms, such as that described in [5, 3]. The k-nearest neighbor was explored in [33].
The computational complexity of nearest neighbor (Voronoi) is described in [35]; work
on search, as described in [27], has proven to be of greater use, in general. Much of
the work on reducing the computational complexity of nearest-neighbor search comes
52 CHAPTER 4. NONPARAMETRIC TECHNIQUES
from the vector quantization and compression community; for instance partial dis-
tance calculation are described in [21]. Friedman has an excellent analysis of some of
the unintuitive properties of high dimensional spaces, and indirectly nearest neighbor
classifiers, an inspiration for several problems here [19]. The definitive collection of
seminal papers in nearest-neighbor classification is [12].
The notion of tangent distance was introduced by Simard and colleagues [38], and
explored by a number of others [24]. Sperduti and Stork introduced a prestructuring
and novel search criterion which speeds search in tangent based classifiers [41]. The
greatest successes of tangent methods have been in optical character recognition, but
the method can be applied in other domains, so long as the invariances are known.
The study of general invariance has been most profitable when limited to a particular
domain, and readers seeking further background should consult [31] for computer
vision and [34] for speech. Background on image transformations is covered in [18].
The philosophical debate concerning frequency, probability, graded category mem-
bership, and so on, has a long history [29]. Keynes espoused a theory of probability as
the logic of probable inference, and did not need to rely on the notion of repeatability,
frequency, etc. We subscribe to the traditional view that probability is a conceptual
and formal relation between hypotheses and conclusions — here, specifically between
data and category. The limiting cases of such rational belief are certainty (on the
one hand), and impossibility (on the other). Classical theory of probability cannot be
based solely on classical logic, which has no formal notions for the probability of an
event. While the rules in Keynes’ probability [26] were taken as axiomatic, Cox [11]
and later Jayne[?] sought to place a formal underpinning.
Many years after these debates, “fuzzy” methods were proposed from the com-
puter science [43]. A formal equivalence of fuzzy category membership functions and
probability is given in [22], which in turn is based on Cox [11]. Cheeseman has made
some remarkably clear and forceful rebuttals to the assertions that fuzzy methods
represent something beyond the notion of subjective probability [7, 8]; representative
expositions to the contrary include [28, 4]. Readers unconcerned with foundational
issues, and whether fuzzy methods provide any representational power or other ben-
efits above standard probability (including subjective probability) can consult [25],
which is loaded with over 3000 references. , many connectives for fuzzy logic [2]
Early reference on the use of potential functions for pattern classification is [1, 6].
This is closely allied with later work such as the RCE network described in [37, 36].
Fisher’s early work on linear discriminants [15], is well described in [30] and a
number of standard textbooks [9, 13, 20, 30, 42].
Problems
⊕
Section 4.3
1. Show that Eqs. 19–22 are sufficient to assure convergence in Eqs. 17 & 18.
2. Consider a normal p(x) ∼ N(µ, σ2) and Parzen-window function ϕ(x) ∼ N(0, 1).
Show that the Parzen-window estimate
pn(x) =
1
nhn
n∑
i=1
ϕ
(
x− xi
hn
)
,
has the following properties:
4.11. PROBLEMS 53
(a) p̄n(x) ∼ N(µ, σ2 + h2n)
(b) Var[pn(x)] � 12nhn√πp(x)
(c) p(x)− p̄n(x) � 12
(
hn
σ
)2[
1−
(
x−µ
σ
)2]
p(x)
for small hn. (Note: if hn = h1/
√
n, this shows that the error due to bias goes to zero
as 1/n, whereas the standard deviation of the noise only goes to zero as 4
√
n.)
3. Let p(x) ∼ U(0, a) be uniform from 0 to a, and let a Parzen window be defined
as ϕ(x) = e−x for x > 0 and 0 for x ≤ 0.
(a) Show that the mean of such a Parzen-window estimate is given by
p̄n(x) =
0 x < 0 1 a (1− e−x/hn) 0 ≤ x ≤ a 1 a (ea/hn − 1)e−x/hn a ≤ x. (b) Plot p̄n(x) versus x for a = 1 and hn = 1, 1/4, and 1/16. (c) How small does hn have to be to have less than one percent bias over 99 percent of the range 0 < x < a? (d) Find hn for this condition if a = 1, and plot p̄n(x) in the range 0 ≤ x ≤ 0.05. 4. Suppose in a c-category supervised learning environment we sample the full distribution p(x), and train a PNN classifier according to Algorithm ??. (a) Show that even if there are unequal category priors and hence unequal numbers of points in each category, the recognition method gives the right solution. (b) Suppose we have trained a PNN with the assumption of equal category priors, but later wish to use it for a problem having the cost matrix λij , representing the cost of choosing category ωi when in fact the pattern came from ωj . How should we do this? (c) Suppose instead we know a cost matrix λij before training. How shall we train a PNN for minimum risk? ⊕ Section 4.4 5. Show that Eq. 31 converges in probability to p(x) given the conditions lim n→∞ kn → ∞ and lim n→∞ kn/n→ 0. 6. Let D = {x1, ...,xn} be a set of n independent labelled samples and let Dk(x) = {x′1, ...,x′k} be the k nearest neighbors of x. Recall that the k-nearest-neighbor rule for classifying x is to give x the label most frequently represented in Dk(x). Consider a two-category problem with P (ω1) = P (ω2) = 1/2. Assume further that the condi- tional densities p(x|ωi) are uniform within unit hyperspheres a distance of ten units apart. (a) Show that if k is odd the average probability of error is given by Pn(e) = 1 2n (k−1)/2∑ j=0 ( n j ) . 54 CHAPTER 4. NONPARAMETRIC TECHNIQUES (b) Show that for this case the single-nearest neighbor rule has a lower error rate than the k-nearest-neighbor error rate for k > 1.
(c) If k is allowed to increases with n but is restricted by k < a
√
n, show that
Pn(e)→ 0 as n→∞.
⊕
Section 4.5
7. Prove that the Voronoi cells induced by the single-nearest neighbor algorithm
must always be convex. That is, for any two points x1 and x2 in a cell, all points on
the line linking x1 and x2 must also lie in the cell.
8. It is easy to see that the nearest-neighbor error rate P can equal the Bayes rate
P ∗ if P ∗ = 0 (the best possibility) or if P ∗ = (c − 1)/c (the worst possibility). One
might ask whether or not there are problems for which P = P ∗ when P ∗ is between
these extremes.
(a) Show that the Bayes rate for the one-dimensional case where P (ωi) = 1/c and
P (x|ωi) =
1 0 ≤ x ≤ cr
c−1
1 i ≤ x ≤ i+ 1− cr
c−1
0 elsewhere
is P ∗ = r.
(b) Show that for this case that the nearest-neighbor rate is P = P ∗.
9. Consider the following set of two-dimensional vectors:
ω1 ω2 ω3
x1 x2 x1 x2 x1 x2
10 0 5 10 2 8
0 -10 0 5 -5 2
5 -2 5 5 10 -4
(a) Plot the decision boundary resulting from the nearest-neighbor rule just for
categorizing ω1 and ω2. Find the sample means m1 and m2 and on the same
figure sketch the decision boundary corresponding to classifying x by assigning
it to the category of the nearest sample mean.
(b) Repeat part (a) for categorizing only ω1 and ω3.
(c) Repeat part (a) for categorizing only ω2 and ω3.
(d) Repeat part (a) for a three-category classifier, classifying ω1, ω2 and ω3.
10. Prove that the computational complexity of the basic nearest-neighbor editing
algorith (Algorithm ??) for n points in d dimension is O(d3n�d/2
lnn).
11. To understand the “curse of dimensionality” in greater depth, consider the
effects of high dimensions on the simple nearest-neighbor algorithm. Suppose we
need to estimate a density function f(x) in the unit hypercube in Rd based on n
samples. If f(x) is complicated, we need dense samples to learn it well.
4.11. PROBLEMS 55
(a) Let n1 denote the number of samples in a “dense” sample in R1. What is the
sample size for the “same density” in Rd? If n1 = 100, what sample size is
needed in a 20-dimensional space?
(b) Show that the interpoint distances are all large and roughly equal in Rd, and
that neighborhoods that have even just a few points must have large radii.
(c) Find ld(p), the length of a hypercube edge in d dimensions that contains the
fraction p of points (0 ≤ p ≤ 1). To better appreciate the implications of your
result, calculate: l5(0.01), l5(0.1), l20(0.01), and l20(0.1).
(d) Show that nearly all points are close to an edge of the full space (e.g., the unit
hypercube in d dimensions). Do this by calculating the L∞ distance from one
point to the closest other point. This shows that nearly all points are closer to
an edge than to another training point. (Argue that L∞ is more favorable than
L2 distance, even though it is easier to calculate here.) The result shows that
most points are on or near the convex hull of training samples and that nearly
every point is an “outlier” with respects to all the others.
12. Show how the “curse of dimensionality” (Problem 11) can be “overcome” by
choosing or assuming that your model is of a particular sort. Suppose that we are
estimating a function of the form y = f(x) +N(0, σ2).
(a) Suppose the true function is linear, f(x) =
n∑
j=1
ajxj , and that the approximation
is f̂(x) =
n∑
j=1
âjxj . Of course, the fit coefficients are:
âj = arg min
aj
n∑
i=1
yi −
d∑
j=1
ajxij
2
,
for j = 1, . . . , d. Prove that E [f(x) − f̂(x)]2 = dσ2/n, i.e., that it increases
linearly with d, and not exponentially as the curse of dimensionality might
otherwise suggest.
(b) Generalize your result from part (a) to the case where a function is expressed
in a different basis set, i.e., f(x) =
n∑
i=1
aiBi(x) for some well-behaved basis set
Bi(x), and hence that the result does not depend on the fact that we have used
a linear basis.
13. Consider classifiers based on samples from the distributions
p(x|ω1) =
{
2x for 0 ≤ x ≤ 1
0 otherwise,
and
p(x|ω2) =
{
2− 2x for 0 ≤ x ≤ 1
0 otherwise.
(a) What is the Bayes decision rule and the Bayes classification error?
56 CHAPTER 4. NONPARAMETRIC TECHNIQUES
(b) Suppose we randomly select a single point from ω1 and a single point from ω2,
and create a nearest-neighbor classifier. Suppose too we select a test point from
one of the categories (ω1 for definiteness). Integrate to find the expected error
rate P1(e).
(c) Repeat with two training samples from each category and a single test point in
order to find P2(e).
(d) Generalize to find the arbitrary Pn(e).
(e) Compare lim
n→∞
Pn(e) with the Bayes error.
14. Repeat Problem 13 but with
p(x|ω1) =
{
3/2 for 0 ≤ x ≤ 2/3
0 otherwise,
and
p(x|ω2) =
{
3/2 for 1/3 ≤ x ≤ 1
0 otherwise.
15. Expand in greater detail Algorithm 3 and add a conditional branch that will
speed it. Assuming the data points come from c categories and there are, on average,
k Voronoi neighbors of any point x, on average how much faster will your improved
algorithm be?
16. Consider the simple nearest-neighbor editing algorithm (Algorithm 3).
(a) Show by counterexample that this algorithm does not yield the minimum set of
points. (Hint: consider a problem where the points from each of two-categories
are constrained to be on the intersections of a two-dimensional Cartesian grid.)
(b) Create a sequential editing algorithm, in which each point is considered in turn,
and retained or rejected before the next point is considered. Prove that your
algorithm does or does not depend upon the sequence the points are considered.
17. Consider classification problem where each of the c categories possesses the same
distribution as well as prior P (ωi) = 1/c. Prove that the upper bound in Eq. 53, i.e.,
P ≤ P ∗
(
2− c
c− 1
P ∗
)
,
is achieved in this “zero-information” case.
18. Derive Eq. 55.
⊕
Section 4.6
19. Consider the Euclidean metric in d dimensions:
D(a,b) =
√√√√ d∑
k=1
(ak − bk)2.
4.11. PROBLEMS 57
Suppose we rescale each axis by a fixed factor, i.e., let x′k = αkxk for real, non-zero
constants αk, k = 1, 2, ..., d. Prove that the resulting space is a metric space. Discuss
the import of this fact for standard nearest-neighbor classification methods.
20. Prove that the Minkowski metric indeed possesses the four properties required
of all metrics.
21. Consider a non-iterative method for finding the tangent distance between x′ and
x, given the matrix T consisting of the r (column) tangent vectors TVi at x′.
(a) As given in the text, take the gradient of the squared Euclidean distance in the
a parameter space to find an equation that must be solved for the optimal a.
(b) Solve your first derivative equation to find the optimizing a.
(c) Compute the second derivative of D2(·, ·) to prove that your solution must be
a minimum squared distance, and not a maximum or inflection point.
(d) If there are r tangent vectors (invariances) in a d-dimensional space, what is the
computational complexity of your method?
(e) In practice, the iterative method described in the text requires only a few
(roughly 5) iterations for problems in handwritten OCR. Compare the com-
plexities of your analytic solution to that of the iterative scheme.
22. Consider a tangent-distance based classifier based on n prototypes, each rep-
resenting a k × k pixel pattern of a handwritten character. Suppose there are r
invariances we believe characterize the problem. What is the storage requirements
(space complexity) of such a tangent-based classifier?
23. The two-sided tangent distance allows both the stored prototype x′ and the test
point x to be transformed. Thus if T is the matrix of the r tangent vectors for x′ and
S likewise at x, the two-sided tangent distance is
D2tan(x
′,x) = min
a,b
[‖(x′ + Ta)− (x + Sb)‖].
(a) Follow the logic in Problem 21 and calculate the gradient with respect to the a
parameter vector and to the b parameter vector.
(b) What are the two update rules for an iterative scheme analogous to Eq. 64?
(c) Prove that there is a unique minium as a function of a and b. Describe this
geometrically.
(d) In an iterative scheme, we would alternatively take steps in the a parameter
space, then the b parameter space. What is the computational complexity to
this approach to the two-sided tangent distance classifier?
(e) Why is the actual complexity for classification in a 2-sided tangent distance
classifier even more sever than your result in (d) would suggest?
24. Consider the two-sided tangent distance described in Problem 23. Suppose we
restrict ourselves to n prototypes x′ in d dimensions, each with an associated matrix
T of r tangent vectors, which we assume are linearly independent. Determine whether
the two-sided tangent distance does or does not satisfy each of the requirements of a
metric: non-negativity, reflexivity, symmetry and the triangle inequality.
58 CHAPTER 4. NONPARAMETRIC TECHNIQUES
25. Consider the computational complexity of nearest neighbor classifier for k ×
k pixel grayscale images of handwritten digits. Instead of using tangent distance,
we will search for the parameters of full nonlinear transforms before computing a
Euclidean distance. Suppose the number of operations needed to perform each of our
r transformations (e.g., rotation, line thinning, shear, and so forth) is aik2, where
for the sake of simplicity we assume ai � 10. Suppose too that for the test of each
prototype we must search though A � 5 such values, and judge it by the Euclidean
distance.
(a) Given a transformed image, how many operations are required to calculate the
Euclidean distance to a stored prototype?
(b) Find the number of operations required per search.
(c) Suppose there are n prototypes. How many operations are required to find the
nearest neighbor, given such transforms?
(d) Assume for simplicity that no complexity reduction methods have been used
(such as editing, partial distance, graph creation). If the number of prototypes
is n = 106 points, and there are r = 6 transformations, and basic operations on
our computer require 10−9 seconds, how long does it take to classify a single
point?
26. Explore the effect of r on the accuracy of nearest-neighbor search based on
partial distance. Assume we have a large number n of points randomly placed in a
d-dimensional hypercube. Suppose we have a test point x, also selected randomly
in the hypercume, and find its nearest neighbor. By definition, if we use the full
d-dimensional Euclidean distance, we are guaranteed to find its nearest neighbor.
Suppose though we use the partial distance
Dr(x,x
′) =
(
r∑
i=1
(xi − x′i)2
)1/2
.
(a) Plot the probability that a partial distance search finds the true closest neighbor
of an arbitrary point x as a function of r for fixed n (1 ≤ r ≤ d) for d = 10.
(b) Consider the effect of r on the accuracy of a nearest-neighbor classifier. Assume
we have have n/2 prototypes from each two categories in a hypercube of length 1
on a side. The density for each category is separable into the product of (linear)
ramp functions, highest at one side, and zero at the other side of the range.
Thus the density for category ω1 is highest at (0, 0, ...0)t and zero at (1, 1, ..., 1)t,
while the density for ω2 is highest at (1, 1, ..., 1)t and zero at (0, 0, ...0)t. State
by inspection the Bayesian decision boundary.
(c) Calculate the Bayes error rate.
(d) Calculate the probability of correct classification of a point x, randomly selected
from one of the category densities, as a function of r in a partial distance metric.
(e) If n = 10, what must r be for the partial distance nearest neighbor classifier to
be within 1% of the Bayes rate?
4.11. PROBLEMS 59
27. Consider the Tanimoto metric applied to sets having discrete elements.
(a) Determine whether the four properties of a metric are obeyed by DTanimoto(·, ·)
as given in Eq. 59.
(b) Consider the following six words as mere sets of unordered letters: pattern,
pat, pots, stop, taxonomy and elementary. Use the Tanimoto metric to rank
order all
(
6
2
)
= 30 possible pairings of these sets.
(c) Is the triangle inequality obeyed for these six patterns?
⊕
Section 4.7
28. Suppose someone asks you whether a cup of water is hot or cold, and you respond
that it is warm. Explain why this exchange in no way indicates that the membership
of the cup in some “hot” class is a graded value less than 1.0.
29. Consider the design a fuzzy classifier for three types of fish based on two features:
length and lightness. The designer feels that there are five ranges of length: short,
medium-short, medium, medium-large and large. Similarly, lightness falls into three
ranges: dark, medium and light. The designer uses the traingle function
T̂ (x; µi, δi) =
{
1− |x−µi|
δi
x ≤ |µi − δi|
0 otherwise.
for the intermediate values, and an open triangle function for the extremes, i.e.,
Ĉ(x, µi, δi) =
1 x > µi
1− x−µi
δi
µi − δi ≤ x ≤ µi
0 otherwise,
and its symmetric version.
Suppose we have for the length δi = 5 and µ1 = 5, µ2 = 7, µ3 = 9, µ4 = 11
and µ5 = 13, and for lightness δj = 30, µ1 = 30, µ2 = 50, and µ3 = 70. Suppose
the designer feels that ω1 = medium-light and long, ω2 = dark and short and ω3 =
medium dark and long, where the conjunction rule “and” is defined in Eq. 65.
(a) Write the algebraic form of the discriminant functions.
(b) If every “category membership function” were rescaled by a constant, would
classification change?
(c) Classify the pattern x = 7.5, 60.
(d) Suppose that instead we knew that pattern is ωx. Would we have any principled
way to know whether the error was due to the number of category membership
functions? their functional form? the conjunction rule?
⊕
Section 4.8
60 CHAPTER 4. NONPARAMETRIC TECHNIQUES
30. Suppose that through standard training of an RCE network (Algorithm 4), all
the radii have been reduced to values less than λm. Prove that there is no subset of
the training data that will yield the same category decision boundary.⊕
Section 4.9
31. Consider a window function ϕ(x) ∼ N(0, 1) and a density estimate
pn(x) =
1
nhn
n∑
i=1
ϕ
(x− xi
hn
)
.
Approximate this estimate by factoring the window function and expanding the factor
ex−xi/h
2
n in a Taylor series about the origin as follows:
(a) Show that in terms of the normalized variable u = x/hn the m-term approxi-
mation is given by
pnm(x) =
1√
2πhn
e−u
2/2
m−1∑
j=0
bju
j
where
bj =
1
n
n∑
i=1
1
j!
u
j
ie
−u2i /2.
(b) Suppose that the n samples happen to be extremely tightly clustered about
u = u0. Show that the two-term approximation peaks at the two points where
u2 + u/u0 − 1 = 0.
(c) Show that one peak occurs approximately at u = u0, as desired, if u0 � 1, but
that it moves only to u = 1 for u0 � 1.
(d) Confirm your answer to part (c) by plotting pn2(u) versus u for u0 = 0.01, 1,
and 10. (Note: you may need to rescale the graphs vertically.)
⊕
Section 4.10
32. Let px(x|ωi) be arbitrary densities with means µi and covariance matrices Σi
— not necessarily normal — for i = 1, 2. Let y = wtx be a projection, and let the
induced one-dimensional densities p(y|ωi) have means µi and variances σ2i .
(a) Show that the criterion function
J1(w) =
(µ1 − µ2)2
σ21 + σ
2
2
is maximized by
w = (Σ1 + Σ2)
−1(µ1 − µ2).
4.11. PROBLEMS 61
(b) If P (ωi) is the prior probability for ωi, show that
J2(w) =
(µ1 − µ2)2
P (ω1)σ21 + P (ω2)σ
2
2
and
w = [P (ω1)Σ1 + P (ω2)Σ2]
−1(µ1 − µ2).
(c) To which of these criterion functions is the J(w) of Eq. ?? more closely related?
Explain.
33. The expression
J1 =
1
n1n2
∑
yi∈Y1
∑
yj∈Y2
(yi − yj)2
clearly measures the total within-group scatter.
(a) Show that this within-group scatter can be written as
J1 = (m1 −m2)2 +
1
n1
s21 +
1
n2
s22.
(b) Show that the total scatter is
J2 =
1
n1
s21 +
1
n2
s22.
(c) If y = wtx, show that the w optimizing J1 subject to the constraint that J2 = 1
is given by
w = λ
( 1
n1
S1 +
1
n2
S2
)−1
(m1 −m2),
where
λ =
[
(m1 −m2)t
( 1
n1
S1 +
1
n2
S2
)
(m1 −m2)
]1/2
,
mi =
1
ni
∑
x∈Di
x,
and
Si =
∑
x∈Di
ni(mi −m)(mi −m)t.
34. If SB and SW are two real, symmetric, d-by-dmatrices, it is well known that there
exists a set of n eigenvalues λ1, …, λn satisfying |SB−λSW | = 0, and a corresponding
set of n eigenvectors e1, …, en satisfying SBei = λiSWei. Furthermore, if SW is
positive definite, the eigenvectors can always be normalized so that etiSWej = δij and
etiSBej = λiδij . Let S̃W = W
tSWW and S̃B = WtSBW, where W is a d-by-n
matrix whose columns correspond to n distinct eigenvectors.
62 CHAPTER 4. NONPARAMETRIC TECHNIQUES
(a) Show that S̃W is the n-by-n identity matrix I, and that S̃B is a diagonal ma-
trix whose elements are the corresponding eigenvalues. (This shows that the
discriminant functions in multiple discriminant analysis are uncorrelated.)
(b) What is the value of J = |S̃B |/|S̃W | ?
(c) Let y = Wtx be transformed by scaling the axes with a nonsingular n-by-n
diagonal matrix D and by rotating this result with an orthogonal matrix Q
where y′ = QDy. Show that J is invariant to this transformation.
35. Consider two normal distributions with arbitrary but equal covariances. Prove
that the Fisher linear discriminant, for suitable threshold, can be derived from the
negative of the log-likelihood ratio.
36. Consider the criterion function J(w) required for the Fisher linear discriminant.
(a) Fill in the steps leading from Eqs. 77, 79 & 83 to Eq. 84.
(b) Use matrix methods to show that the solution to Eq. 84 is indeed given by
Eq. 85.
(c) At the extreme of J(w), a small change in w must leave J(w) unchanged.
Consider a small perturbation away from the optimal, w + ∆w, and derive the
solution condition of Eq. 85.
⊕
Section 4.11
37. Consider multidiscriminant versions of Fisher’s method for the case of c Gaussian
distributions in d dimensions, each having the same covariance Σ (otherwise arbitrary)
but different means. Solve for the optimal subspace in terms of Σ and the d mean
vectors.
Computer exercises
Several exercises will make use of the following three-dimensional data sampled from
three categories, denoted ωi.
ω1 ω2 ω3
sample x1 x2 x3 x1 x2 x3 x1 x2 x3
1 0.28 1.31 -6.2 0.011 1.03 -0.21 1.36 2.17 0.14
2 0.07 0.58 -0.78 1.27 1.28 0.08 1.41 1.45 -0.38
3 1.54 2.01 -1.63 0.13 3.12 0.16 1.22 0.99 0.69
4 -0.44 1.18 -4.32 -0.21 1.23 -0.11 2.46 2.19 1.31
5 -0.81 0.21 5.73 -2.18 1.39 -0.19 0.68 0.79 0.87
6 1.52 3.16 2.77 0.34 1.96 -0.16 2.51 3.22 1.35
7 2.20 2.42 -0.19 -1.38 0.94 0.45 0.60 2.44 0.92
8 0.91 1.94 6.21 -0.12 0.82 0.17 0.64 0.13 0.97
9 0.65 1.93 4.38 -1.44 2.31 0.14 0.85 0.58 0.99
10 -0.26 0.82 -0.96 0.26 1.94 0.08 0.66 0.51 0.88
⊕
Section 4.2
1. Explore some of the properties of density estimation in the following way.
4.11. COMPUTER EXERCISES 63
(a) Write a program to generate points according to a uniform distribution in a unit
cube, −1/2 ≤ xi ≤ 1/2 for i = 1, 2, 3. Generate 104 such points.
(b) Write a program to estimate the density at the origin based on your 104 points as
a function of the size of a cubical window function of size h. Plot your estimate
as a function of h, for 0 < h ≤ 1.
(c) Evaluate the density at the origin using n of your points and the volume of a
cube window which just encloses n points. Plot your estimate as a function of
n = 1, ..., 104.
(d) Write a program to generate 104 points from a spherical Gaussian density (with
Σ = I) centered on the origin. Repeat (b) & (c) with your Gaussian data.
(e) Discuss any qualitative differences between the functional dependencies of your
estimation results for the uniform and Gaussian densities.
⊕
Section 4.3
2. Consider Parzen-window estimates and classifiers for points in the table above.
Let your window function be a spherical Gaussian, i.e.,
ϕ((x− xi)/h) ∝ Exp[−(x− xi)t(x− xi)/(2h2)].
(a) Write a program to classify an arbitrary test point x based on the Parzen window
estimates. Train your classifier using the three-dimensional data from your three
categories in the table above. Set h = 1 and classify the following three points:
(0.50, 1.0, 0.0)t, (0.31, 1.51,−0.50)t and (−0.3, 0.44,−0.1)t.
(b) Repeat with h = 0.1.
⊕
Section 4.4
3. Consider k-nearest-neighbor density estimations in different numbers of dimen-
sions
(a) Write a program to find the k-nearest-neighbor density for n (unordered) points
in one dimension. Use your program to plot such a density estimate for the x1
values in category ω3 in the table above for k = 1, 3 and 5.
(b) Write a program to find the k-nearest-neighbor density estimate for n points
in two dimensions. Use your program to plot such a density estimate for the
x1 − x2 values in ω2 for k = 1, 3 and 5.
(c) Write a program to form a k-nearest-neighbor classifier for the three-dimensional
data from the three categories in the table above. Use your program with k =
1, 3 and 5 to estimate the relative densities at the following points: (−0.41, 0.82, 0.88)t,
(0.14, 0.72, 4.1)t and (−0.81, 0.61,−0.38)t.
⊕
Section 4.5
4. Write a program to create a Voronoi tesselation in two dimensions as follows.
64 CHAPTER 4. NONPARAMETRIC TECHNIQUES
(a) First derive analytically the equation of a line separating two arbitrary points.
(b) Given the full data set D of prototypes and a particular point x ∈ D, write a
program to create a list of line segments comprising the Voronoi cell of x.
(c) Use your program to form the Voronoi tesselation of the x1 − x2 features from
the data of ω1 and ω3 in the table above. Plot your Voronoi diagram.
(d) Write a program to find the category decision boundary based on this full set
D.
(e) Implement a version of the pruning method described in Algorithm 3. Prune
your data set from (c) to form a condensed set.
(f) Apply your programs from (c) & (d) to form the Voronoi tesselation and bound-
ary for your condensed data set. Compare the decision boundaries you found
for the full and the condensed sets.
5. Explore the tradeoff between computational complexity (as it relates to par-
tial distance calculations) and search accuracy in nearest-neighbor classifiers in the
following exercise.
(a) Write a program to generate n prototypes from a uniform distributions in a
6-dimensional hypercube centered on the origin. Use your program to generate
106 points for category ω1, 106 different points for category ω2, and likewise for
ω3 and ω4. Denote this full set D.
(b) Use your program to generate a test set Dt of n = 100 points, also uniformly
distributed in the 6-dimensional hypercube.
(c) Write a program to implement the nearest-neighbor neighbor algorithm. Use
this program to label each of your points in Dt by the category of its nearest
neighbor in D. From now on we will assume that the labels you find are in fact
the true ones, and thus the “test error” is zero.
(d) Write a program to perform nearest-neighbor classification using partial dis-
tance, based on just the first r features of each vector. Suppose we define the
search accuracy as the percentage of points in Dt that are associated with their
particular closest prototype in D. (Thus for r = 6, this accuracy is 100%, by
construction.) For 1 ≤ r ≤ 6 in your partial distance classifier, estimate the
search accuracy. Plot a curve of this search accuracy versus r. What value of r
would give a 90% search accuracy? (Round r to the nearest integer.)
(e) Estimate the “wall clock time” — the overall time required by your computer
to perform the search — as a function of r. If T is the time for a full search
in six dimensions, what value of r requires roughly T/2? What is the search
accuracy in that case?
(f) Suppose instead we define search accuracy as the classification accuracy. Esti-
mate this classification accuracy for a partial distance nearest-neighbor classifier
using your points of Dt. Plot this accuracy for 1 ≤ r ≤ 6. Explain your result.
(g) Repeat (e) for this classification accuracy. If T is the time for full search in d
dimensions, what value of r requires roughly T/2? What is the classification
search accuracy in this case?
4.11. COMPUTER EXERCISES 65
⊕
Section 4.6
6. Consider nearest-neighbor classifiers employing different values of k in the Lk
norm or Minkowski metric.
(a) Write a program to implement a nearest-neighbor classifier for c categories, using
the Minkowski metric or Lk norm, where k can be selected at classification time.
(b) Use the three dimensional data in the table above to classify the following points
using the Lk norm for k = 1, 2, 4 and ∞: (2.21, 1.9, 0.43)t, (−0.15, 1.17, 6.19)t
and (0.01, 1.34, 2.60)t.
7. Create a 10 × 10 pixel grayscale pattern x′ of a handwritten 4.
(a) Plot the Euclidean distance between the 100-dimensional vectors corresponding
to x′ and a horizontally shifted version of it as a function of the horizontal offset.
(b) Shift x′ by two pixels to the right to form the tangent vector TV1. Write a
program to calculate the tangent distance for shifted patterns using your TV1.
Plot the tangent distance as a function of the displacement of the test pattern.
Compare your graphs and explain the implications.
8. Repeat Computer exercise 7 but for a handwritten 7, and vertical translations.⊕
Section 4.7
9. Assume that size, color and shape are appropriate descriptions of fruit, and
use fuzzy methods to classify fruit. In particular, assume all “category membership”
functions are either triangular (with center µ and full half-width δ) or, at the extremes,
are left- or right-open triangular functions.
Suppose the size features (measured in cm) are: small (µ = 2), medium (µ =
4), large (µ = 6), and extra-large (µ = 8). In all cases we assume the category
membership tions have δ = 3. Suppose shape is described by the excentricity, here
the ratio of the major axis to minor axis lengths: thin (µ = 2, δ = .6), oblong
(µ = 1.6, δ = .3), oval (µ = 1.4, δ = .2) and spherical (µ = 1.1, δ = .2). Suppose
color here is represented by some measure of the mixture of red to yellow: yellow
(µ = .1, δ = .1), yellow-orange (µ = 0.3, δ = 0.3), orange (µ = 0.5, δ = 0.3), orange-
red (µ = 0.7, δ = 0.3) and red (µ = 0.9, δ = 0.3). The fuzzy practitioner believes the
following are good descriptions of some common fruit:
• ω1 = cherry = {small and spherical and red}
• ω2 = orange = {medium and spherical and orange}
• ω3 = banana = {large and thin and yellow}
• ω4 = peach = {medium and spherical and orange-red}
• ω5 = plum = {medium and spherical and red}
• ω6 = lemon = {medium and oblong and yellow}
• ω7 = grapefruit = {medium and spherical and yellow}
(a) Write a program to take any objective pattern and classify it.
66 CHAPTER 4. NONPARAMETRIC TECHNIQUES
(b) Classify each of these {size, shape, color}: {2.5, 1.0, 0.95}, {7.5, 1.9, 0.2} and
{5.0, 0.5, 0.4}.
(c) Suppose there is a cost associated with classification, as described by a cost
matrix λij — the cost of selecting ωi given that the true category is ωj . Suppose
the cost matrix is
λij =
0 1 1 0 2 2 1
1 0 2 2 0 0 1
1 2 0 1 0 0 2
0 2 1 0 2 2 2
2 0 0 2 0 1 1
2 0 0 2 1 0 2
1 1 2 2 1 2 0
.
Reclassify the patterns in (b) for minimum cost.
⊕
Section 4.8
10. Explore relaxation networks in the following way.
(a) Write a program to implement an RCE classifier in three dimensions. Let the
starting radius be λm = 0.5. Train your classifier with the data from the three
categories in the table above. For this data, how many times was any sphere
reduced in size? (If the same sphere is reduced two times, count that as twice.)
(b) Use your classifier to classify the following: (0.53,−0.44, 1.1)t, (−0.49, 0.44, 1.11)t
and (0.51,−0.21, 2.15)t. If the classification of any point is ambiguous, state
which are the candidate categories.
⊕
Section 4.9
11. Consider a classifier based on a Taylor series expansion of a Gaussian window
function. Let k be the highest power of xi in a Taylor series expansion of each of the
independent features of a two-dimensional Gaussian. Below, consider just the x1−x2
features of categories ω2 and ω3 in the table above.
(a) For each value k = 2, 4, and 6, classify the following three points: (0.56, 2.3, 0.10)t,
(0.60, 5.1, 0.86)t and (−0.95, 1.3, 0.16)t.
⊕
Section 4.10
12. Consider the Fisher linear discriminant method.
(a) Write a general program to calculate the optimal direction w for a Fisher linear
discriminant based on three-dimensional data.
(b) Find the optimal w for categories ω2 and ω3 in the table above.
(c) Plot a line representing your optimal direction w and mark on it the positions
of the projected points.
4.11. COMPUTER EXERCISES 67
(d) In this subspace, fit each distribution with a (univariate) Gaussian, and find the
resulting decision boundary.
(e) What is the training error (the error on the training points themselves) in the
optimal subspace you found in (b)?
(f) For comparison, repeat (d) & (c) using instead the non-optimal direction w =
(1.0, 2.0,−1.5)t. What is the training error in this non-optimal subspace?
⊕
Section 4.11
13. Consider the multicategory generalization of the Fisher linear discriminant,
applied to the data in the table above.
(a) Write a general program to calculate the optimal w for multiple discriminant.
Use your program to find the optimal two-dimensional plane (described by nor-
mal vector w) for the three-dimensional data in the table.
(b) In the subspace, fit a circularly symmetric Gaussian to the data, and use a
simple linear classifier in each to find the decision boundaries in the subspace.
(c) What is the error on the training set?
(d) Classify following points : (1.40,−0.36,−0.41)t, (0.62, 1.30, 1.11)t and (−0.11, 1.60, 1.51)t.
(e) For comparison, repeat (b) & (c) for the non-optimal direction w = (−0.5,−0.5, 1.0)t.
Explain the difference between your training errors in the two cases.
68 CHAPTER 4. NONPARAMETRIC TECHNIQUES
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Index
L2 norm, 32
L∞ norm, 33
Lk norm, 32
k–nearest neighbor
Bayes relation, 23
k–nearest neighbor
Bayes relation, 22
k–nearest neighbor, 8, 19–22
accuracy
complexity tradeoff, 31
Bayes
nearest-neighbor relation, 23
between-class scatter, see scatter, between-
class
binomial distribution, see distribution,
binomial
category unit
PNN, 16
computational complexity
accuracy tradeoff, see accuracy, com-
putational complexity trade-
off
conjunction rule, 37
convolution
Parzen, 11
curse of dimensionality, 16
decision function
direct estimation, 5
density
estimation
convergence, 7
divergence, 7
nonparametric, 5
Parzen, 8–19
function
average, 6
multimodal, 5
space average, 7
unimodal, 5
dimensionality
curse of, see curse of dimensional-
ity
discriminant
multiple, 48–51
distance, see metric
city block, see distance, Manhat-
tan
tangent
two-sided, 58
distribution
binomial, 6
separable, 5
eigenfunction expansion, 43
eigenvalue
Fisher discriminant, 47
error
test, 15
training
Parzen window, 15
minimum
k–nearest neighbor
error, 22
estimation
Parzen, 8, 22
convergence, 9
illustration, 12
expansion
eigenfunction, 43
series, 44
Taylor series, 43
expected value
probability, 6
feature space
transform, 5
Fisher linear discriminant, see linear
discriminant, Fisher
hypercube, 8
72
INDEX 73
i.i.d., 6
inner product
PNN
pattern unit, 18
input unit
PNN, 16
interpolation
Parzen, 8
invariance
tangent distance method, 33
kernel, see Parzen window
linear discriminant, 44
Fisher, 5, 44–51
Manhattan distance, see metric, Man-
hattan
matrix
scatter, 46
total, see scatter matrix, total
singular, 47
mean
Parzen estimate
convergence, 11
vector
total, 48
metric, 32
L1 norm, 32
L∞ norm, 33
Lk norm, 32
city block, see metric, Manhattan
Euclidean, 32
Manhattan, 32
Minkowski, 32
non-negativity, 32
properties, 32
reflexivity, 32
symmetry, 32
tangent distance, 33–37
triangle inequality, 32
nearest-neighbor
condensingsee nearest-neighbor, edit-
ing 31
convergence, 24
editing, 30–31
Algorithm, 31
minimal set, 31
error, 24–27
bound, 25–27
finite sample, 27, 56
variance, 26
partical distance, 30
prestructuring, 30
pruningsee nearest-neighbor, edit-
ing 31
neural network
Probabilistic, 16–19
nonparametric method
benefits, 15
outer product, 47
Parzen
estimate
widow width effect, 8
Parzen window, 8
classification, 15–16
estimate, 8
function, 19
prior, 15
pattern unit
PNN, 16
PNN, see neural network, probabilistic
space complexity, 18
time complexity, 18
weight, 16
potential function, 40
probabilistic neural network, see neu-
ral network, probabilistic
benefits, 18
recall
Algorithm, 18
training
Algorithm, 17
probability
posterior
nonparametric estimation, 5
subjective, 38
prototype, 19
Rayleigh quotient, 47
RCE, see reduced coulomb energy
classification
Algorithm, 41
training
Algorithm, 40
reduced coulomb energy, 40
scatter
between-class, 47
74 INDEX
within-class, 46, 47
scatter matrix, see matrix, scatter
search
tree, 30
subjective probability, see probability,
subjective
tangent vector, 34
training data
limited, 7
triangle inequality, see metric, triangle
inequality
variance
Parzen estimate
convergence, 11
vector
mean
total, see mean vector, total
Voronoi
cell, 23
tesselation, 23
window function
Gaussian, 43
within-class scatter, see scatter, within-
class
zero-information distribution, 27
Contents
5 Linear Discriminant Functions 3
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
5.2 Linear Discriminant Functions and Decision Surfaces . . . . . . . . . . 4
5.2.1 The Two-Category Case . . . . . . . . . . . . . . . . . . . . . . 4
5.2.2 The Multicategory Case . . . . . . . . . . . . . . . . . . . . . . 6
5.3 Generalized Linear Discriminant Functions . . . . . . . . . . . . . . . 6
5.4 The Two-Category Linearly-Separable Case . . . . . . . . . . . . . . . 11
5.4.1 Geometry and Terminology . . . . . . . . . . . . . . . . . . . . 11
5.4.2 Gradient Descent Procedures . . . . . . . . . . . . . . . . . . . 12
Algorithm 1: Gradient descent . . . . . . . . . . . . . . . . . . . . . . . 13
Algorithm 2: Newton descent . . . . . . . . . . . . . . . . . . . . . . . 14
5.5 Minimizing the Perceptron Criterion Function . . . . . . . . . . . . . . 14
5.5.1 The Perceptron Criterion Function . . . . . . . . . . . . . . . . 14
Algorithm 3: Batch Perceptron . . . . . . . . . . . . . . . . . . . . . . 15
5.5.2 Convergence Proof for Single-Sample Correction . . . . . . . . 17
Algorithm 4: Fixed increment descent . . . . . . . . . . . . . . . . . . 18
5.5.3 Some Direct Generalizations . . . . . . . . . . . . . . . . . . . 21
Algorithm 5: Fixed increment descent . . . . . . . . . . . . . . . . . . 21
Algorithm 6: Batch variable increment Perceptron . . . . . . . . . . . 22
Algorithm 7: Balanced Winnow algorithm . . . . . . . . . . . . . . . . 23
5.6 Relaxation Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.6.1 The Descent Algorithm . . . . . . . . . . . . . . . . . . . . . . 23
Algorithm 8: Relaxation training with margin . . . . . . . . . . . . . . 24
Algorithm 9: Relaxation rule . . . . . . . . . . . . . . . . . . . . . . . 25
5.6.2 Convergence Proof . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.7 Nonseparable Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.8 Minimum Squared Error Procedures . . . . . . . . . . . . . . . . . . . 28
5.8.1 Minimum Squared Error and the Pseudoinverse . . . . . . . . . 28
Example 1: Constructing a linear classifier by matrix pseudoinverse . 29
5.8.2 Relation to Fisher’s Linear Discriminant . . . . . . . . . . . . . 30
5.8.3 Asymptotic Approximation to an Optimal Discriminant . . . . 32
5.8.4 The Widrow-Hoff Procedure . . . . . . . . . . . . . . . . . . . . 34
Algorithm 10: LMS algorithm . . . . . . . . . . . . . . . . . . . . . . . 34
5.8.5 Stochastic Approximation Methods . . . . . . . . . . . . . . . . 35
5.9 *The Ho-Kashyap Procedures . . . . . . . . . . . . . . . . . . . . . . . 37
5.9.1 The Descent Procedure . . . . . . . . . . . . . . . . . . . . . . 37
Algorithm 11: Ho-Kashyap . . . . . . . . . . . . . . . . . . . . . . . . 39
5.9.2 Convergence Proof . . . . . . . . . . . . . . . . . . . . . . . . . 39
1
2 CONTENTS
5.9.3 Nonseparable Behavior . . . . . . . . . . . . . . . . . . . . . . . 41
5.9.4 Some Related Procedures . . . . . . . . . . . . . . . . . . . . . 42
Algorithm 12: Modified Ho-Kashyap . . . . . . . . . . . . . . . . . . . 42
5.10 *Linear Programming Algorithms . . . . . . . . . . . . . . . . . . . . . 44
5.10.1 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . 44
5.10.2 The Linearly Separable Case . . . . . . . . . . . . . . . . . . . 45
5.10.3 Minimizing the Perceptron Criterion Function . . . . . . . . . . 46
5.11 *Support Vector Machines . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.11.1 SVM training . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Example 2: SVM for the XOR problem . . . . . . . . . . . . . . . . . . 51
5.12 Multicategory Generalizations . . . . . . . . . . . . . . . . . . . . . . . 52
5.12.1 Kesler’s Construction . . . . . . . . . . . . . . . . . . . . . . . 52
5.12.2 Convergence of the Fixed-Increment Rule . . . . . . . . . . . . 53
5.12.3 Generalizations for MSE Procedures . . . . . . . . . . . . . . . 55
Bibliographical and Historical Remarks . . . . . . . . . . . . . . . . . . . . 57
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Computer exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Chapter 5
Linear Discriminant Functions
5.1 Introduction
I n Chap. ?? we assumed that the forms for the underlying probability densities wereknown, and used the training samples to estimate the values of their parameters.
In this chapter we shall instead assume we know the proper forms for the discriminant
functions, and use the samples to estimate the values of parameters of the classifier.
We shall examine various procedures for determining discriminant functions, some of
which are statistical and some of which are not. None of them, however, requires
knowledge of the forms of underlying probability distributions, and in this limited
sense they can be said to be nonparametric.
Throughout this chapter we shall be concerned with discriminant functions that
are either linear in the components of x, or linear in some given set of functions
of x. Linear discriminant functions have a variety of pleasant analytical properties.
As we have seen in Chap. ??, they can be optimal if the underlying distributions
are cooperative, such as Gaussians having equal covariance, as might be obtained
through an intelligent choice of feature detectors. Even when they are not optimal,
we might be willing to sacrifice some performance in order to gain the advantage of
their simplicity. Linear discriminant functions are relatively easy to compute and in
the absense of information suggesting otherwise, linear classifiers are an attractive
candidates for initial, trial classifiers. They also illustrate a number of very important
principles which will be used more fully in neural networks (Chap. ??).
The problem of finding a linear discriminant function will be formulated as a prob-
lem of minimizing a criterion function. The obvious criterion function for classification
purposes is the sample risk, or training error — the average loss incurred in classifying training
errorthe set of training samples. We must emphasize right away, however, that despite the
attractiveness of this criterion, it is fraught with problems. While our goal will be to
classify novel test patterns, a small training error does not guarantee a small test error
— a fascinating and subtle problem that will command our attention in Chap. ??.
As we shall see here, it is difficult to derive the minimum-risk linear discriminant
anyway, and for that reason we investigate several related criterion functions that are
analytically more tractable.
Much of our attention will be devoted to studying the convergence properties
3
4 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
and computational complexities of various gradient descent procedures for minimizing
criterion functions. The similarities between many of the procedures sometimes makes
it difficult to keep the differences between them clear and for this reason we have
included a summary of the principal results in Table 5.1 at the end of Sect. 5.10.
5.2 Linear Discriminant Functions and Decision Sur-
faces
5.2.1 The Two-Category Case
A discriminant function that is a linear combination of the components of x can be
written as
g(x) = wtx + w0, (1)
where w is the weight vector and w0 the bias or threshold weight. A two-categorythreshold
weight linear classifier implements the following decision rule: Decide ω1 if g(x) > 0 and ω2
if g(x) < 0. Thus, x is assigned to ω1 if the inner product wtx exceeds the threshold −w0 and ω2 otherwise. If g(x) = 0, x can ordinarily be assigned to either class, but in this chapter we shall leave the assignment undefined. Figure 5.1 shows a typical implementation, a clear example of the general structure of a pattern recognition system we saw in Chap. ??. x0 = 1 x1 x2 . . . xd . . . w2 w0 w1 wd g(x) Figure 5.1: A simple linear classifier having d input units, each corresponding to the values of the components of an input vector. Each input feature value xi is multiplied by its corresponding weight wi; the output unit sums all these products and emits a +1 if wtx + w0 > 0 or a −1 otherwise.
The equation g(x) = 0 defines the decision surface that separates points assigned
to ω1 from points assigned to ω2. When g(x) is linear, this decision surface is a
hyperplane. If x1 and x2 are both on the decision surface, then
wtx1 + w0 = w
tx2 + w0
or
wt(x1 − x2) = 0,
5.2. LINEAR DISCRIMINANT FUNCTIONS AND DECISION SURFACES 5
and this shows that w is normal to any vector lying in the hyperplane. In general,
the hyperplane H divides the feature space into two halfspaces, decision region R1
for ω1 and region R2 for ω2. Since g(x) > 0 if x is in R1, it follows that the normal
vector w points into R1. It is sometimes said that any x in R1 is on the positive side
of H, and any x in R2 is on the negative side.
The discriminant function g(x) gives an algebraic measure of the distance from x
to the hyperplane. Perhaps the easiest way to see this is to express x as
x = xp + r
w
‖w‖
,
where xp is the normal projection of x onto H, and r is the desired algebraic distance
— positive if x is on the positive side and negative if x is on the negative side. Then,
since g(xp) = 0,
g(x) = wtx + w0 = r‖w‖,
or
r =
g(x)
‖w‖
.
In particular, the distance from the origin to H is given by w0/‖w‖. If w0 > 0 the
origin is on the positive side of H, and if w0 < 0 it is on the negative side. If w0 = 0,
then g(x) has the homogeneous form wtx, and the hyperplane passes through the
origin. A geometric illustration of these algebraic results is given in Fig. 5.2.
R2
R1
x
g(x)=0
w
x1
x2
x3
w0
/||w
||
r
H
xp
Figure 5.2: The linear decision boundary H, where g(x) = wtx + w0 = 0, separates
the feature space into two half-spaces R1 (where g(x) > 0) and R2 (where g(x) < 0).
To summarize, a linear discriminant function divides the feature space by a hy-
perplane decision surface. The orientation of the surface is determined by the normal
vector w, and the location of the surface is determined by the bias w0. The discrim-
inant function g(x) is proportional to the signed distance from x to the hyperplane,
with g(x) > 0 when x is on the positive side, and g(x) < 0 when x is on the negative
side.
6 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
5.2.2 The Multicategory Case
There is more than one way to devise multicategory classifiers employing linear dis-
criminant functions. For example, we might reduce the problem to c − 1 two-class
problems, where the ith problem is solved by a linear discriminant function that
separates points assigned to ωi from those not assigned to ωi. A more extravagant
approach would be to use c(c−1)/2 linear discriminants, one for every pair of classes.
As illustrated in Fig. 5.3, both of these approaches can lead to regions in which the
classification is undefined. We shall avoid this problem by adopting the approach
taken in Chap. ??, defining c linear discriminant functions
gi(x) = w
txi + wi0 i = 1, ..., c, (2)
and assigning x to ωi if gi(x) > gj(x) for all j �= i; in case of ties, the classification
is left undefined. The resulting classifier is called a linear machine. A linear machinelinear
machine divides the feature space into c decision regions, with gi(x) being the largest discrim-
inant if x is in region Ri. If Ri and Rj are contiguous, the boundary between them
is a portion of the hyperplane Hij defined by
gi(x) = gj(x)
or
(wi −wj)tx + (wi0 − wj0) = 0.
It follows at once that wi −wj is normal to Hij , and the signed distance from x
to Hij is given by (gi − gj)/‖wi −wj‖. Thus, with the linear machine it is not the
weight vectors themselves but their differences that are important. While there are
c(c− 1)/2 pairs of regions, they need not all be contiguous, and the total number of
hyperplane segments appearing in the decision surfaces is often fewer than c(c−1)/2,
as shown in Fig. 5.4.
It is easy to show that the decision regions for a linear machine are convex and this
restriction surely limits the flexibility and accuracy of the classifier (Problems 1 & 2).
In particular, for good performance every decision region should be singly connected,
and this tends to make the linear machine most suitable for problems for which the
conditional densities p(x|ωi) are unimodal.
5.3 Generalized Linear Discriminant Functions
The linear discriminant function g(x) can be written as
g(x) = w0 +
d∑
i=1
wixi, (3)
where the coefficients wi are the components of the weight vector w. By adding
additional terms involving the products of pairs of components of x, we obtain the
quadratic discriminant functionquadratic
discriminant
g(x) = w0 +
d∑
i=1
wixi +
d∑
i=1
d∑
j=1
wijxixj . (4)
5.3. GENERALIZED LINEAR DISCRIMINANT FUNCTIONS 7
ω1
not ω1
ω1
not ω2ω2
not ω3 ω3
not ω4
ω4
ambiguous
region
ambiguous
region
ω1 ω1
ω1
ω2
ω2
ω2
ω3
ω3
ω3
ω4
ω4 ω4
ω2
ω4
ω3
ω3ω2
ω1
ω4
Figure 5.3: Linear decision boundaries for a four-class problem. The top figure shows
ωi/not ωi dichotomies while the bottom figure shows ωi/ωj dichotomies. The pink
regions have ambiguous category assigments.
Since xixj = xjxi, we can assume that wij = wji with no loss in generality. Thus, the
quadratic discriminant function has an additional d(d+1)/2 coefficients at its disposal
with which to produce more complicated separating surfaces. The separating surface
defined by g(x) = 0 is a second-degree or hyperquadric surface. The linear terms
in g(x) can be eliminated by translating the axes. We can define W = [wij ], a
symmetric, nonsingular matrix and then the basic character of the separating surface
can be described in terms of the scaled matrix W̄ = W/(wtW−1w − 4w0). If W̄
is a positive multiple of the identity matrix, the separating surface is a hypersphere.
If W̄ is positive definite, the separating surfaces is a hyperellipsoid. If some of the
eigenvalues of W̄ are positive and others are negative, the surface is one of the variety
of types of hyperhyperboloids (Problem 11). As we observed in Chap. ??, these are
the kinds of separating surfaces that arise in the general multivariate Gaussian case.
By continuing to add terms such as wijkxixjxk we can obtain the class of polyno-
mial discriminant functions. These can be thought of as truncated series expansions polynomial
discriminantof some arbitrary g(x), and this in turn suggest the generalized linear discriminant
function
g(x) =
d̂∑
i=1
aiyi(x) (5)
8 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
R1
R2
R3
R4
R5
ω1
R2
R3
R1
ω2 ω1
ω3
ω5
ω2ω3
ω4
Figure 5.4: Decision boundaries produced by a linear machine for a three-class prob-
lem and a five-class problem.
or
g(x) = aty, (6)
where a is now a d̂-dimensional weight vector, and where the d̂ functions yi(x) — some-
times called ϕ functions — can be arbitrary functions of x. Such functions might bephi
function computed by a feature detecting subsystem. By selecting these functions judiciously
and letting d̂ be sufficiently large, one can approximate any desired discriminant func-
tion by such an expansion. The resulting discriminant function is not linear in x, but
it is linear in y. The d̂ functions yi(x) merely map points in d-dimenional x-space
to points in d̂-dimensional y-space. The homogeneous discriminant aty separates
points in this transformed space by a hyperplane passing through the origin. Thus,
the mapping from x to y reduces the problem to one of finding a homogeneous linear
discriminant function.
Some of the advantages and disadvantages of this approach can be clarified by
considering a simple example. Let the quadratic discriminant function be
g(x) = a1 + a2x + a3x
2, (7)
so that the three-dimensional vector y is given by
y =
1x
x2
. (8)
The mapping from x to y is illustrated in Fig. 5.5. The data remain inherently one-
dimensional, since varying x causes y to trace out a curve in three dimensions. Thus,
one thing to notice immediately is that if x is governed by a probability law p(x), the
induced density p̂(y) will be degenerate, being zero everywhere except on the curve,
where it is infinite. This is a common problem whenever d̂ > d, and the mapping
takes points from a lower-dimensional space to a higher-dimensional space.
The plane Ĥ defined by aty = 0 divides the y-space into two decision regions R̂1
and R̂2. Figure ?? shows the separating plane corresponding to a = (−1, 1, 2)t, the
decision regions R̂1 and R̂2, and their corresponding decision regions R1 and R2 in
the original x-space. The quadratic discriminant function g(x) = −1 + x + 2×2 is
5.3. GENERALIZED LINEAR DISCRIMINANT FUNCTIONS 9
0
0.5
1
1.5
2
2.5
-1
0
1
2
y2
0
2
4
y3
0
0.5
1
1.5
2
2.5
y1
R1
R2
1-1 20-2
x
R1R1 R2
y = ( )
1
x
x2
Figure 5.5: The mapping y = (1, x, x2)t takes a line and transforms it to a parabola
in three dimensions. A plane splits the resulting y space into regions corresponding
to two categories, and this in turn gives a non-simply connected decision region in the
one-dimensional x space.
positive if x < −1 or if x > 0.5, and thus R1 is multiply connected. Thus although
the decision regions in y-space are convex, this is by no means the case in x-space.
More generally speaking, even with relatively simple functions yi(x), decision surfaces
induced in an x-space can be fairly complex (Fig. 5.6).
Unfortunately, the curse of dimensionality often makes it hard to capitalize on
this flexibility in practice. A complete quadratic discriminant function involves d̂ =
(d + 1)(d + 2)/2 terms. If d is modestly large, say d = 50, this requires the com-
putation of a great many terms; inclusion of cubic and higher orders leads to O(d̂3)
terms. Furthermore, the d̂ components of the weight vector a must be determined
from training samples. If we think of d̂ as specifying the number of degrees of freedom
for the discriminant function, it is natural to require that the number of samples be
not less than the number of degrees of freedom (cf., Chap. ??). Clearly, a general
series expansion of g(x) can easily lead to completely unrealistic requirements for
computation and data. We shall see in Sect. ?? that this drawback can be accom-
modated by imposing a constraint of large margins, or bands between the training
patterns, however. In this case, we are not technically speaking fitting all the free
parameters; instead, we are relying on the assumption that the mapping to a high-
dimensional space does not impose any spurious structure or relationships among the
training points. Alternatively, multilayer neural networks approach this problem by
employing multiple copies of a single nonlinear function of the input features, as we
shall see in Chap. ??.
While it may be hard to realize the potential benefits of a generalized linear dis-
criminant function, we can at least exploit the convenience of being able to write
g(x) in the homogeneous form aty. In the particular case of the linear discriminant
function
10 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
y2
w
R2R1
R1
R2
^
R1
^
x1
x2
x1
x2
y1
y3
y
=
(
)x 1
x 2
αx 1
x 2
Ĥ
Figure 5.6: The two-dimensional input space x is mapped through a polynomial
function f to y. Here the mapping is y1 = x1, y2 = x2 and y3 ∝ x1x2. A linear
discriminant in this transformed space is a hyperplane, which cuts the surface. Points
to the positive side of the hyperplane Ĥ correspond to category ω1, and those beneath
it ω2. Here, in terms of the x space, R1 is a not simply connected.
g(x) = w0 +
d∑
i=1
wixi =
d∑
i=0
wixi (9)
where we set x0 = 1. Thus we can write
y =
1
x1
…
xd
=
1
x
, (10)
and y is sometimes called an augmented feature vector. Likewise, an augmented weightaugmented
vector vector can be written as:
a =
w0
w1
…
wd
=
w0
w
. (11)
This mapping from d-dimensional x-space to (d+1)-dimensional y-space is mathe-
matically trivial but nonetheless quite convenient. The addition of a constant compo-
nent to x preserves all distance relationships among samples. The resulting y vectors
all lie in a d-dimensional subspace, which is the x-space itself. The hyperplane deci-
sion surface Ĥ defined by aty = 0 passes through the origin in y-space, even though
the corresponding hyperplane H can be in any position in x-space. The distance from
y to Ĥ is given by |aty|/‖a‖, or |g(x)|/‖a‖. Since ‖a‖ > ‖w‖, this distance is less
5.4. THE TWO-CATEGORY LINEARLY-SEPARABLE CASE 11
than, or at most equal to the distance from x to H. By using this mapping we reduce
the problem of finding a weight vector w and a threshold weight w0 to the problem
of finding a single weight vector a (Fig. 5.7).
y1
y2
y0
a
y0 = 1
R1
R2
y0 = 0
Figure 5.7: A three-dimensional augmented feature space y and augmented weight
vector a (at the origin). The set of points for which aty = 0 is a plane (or more
generally, a hyperplane) perpendicular to a and passing through the origin of y-
space, as indicated by the red disk. Such a plane need not pass through the origin of
the two-dimensional x-space at the top, of course, as shown by the dashed line. Thus
there exists an augmented weight vector a that will lead to any straight decision line
in x-space.
5.4 The Two-Category Linearly-Separable Case
5.4.1 Geometry and Terminology
Suppose now that we have a set of n samples y1, …,yn, some labelled ω1 and some
labelled ω2. We want to use these samples to determine the weights a in a linear
discriminant function g(x) = aty. Suppose we have reason to believe that there
exists a solution for which the probability of error is very low. Then a reasonable
approach is to look for a weight vector that classifies all of the samples correctly. If
such a weight vector exists, the samples are said to be linearly separable. linearly
separableA sample yi is classified correctly if atyi > 0 and yi is labelled ω1 or if atyi < 0
and yi is labelled ω2. This suggests a “normalization” that simplifies the treatment
of the two-category case, viz., the replacement of all samples labelled ω2 by their
negatives. With this “normalization” we can forget the labels and look for a weight
vector a such that atyi > 0 for all of the samples. Such a weight vector is called a
separating vector or more generally a solution vector. separating
vectorThe weight vector a can be thought of as specifying a point in weight space. Each
sample yi places a constraint on the possible location of a solution vector. The
equation atyi = 0 defines a hyperplane through the origin of weight space having yi
as a normal vector. The solution vector — if it exists — must be on the positive side
12 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
of every hyperplane. Thus, a solution vector must lie in the intersection of n half-
spaces; indeed any vector in this region is a solution vector. The corresponding region
is called the solution region, and should not be confused with the decision region insolution
region feature space corresponding to any particular category. A two-dimensional example
illustrating the solution region for both the normalized and the unnormalized case is
shown in Fig. 5.8.
y1
y2
separ
ating
plan
e
solution
region
y1
y2
“sepa
rating
” pla
ne
solution
region
aa
Figure 5.8: Four training samples (black for ω1, red for ω2) and the solution region
in feature space. The figure on the left shows the raw data; the solution vectors leads
to a plane that separates the patterns from the two categories. In the figure on the
right, the red points have been “normalized” — i.e., changed in sign. Now the solution
vector leads to a plane that places all “normalized” points on the same side.
From this discussion, it should be clear that the solution vector — again, if it
exists — is not unique. There are several ways to impose additional requirements to
constrain the solution vector. One possibility is to seek a unit-length weight vector
that maximizes the minimum distance from the samples to the separating plane.
Another possibility is to seek the minimum-length weight vector satisfying atyi ≥ b
for all i, where b is a positive constant called the margin. As shown in Fig. 5.9, themargin
solution region resulting form the intersections of the halfspaces for which atyi ≥ b > 0
lies within the previous solution region, being insultated from the old boundaries by
the distance b/‖yi‖.
The motivation behind these attempts to find a solution vector closer to the “mid-
dle” of the solution region is the natural belief that the resulting solution is more likely
to classify new test samples correctly. In most of the cases we shall treat, however,
we shall be satisfied with any solution strictly within the solution region. Our chief
concern will be to see that any iterative procedure used does not converge to a limit
point on the boundary. This problem can always be avoided by the introduction of a
margin, i.e., by requiring that atyi ≥ b > 0 for all i.
5.4.2 Gradient Descent Procedures
The approach we shall take to finding a solution to the set of linear inequalities
atyi > 0 will be to define a criterion function J(a) that is minimized if a is a solution
vector. This reduces our problem to one of minimizing a scalar function — a problem
that can often be solved by a gradient descent procedure. Basic gradient descent is
very simple. We start with some arbitrarily chosen weight vector a(1) and compute
the gradient vector ∇J(a(1)). The next value a(2) is obtained by moving some
5.4. THE TWO-CATEGORY LINEARLY-SEPARABLE CASE 13
solution
region
y1
y2
y3
a1
a2
solution
region
a2
a1
y1
y2
y3
b/||y
2 ||
b/||
y 1||
b/
||y
3
||
}
}
}
Figure 5.9: The effect of the margin on the solution region. At the left, the case of
no margin (b = 0) equivalent to a case such as shown at the left in Fig. 5.8. At the
right is the case b > 0, shrinking the solution region by margins b/‖yi‖.
distance from a(1) in the direction of steepest descent, i.e., along the negative of the
gradient. In general, a(k + 1) is obtained from a(k) by the equation
a(k + 1) = a(k)− η(k)∇J(a(k)), (12)
where η is a positive scale factor or learning rate that sets the step size. We hope learning
ratethat such a sequence of weight vectors will converge to a solution minimizing J(a).
In algorithmic form we have:
Algorithm 1 (Basic gradient descent)
1 begin initialize a, criterion θ, η(·), k = 0
2 do k ← k + 1
3 a← a− η(k)∇J(a)
4 until η(k)∇J(a) < θ
5 return a
6 end
The many problems associated with gradient descent procedures are well known.
Fortunately, we shall be constructing the functions we want to minimize, and shall be
able to avoid the most serious of these problems. One that will confront us repeatedly,
however, is the choice of the learning rate η(k). If η(k) is too small, convergence is
needlessly slow, whereas if η(k) is too large, the correction process will overshoot and
can even diverge (Sect. 5.6.1).
We now consider a principled method for setting the learning rate. Suppose that
the criterion function can be well approximated by the second-order expansion around
a value a(k) as
J(a)
J(a(k)) + ∇J t(a− a(k)) + 1
2
(a− a(k))tH (a− a(k)), (13)
where H is the Hessian matrix of second partial derivatives ∂2J/∂ai∂aj evaluated at Hessian
matrixa(k). Then, substituting a(k + 1) from Eq. 12 into Eq. 13 we find:
14 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
J(a(k + 1))
J(a(k))− η(k)‖∇J‖2 + 1
2
η2(k)∇J tH∇J.
From this it follows (Problem 12) that J(a(k + 1)) can be minimized by the choice
η(k) =
‖∇J‖2
∇J tH∇J
, (14)
where H depends on a, and thus indirectly on k. This then is the optimal choice
of η(k) given the assumptions mentioned. Note that if the criterion function J(a) is
quadratic throughout the region of interest, then H is constant and η is a constant
independent of k.
An alternative approach, obtained by ignoring Eq. 12 and by choosing a(k +
1) to minimize the second-order expansion, is Newton’s algorithm where line 3 inNewton’s
algorithm Algorithm 1 is replaced by
a(k + 1) = a(k)−H−1∇J, (15)
leading to the following algorithm:
Algorithm 2 (Newton descent)
1 begin initialize a, criterion θ
2 do
3 a← a−H−1∇J(a)
4 until H−1∇J(a) < θ
5 return a
6 end
Simple gradient descent and Newton’s algorithm are compared in Fig. 5.10.
Generally speaking, Newton’s algorithm will usually give a greater improvement
per step than the simple gradient descent algorithm, even with the optimal value
of η(k). However, Newton’s algorithm is not applicable if the Hessian matrix H is
singular. Furthermore, even when H is nonsingular, the O(d3) time required for
matrix inversion on each iteration can easily offset the descent advantage. In fact,
it often takes less time to set η(k) to a constant η that is smaller than necessary
and make a few more corrections than it is to compute the optimal η(k) at each step
(Computer exercise 1).
5.5 Minimizing the Perceptron Criterion Function
5.5.1 The Perceptron Criterion Function
Consider now the problem of constructing a criterion function for solving the linear
inequalities atyi > 0. The most obvious choice is to let J(a; y1, …,yn) be the number
of samples misclassified by a. However, because this function is piecewise constant, it
is obviously a poor candidate for a gradient search. A better choice is the Perceptron
criterion function
Jp(a) =
∑
y∈Y
(−aty), (16)
5.5. MINIMIZING THE PERCEPTRON CRITERION FUNCTION 15
-2
-1
0
1
2
1
2
3
4
0
5
10
-2
-1
0
1
2
a1
a2
J(a)
Figure 5.10: The sequence of weight vectors given by a simple gradient descent method
(red) and by Newton’s (second order) algorithm (black). Newton’s method typically
leads to greater improvement per step, even when using optimal learning rates for both
methods. However the added computational burden of inverting the Hessian matrix
used in Newton’s method is not always justified, and simple descent may suffice.
where Y(a) is the set of samples misclassified by a. (If no samples are misclassified,
Y is empty and we define Jp to be zero.) Since aty ≤ 0 if y is misclassified, Jp(a)
is never negative, being zero only if a is a solution vector, or if a is on the decision
boundary. Geometrically, Jp(a) is proportional to the sum of the distances from the
misclassified samples to the decision boundary. Figure 5.11 illustrates Jp for a simple
two-dimensional example.
Since the jth component of the gradient of Jp is ∂Jp/∂aj , we see from Eq. 16 that
∇Jp =
∑
y∈Y
(−y), (17)
and hence the update rule becomes
a(k + 1) = a(k) + η(k)
∑
y∈Yk
y, (18)
where Yk is the set of samples misclassified by a(k). Thus the Perceptron algorithm
is:
Algorithm 3 (Batch Perceptron)
1 begin initialize a, η(·), criterion θ, k = 0
2 do k ← k + 1
3 a← a + η(k)
∑
y∈Yk
y
4 until η(k)
∑
y∈Yk
y < θ
5 return a
6 end
16 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
-2
0
2
4
-2
0
2
4
0
100
-2
0
2
4
0
100
-2
0
2
4
-2
0
2
4
0
5
-2
0
2
4
0
5
-2
0
2
4
-2
0
2
4
0
1
2
3
-2
0
2
4
0
1
2
3
-2
0
2
4
-2
0
2
4
0
5
10
-2
0
2
4
0
5
10
y1
y1
y1y1
y2
y2
y2 y2
y3 y3
y3 y3
solution
region
solution
region
solution
region
solution
region
a2a2
a2a2
a1 a1
a1 a1
Jp(a)
Jq(a) Jr(a)
J(a)
Figure 5.11: Four learning criteria as a function of weights in a linear classifier. At the
upper left is the total number of patterns misclassified, which is piecewise constant
and hence unacceptable for gradient descent procedures. At the upper right is the
Perceptron criterion (Eq. 16), which is piecewise linear and acceptable for gradient
descent. The lower left is squared error (Eq. 32), which has nice analytic properties
and is useful even when the patterns are not linearly separable. The lower right is
the square error with margin (Eq. 33). A designer may adjust the margin b in order
to force the solution vector to lie toward the middle of the b = 0 solution region in
hopes of improving generalization of the resulting classifier.
Thus, the batch Perceptron algorithm for finding a solution vector can be stated
very simply: the next weight vector is obtained by adding some multiple of the sum
of the misclassified samples to the present weight vector. We use the term “batch”batch
training to refer to the fact that (in general) a large group of samples is used when com-
puting each weight update. (We shall soon see alternate methods based on single
samples.) Figure 5.12 shows how this algorithm yields a solution vector for a simple
two-dimensional example with a(1) = 0, and η(k) = 1. We shall now show that it
will yield a solution for any linearly separable problem.
5.5. MINIMIZING THE PERCEPTRON CRITERION FUNCTION 17
a2
a1
Jp
y1
y2
solution
region
-2
0
2
4
-2
0
2
4
0
5
10
-2
0
2
4
0
5
10
y3
y1
y3
y3
Figure 5.12: The Perceptron criterion, Jp is plotted as a function of the weights a1
and a2 for a three-pattern problem. The weight vector begins at 0, and the algorithm
sequentially adds to it vectors equal to the “normalized” misclassified patterns them-
selves. In the example shown, this sequence is y2,y3,y1,y3, at which time the vector
lies in the solution region and iteration terminates. Note that the second update (by
y3) takes the candidate vector farther from the solution region than after the first
update (cf. Theorem 5.1. (In an alternate, batch method, all the misclassified points
are added at each iteration step leading to a smoother trajectory in weight space.)
5.5.2 Convergence Proof for Single-Sample Correction
We shall begin our examination of convergence properties of the Perceptron algo-
rithm with a variant that is easier to analyze. Rather than testing a(k) on all of the
samples and basing our correction of the set Yk of misclassified training samples, we
shall consider the samples in a sequence and shall modify the weight vector when-
ever it misclassifies a single sample. For the purposes of the convergence proof, the
detailed nature of the sequence is unimportant as long as every sample appears in
the sequence infinitely often. The simplest way to assure this is to repeat the sam-
ples cyclically, though from a practical point of view random selection is often to be
preferred (Sec. 5.8.5). Clearly neither the batch nor this single-sample version of the
Perceptron algorithm are on-line since we must store and potentially revisit all of the
training patterns.
Two further simplifications help to clarify the exposition. First, we shall tem-
porarily restrict our attention to the case in which η(k) is constant — the so-called
fixed-increment case. It is clear from Eq. 18 that if η(t) is constant it merely serves to fixed
incrementscale the samples; thus, in the fixed-increment case we can take η(t) = 1 with no loss
in generality. The second simplification merely involves notation. When the samples
18 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
are considered sequentially, some will be misclassified. Since we shall only change the
weight vector when there is an error, we really need only pay attention to the mis-
classified samples. Thus we shall denote the sequence of samples using superscripts,
i.e., by y1, y2, ..., yk, ..., where each yk is one of the n samples y1, ...,yn, and where
each yk is misclassified. For example, if the samples y1, y2, and y3 are considered
cyclically, and if the marked samples
↓
y1, y2,
↓
y3,
↓
y1,
↓
y2, y3, y1,
↓
y2, ... (19)
are misclassified, then the sequence y1, y2, y3, y4, y5, ... denotes the sequence
y1, y3, y1, y2, y2, ... With this understanding, the fixed-increment rule for generatingfixed-
increment
rule
a sequence of weight vectors can be written as
a(1) arbitrary
a(k + 1) = a(k) + yk k ≥ 1
}
(20)
where at(k)yk ≤ 0 for all k. If we let n denote the total number of patterns, the
algorithm is:
Algorithm 4 (Fixed-increment single-sample Perceptron)
1 begin initialize a, k = 0
2 do k ← (k + 1)modn
3 if yk is misclassified by a then a← a− yk
4 until all patterns properly classified
5 return a
6 end
The fixed-increment Perceptron rule is the simplest of many algorithms that have
been proposed for solving systems of linear inequalities. Geometrically, its interpre-
tation in weight space is particularly clear. Since a(k) misclassifies yk, a(k) is not on
the positive side of the yk hyperplane atyk = 0. The addition of yk to a(k) moves
the weight vector directly toward and perhaps across this hyperplane. Whether the
hyperplane is crossed or not, the new inner product at(k+1)yk is larger than the old
inner product at(k)yk by the amount ‖yk‖2, and the correction is clearly moving the
weight vector in a good direction (Fig. 5.13).
5.5. MINIMIZING THE PERCEPTRON CRITERION FUNCTION 19
7 8 9
4 5 6
1 2 3
Figure 5.13: Samples from two categories, ω1 (black) and ω2 (red) are shown in
augmented feature space, along with an augmented weight vector a. At each step
in a fixed-increment rule, one of the misclassified patterns, yk, is shown by the large
dot. A correction ∆a (proportional to the pattern vector yk) is added to the weight
vector — towards an ω1 point or away from an ω2 point. This changes the decision
boundary from the dashed position (from the previous update) to the solid position.
The sequence of resulting a vectors is shown, where later values are shown darker. In
this example, by step 9 a solution vector has been found and the categories successfully
separated by the decision boundary shown.
Clearly this algorithm can only terminate if the samples are linearly separable; we
now prove that indeed it terminates so long as the samples are linearly separable.
Theorem 5.1 (Perceptron Convergence) If training samples are linearly sepa-
rable then the sequence of weight vectors given by Algorithm 4 will terminate at a
solution vector.
Proof:
In seeking a proof, it is natural to try to show that each correction brings the weight
vector closer to the solution region. That is, one might try to show that if â is any
solution vector, then ‖a(k + 1)− â‖ is smaller than ‖a(k)− â‖. While this turns out
20 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
not to be true in general (cf. steps 6 & 7 in Fig. 5.13), we shall see that it is true for
solution vectors that are sufficiently long.
Let â be any solution vector, so that âtyi is strictly positive for all i, and let α be
a positive scale factor. From Eq. 20,
a(k + 1)− αâ = (a(k)− αâ) + yk
and hence
‖a(k + 1)− αâ‖2 = ‖a(k)− αâ‖2 + 2(a(k)− αâ)tyk + ‖yk‖2.
Since yk was misclassified, at(k)yk ≤ 0, and thus
‖a(k + 1)− αâ‖2 ≤ ‖a(k)− αâ‖2 − 2αâtyk + ‖yk‖2.
Because âtyk is strictly positive, the second term will dominate the third if α is
sufficiently large. In particular, if we let β be the maximum length of a pattern
vector,
β2 = max
i
‖yi‖2, (21)
and γ be the smallest inner product of the solution vector with any pattern vector,
i.e.,
γ = min
i
[
âtyi
]
> 0, (22)
then we have the inequality
‖a(k + 1)− αâ‖2 ≤ ‖a(k)− αâ‖2 − 2αγ + β2.
If we choose
α =
β2
γ
, (23)
we obtain
‖a(k + 1)− αâ‖2 ≤ ‖a(k)− αâ‖2 − β2.
Thus, the squared distance from a(k) to αâ is reduced by at least β2 at each correction,
and after k corrections
‖a(k + 1)− αâ‖2 ≤ ‖a(k)− αâ‖2 − kβ2. (24)
Since the squared distance cannot become negative, it follows that the sequence of
corrections must terminate after no more than k0 corrections, where
k0 =
‖a(1)− αâ‖2
β2
. (25)
5.5. MINIMIZING THE PERCEPTRON CRITERION FUNCTION 21
Since a correction occurs whenever a sample is misclassified, and since each sample
appears infinitely often in the sequence, it follows that when corrections cease the
resulting weight vector must classify all of the samples correctly.
The number k0 gives us a bound on the number of corrections. If a(1) = 0, we
get the following particularly simple expression for k0:
k0 =
α2‖â‖2
β2
=
β2α2‖â‖2
γ2
=
max
i
‖yi‖2‖â‖2
min
i
[yti â]
2
. (26)
The denominator in Eq. 26 shows that the difficulty of the problem is essentially
determined by the samples most nearly orthogonal to the solution vector. Unfortu-
nately, it provides no help when we face an unsolved problem, since the bound is
expressed in terms of a solution vector which is unknown. In general, it is clear that
linearly-separable problems can be made arbitrarily difficult to solve by making the
samples almost coplanar (Computer exercise 2). Nevertheless, if the training sam-
ples are linearly separable, the fixed-increment rule will yield a solution after a finite
number of corrections.
5.5.3 Some Direct Generalizations
The fixed increment rule can be generalized to provide a variety of related algorithms.
We shall briefly consider two variants of particular interest. The first variant intro-
duces a variable increment η(k) and a margin b, and calls for a correction whenever variable
incrementat(k)yk fails to excede the margin. The update is given by
a(1) arbitrary
a(k + 1) = a(k) + η(k)yk k ≥ 1,
}
(27)
where now at(k)yk ≤ b for all k. Thus for n patterns, our algorithm is:
Algorithm 5 (Variable increment Perceptron with margin)
1 begin initialize a, criterion θ,margin b, η(·), k = 0
2 do k ← k + 1
3 if atyk + b < 0 then a← a− η(k)yk
4 until atyk + b ≤ 0 for all k
5 return a
6 end
It can be shown that if the samples are linearly separable and if
η(k) ≥ 0, (28)
lim
m→∞
m∑
k=1
η(k) =∞ (29)
and
22 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
lim
m→∞
m∑
k=1
η2(k)
(
m∑
k=1
η(k)
)2 = 0, (30)
then a(k) converges to a solution vector a satisfying atyi > b for all i (Problem 18).
In particular, these conditions on η(k) are satisfied if η(k) is a positive constant, or if
it decreases like 1/k.
Another variant of interest is our original gradient descent algorithm for Jp,
a(1) arbitrary
a(k + 1) = a(k) + η(k)
∑
y∈Yk
y,
}
(31)
where Yk is the set of training samples misclassified by a(k). It is easy to see that this
algorithm will also yield a solution once one recognizes that if â is a solution vector
for y1, …,yn, then it correctly classifies the correction vector
yk =
∑
y∈Yk
y.
In greater detail, then, the algorithm is
Algorithm 6 (Batch variable increment Perceptron)
1 begin initialize a, η(·), k = 0
2 do k ← k + 1
3 Yk = {}
4 j = 0
5 do j ← j + 1
6 if yj is misclassified then Append yj to Yk
7 until j = n
8 a← a + η(k)
∑
y∈Yk
y
9 until Yk = {}
10 return a
11 end
The benefit of batch gradient descent is that the trajectory of the weight vector is
smoothed, compared to that in corresponding single-sample algorithms (e.g., Algo-
rithm 5), since at each update the full set of misclassified patterns is used — the
local statistical variations in the misclassified patterns tend to cancel while the large-
scale trend does not. Thus, if the samples are linearly separable, all of the possible
correction vectors form a linearly separable set, and if η(k) satisfies Eqs. 28–30, the
sequence of weight vectors produced by the gradient descent algorithm for Jp(·) will
always converge to a solution vector.
It is interesting to note that the conditions on η(k) are satisfied if η(k) is a positive
constant, if it decreases as 1/k, or even if it increases as k. Generally speaking, one
would prefer to have η(k) become smaller as time goes on. This is particularly true
if there is reason to believe that the set of samples is not linearly separable, since it
reduces the disruptive effects of a few “bad” samples. However, in the separable case
it is a curious fact that one can allow η(k) to become larger and still obtain a solution.
5.6. RELAXATION PROCEDURES 23
This observation brings out one of the differences between theoretical and practical
attitudes. From a theoretical viewpoint, it is interesting that we can obtain a solution
in a finite number of steps for any finite set of separable samples, for any initial weight
vector a(1), for any nonnegative margin b, and for any scale factor η(k) satisfying
Eqs. 28–30. From a practical viewpoint, we want to make wise choices for these
quantities. Consider the margin b, for example. If b is much smaller than η(k)‖yk‖2,
the amount by which a correction increases at(k)yk, it is clear that it will have little
effect at all. If it is much larger than η(k)‖yk‖2, many corrections will be needed
to satisfy the conditions at(k)yk > b. A value close to η(k)‖yk‖2 is often a useful
compromise. In addition to these choices for η(k) and b, the scaling of the components
of yk can also have a great effect on the results. The possession of a convergence
theorem does not remove the need for thought in applying these techniques.
A close descendant of the Perceptron algorithm is the Winnow algorithm, which
has applicability to separable training data. The key difference is that while the Winnow
algorithmweight vector returned by the Perceptron algorithm has components ai (i = 0, …d),
in Winnow they are scaled according to 2sinh[ai]. In one version, the balanced Win-
now algorithm, there are separate “positive” and “negative” weight vectors, a+ and
a−, each associated with one of the two categories to be learned. Corrections on the
positive weight are made if and only if a training pattern in ω1 is misclassified; con-
versely, corrections on the negative weight are made if and only if a training pattern
in ω2 is misclassified.
Algorithm 7 (Balanced Winnow)
1 begin initialize a+,a−, η(·), k ← 0, α > 1
2 if sign[a+tyk − a−tyk] �= zk (pattern misclassified)
3 then if zk = +1 then a
+
i ← α+yia
+
i ; a
−
i ← α−yia
−
i for all i
4 if zk = −1 then a+i ← α−yia
+
i ; a
−
i ← α+yia
−
i for all i
5 return a+,a−
6 end
There are two main benefits of such a version of the Winnow algorithm. The
first is that during training each of the two consituent weight vectors moves in a uni-
form direction and this means the “gap,” determined by these two vectors, can never
increase in size for separable data. This leads to a convergence proof that, while some-
what more complicated, is nevertheless more general than the Perceptron convergence
theorem (cf. Bibliography). The second benefit is that convergence is generally faster
than in a Perceptron, since for proper setting of learning rate, each constituent weight
does not overshoot its final value. This benefit is especially pronounced whenever a
large number of irrelevant or redundant features are present (Computer exercise 6).
5.6 Relaxation Procedures
5.6.1 The Descent Algorithm
The criterion function Jp is by no means the only function we can construct that is
minimized when a is a solution vector. A close but distinct relative is
Jq(a) =
∑
y∈Y
(aty)2, (32)
24 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
where Y(a) again denotes the set of training samples misclassified by a. Like Jp, Jq
focuses attention on the misclassified samples. Its chief difference is that its gradient
is continuous, whereas the gradient of Jp is not. Thus, Jq presents a smoother surface
to search (Fig. 5.11). Unfortunately, Jq is so smooth near the boundary of the solution
region that the sequence of weight vectors can converge to a point on the boundary.
It is particularly embarrassing to spend some time following the gradient merely to
reach the boundary point a = 0. Another problem with Jq is that its value can be
dominated by the longest sample vectors. Both of these problems are avoided by the
criterion function
Jr(a) =
1
2
∑
y∈Y
(aty − b)2
‖y‖2
, (33)
where now Y(a) is the set of samples for which aty ≤ b. (If Y(a) is empty, we define
Jr to be zero.) Thus, Jr(a) is never negative, and is zero if and only if aty ≥ b for all
of the training samples. The gradient of Jr is given by
∇Jr =
∑
y∈Y
aty − b
‖y‖2
y,
and the update rule
a(1) arbitrary
a(k + 1) = a(k) + η(k)
∑
y∈Y
b−aty
‖y‖2 y.
(34)
Thus the relaxation algorithm becomes
Algorithm 8 (Batch relaxation with margin)
1 begin initialize a, η(·), k = 0
2 do k ← k + 1
3 Yk = {}
4 j = 0
5 do j ← j + 1
6 if yj is misclassified then Append yj to Yk
7 until j = n
8 a← a + η(k)
∑
y∈Y
b−aty
‖y‖2 y
9 until Yk = {}
10 return a
11 end
As before, we find it easier to prove convergence when the samples are considered
one at a time rather than jointly, i.e., single-sample rather than batch. We also limit
our attention to the fixed-increment case, η(k) = η. Thus, we are again led to consider
a sequence y1,y2, … formed from those samples that call for the weight vector to be
corrected. The single-sample correction rule analogous to Eq. 33 is
a(1) arbitrary
a(k + 1) = a(k) + η b−a
t(k)yk
‖yk‖2 y
k,
}
(35)
where at(k)yk ≤ b for all k. The algorithm is:
5.6. RELAXATION PROCEDURES 25
Algorithm 9 (Single-sample relaxation with margin)
1 begin initialize a, η(·), k = 0
2 do k ← k + 1
3 if yk is misclassified then a← a + η(k) b−a
ty
‖yk‖2 yk
4 until all patterns properly classified
5 return a
6 end
This algorithm is known as the single-sample relaxation rule with margin, and it
has a simple geometrical interpretation. The quantity
r(k) =
b− at(k)yk
‖yk‖
(36)
is the distance from a(k) to the hyperplane atyk = b. Since yk/‖yk‖ is the unit
normal vector for the hyperplane, Eq. 35 calls for a(k) to be moved a certain fraction
η of the distance from a(k) to the hyperplane. If η = 1, a(k) is moved exactly to the
hyperplane, so that the “tension” created by the inequality at(k)yk ≤ b is “relaxed”
(Fig. 5.14). From Eq. 35, after a correction,
at(k + 1)yk − b = (1− η)(at(k)yk − b). (37)
If η < 1, then at(k + 1)yk is still less than b, while if η > 1, then at(k + 1)yk is
greater than b. These conditions are referred to as underrelaxation and overrelaxation, under-
relaxation
over-
relaxation
respectively. In general, we shall restrict η to the range 0 < η < 2 (Figs. 5.14 & 5.15).
a ty k = b
a(k)
yk
r(
k)
y1
y2
1−
η
η
Figure 5.14: In each step of a basic relaxation algorithm, the weight vector is moved
a proportion η of the way towards the hyperplane defined by atyk = b.
5.6.2 Convergence Proof
When the relaxation rule is applied to a set of linearly separable samples, the number
of corrections may or may not be finite. If it is finite, then of course we have obtained
a solution vector. If it is not finite, we shall see that a(k) converges to a limit vector
on the boundary of the solution region. Since the region in which aty ≥ b is contained
in a larger region where aty > 0 if b > 0, this implies that a(k) will enter this larger
region at least once, eventually remaining there for all k greater than some finite k0.
26 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
a1 a1
J(a) J(a)
Figure 5.15: At the left, underrelaxation (η < 1) leads to needlessly slow descent, or
even failure to converge. Overrelaxation (1 < η < 2, shown in the middle) describes
overshooting; nevertheless convergence will ultimately be achieved.
The proof depends upon the fact that if â is any vector in the solution region —
i.e., any vector satisfying âtyi > b for all i — then at each step a(k) gets closer to â.
This fact follows at once from Eq. 35, since
‖a(k + 1)− â‖2 = ‖a(k)− â‖2 − 2η (b− a
t(k)yk)
‖yk‖2
(â− a(k))tyk
+η2
(b− at(k)yk)2
‖yk‖2
(38)
and
(â− a(k))tyk > b− at(k)yk ≥ 0, (39)
so that
‖a(k + 1)− â‖2 ≤ ‖a(k)− â‖2 − η(2− η) (b− a
t(k)yk)2
‖yk‖2
. (40)
Since we restrict η to the range 0 < η < 2, it follows that ‖a(k+1)−â‖ ≤ ‖a(k)−â‖.
Thus, the vectors in the sequence a(1),a(2), ... get closer and closer to â, and in the
limit as k goes to infinity the distance ‖a(k)− â‖ approaches some limiting distance
r(â). This means that as k goes to infinity a(k) is confined to the surface of a
hypersphere with center â and radius r(â). Since this is true for any â in the solution
region, the limiting a(k) is confined to the intersection of the hyperspheres centered
about all of the possible solution vectors.
We now show that the common intersection of these hyperspheres is a single point
on the boundary of the solution region. Suppose first that there are at least two
points a′ and a′′ on the common intersection. Then ‖a′ − â‖ = ‖a′′ − â‖ for every â
in the solution region. But this implies that the solution region is contained in the
(d̂− 1)-dimensional hyperplane of points equidistant from a′ to a′′, whereas we know
that the solution region is d̂-dimensional. (Stated formally, if âtyi > 0 for i = 1, …, n,
then for any d̂-dimensional vector v, we have (â + “v)ty > 0 for i = 1, …, n if ” is
sufficiently small.) Thus, a(k) converges to a single point a. This point is certainly
5.7. NONSEPARABLE BEHAVIOR 27
not inside the solution region, for then the sequence would be finite. It is not outside
either, since each correction causes the weight vector to move η times its distance
from the boundary plane, thereby preventing the vector from being bounded away
from the boundary forever. Hence the limit point must be on the boundary.
5.7 Nonseparable Behavior
The Perceptron and relaxation procedures give us a number of simple methods for
finding a separating vector when the samples are linearly separable. All of these
methods are called error-correcting procedures, because they call for a modification error-
correcting
procedure
of the weight vector when and only when an error is encountered. Their success on
separable problems is largely due to this relentless search for an error-free solution.
In practice, one would only consider the use of these methods if there was reason to
believe that the error rate for the optimal linear discriminant function is low.
Of course, even if a separating vector is found for the training samples, it does
not follow that the resulting classifier will perform well on independent test data.
A moment’s reflection will show that any set of fewer than 2d̂ samples is likely to
be linearly separable — a matter we shall return to in Chap. ??. Thus, one should
use several times that many design samples to overdetermine the classifier, thereby
ensuring that the performance on training and test data will be similar. Unfortunately,
sufficiently large design sets are almost certainly not linearly separable. This makes it
important to know how the error-correction procedures will behave when the samples
are nonseparable.
Since no weight vector can correctly classify every sample in a nonseparable set (by
definition), it is clear that the corrections in an error-correction procedure can never
cease. Each algorithm produces an infinite sequence of weight vectors, any member
of which may or may not yield a useful “solution.” The exact nonseparable behavior
of these rules has been studied thoroughly in a few special cases. It is known, for
example, that the length of the weight vectors produced by the fixed-increment rule
are bounded. Empirical rules for terminating the correction procedure are often based
on this tendency for the length of the weight vector to fluctuate near some limiting
value. From a theoretical viewpoint, if the components of the samples are integer-
valued, the fixed-increment procedure yields a finite-state process. If the correction
process is terminated at some arbitrary point, the weight vector may or may not be in
a good state. By averaging the weight vectors produced by the correction rule, one can
reduce the risk of obtaining a bad solution by accidentally choosing an unfortunate
termination time.
A number of similar heuristic modifications to the error-correction rules have been
suggested and studied empirically. The goal of these modifications is to obtain ac-
ceptable performance on nonseparable problems while preserving the ability to find a
separating vector on separable problems. A common suggestion is the use of a vari-
able increment η(k), with η(k) approaching zero as k approaches infinity. The rate
at which η(k) approaches zero is quite important. If it is too slow, the results will
still be sensitive to those training samples that render the set nonseparable. If it is
too fast, the weight vector may converge prematurely with less than optimal results.
One way to choose η(k) is to make it a function of recent performance, decreasing
it as performance improves. Another way is to program η(k) by a choice such as
η(k) = η(1)/k. When we examine stochastic approximation techniques, we shall see
that this latter choice is the theoretical solution to an analogous problem. Before we
28 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
take up this topic, however, we shall consider an approach that sacrifices the ability
to obtain a separating vector for good compromise performance on both separable
and nonseparable problems.
5.8 Minimum Squared Error Procedures
5.8.1 Minimum Squared Error and the Pseudoinverse
The criterion functions we have considered thus far have focussed their attention on
the misclassified samples. We shall now consider a criterion function that involves all
of the samples. Where previously we have sought a weight vector a making all of the
inner products atyi positive, now we shall try to make atyi = bi, where the bi are
some arbitrarily specified positive constants. Thus, we have replaced the problem of
finding the solution to a set of linear inequalities with the more stringent but better
understood problem of finding the solution to a set of linear equations.
The treatment of simultaneous linear equations is simplified by introducing matrix
notation. Let Y be the n-by-d̂ matrix (d̂ = d + 1) whose ith row is the vector yti ,
and let b be the column vector b = (b1, …, bn)t. Then our problem is to find a weight
vector a satisfying
Y10 Y11 · · · Y1d
Y20 Y21 · · · Y2d
…
…
…
…
…
…
…
…
…
Yn0 Yn1 · · · Ynd
a0
a1
…
ad
=
b1
b2
…
…
…
bn
or Ya = b. (41)
If Y were nonsingular, we could write a = Y−1b and obtain a formal solution at once.
However, Y is rectangular, usually with more rows than columns. When there are
more equations than unknowns, a is overdetermined, and ordinarily no exact solution
exists. However, we can seek a weight vector a that minimizes some function of the
error between Ya and b. If we define the error vector e by
e = Ya− b (42)
then one approach is to try to minimize the squared length of the error vector. This
is equivalent to minimizing the sum-of-squared-error criterion function
Js(a) = ‖Ya− b‖2 =
n∑
i=1
(atyi − bi)2. (43)
The problem of minimizing the sum of squared error is a classical one. It can be
solved by a gradient search procedure, as we shall see in Sect. ??. A simple closed-form
solution can also be found by forming the gradient
∇Js =
n∑
i=1
2(atyi − bi)yi = 2Yt(Ya− b) (44)
5.8. MINIMUM SQUARED ERROR PROCEDURES 29
and setting it equal to zero. This yields the necessary condition
YtYa = Ytb, (45)
and in this way we have converted the problem of solving Ya = b to that of solving
YtYa = Ytb. This celebrated equation has the great advantage that the d̂-by-d̂
matrix YtY is square and often nonsingular. If it is nonsingular, we can solve for a
uniquely as
a = (YtY)−1Ytb
= Y†b, (46)
where the d̂-by-n matrix
Y† ≡ (YtY)−1Yt (47)
is called the pseudoinverse of Y. Note that if Y is square and nonsingular, the pseu- pseudo-
inversedoinverse coincides with the regular inverse. Note also that Y†Y = I, but YY† �= I
in general. However, a minimum-squared-error (MSE) solution always exists. In
particular, if Y† is defined more generally by
Y† ≡ lim
�→0
(YtY + “I)−1Yt, (48)
it can be shown that this limit always exists, and that a = Y†b is an MSE solution
to Ya = b.
The MSE solution depends on the margin vector b, and we shall see that different
choices for b give the solution different properties. If b is fixed arbitrarily, there is
no reason to believe that the MSE solution yields a separating vector in the linearly
separable case. However, it is reasonable to hope that by minimizing the squared-
error criterion function we might obtain a useful discriminant function in both the
separable and the nonseparable cases. We shall now examine two properties of the
solution that support this hope.
Example 1: Constructing a linear classifier by matrix pseudoinverse
Suppose we have the following two-dimensional points for two categories: ω1:
(1, 2)t and (2, 0)t, and ω2: (3, 1)t and (2, 3)t, as shown in black and red, respectively,
in the figure.
Our matrix Y is therefore
Y =
1 1 2
1 2 0
−1 −3 −1
−1 −2 −3
and after a few simple calculations we find that its pseudoinverse is
Y† ≡ lim
�→0
(YtY + “I)−1Yt =
5/4 13/12 3/4 7/12−1/2 −1/6 −1/2 −1/6
0 −1/3 0 −1/3
30 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
0 1 2 3 4
0
1
2
3
4
R1
R2
x1
x2
Four training points and the decision boundary at
1×1
x2
= 0, where a was found
by means of a pseudoinverse technique.
We arbitrarily let all the margins be equal, i.e., b = (1, 1, 1, 1)t. Our solution is
a = Y†b = (11/3,−4/3,−2/3)t, and leads to the decision boundary shown in the
figure. Other choices for b would typically lead to different decision boundaries, of
course.
5.8.2 Relation to Fisher’s Linear Discriminant
In this section we shall show that with the proper choice of the vector b, the MSE
discriminant function aty is directly related to Fisher’s linear discriminant. To do
this, we must return to the use of linear rather than generalized linear discriminant
functions. We assume that we have a set of n d-dimensional samples x1, …,xn, n1 of
which are in the subset D1 labelled ω1, and n2 of which are in the subset D2 labelled
ω2. Further, we assume that a sample yi is formed from xi by adding a threshold
component x0 = 1 to make an augmented pattern vector. Further, if the sample isaugmented
pattern
vector
labelled ω2, then the entire pattern vector is multiplied by −1 — the “normlization”
we saw in Sect. 5.4.1. With no loss in generality, we can assume that the first n1
samples are labelled ω1 and the second n2 are labelled ω2. Then the matrix Y can
be partitioned as follows:
Y =
[
11 X1
−12 −X2
]
,
where 1i is a column vector of ni ones, and Xi is an ni-by-d matrix whose rows are
the samples labelled ωi. We partition a and b correspondingly, with
a =
[
w0
w
]
and with
5.8. MINIMUM SQUARED ERROR PROCEDURES 31
b =
[
n
n1
11
n
n2
12
]
.
We shall now show that this special choice for b links the MSE solution to Fisher’s
linear discriminant.
We begin by writing Eq. 47 for a in terms of the partitioned matrices:
[ 1t1 −1t2
Xt1 −Xt2
][ 11 X1
−12 −X2
][
w0
w
]
=
[ 1t1 −1t2
Xt1 −Xt2
] [ n
n1
11
n
n2
12
]
. (49)
By defining the sample means mi and the pooled sample scatter matrix SW as
mi =
1
ni
∑
x∈Di
x i = 1, 2 (50)
and
SW =
2∑
i=1
∑
x∈Di
(x−mi)(x−mi)t, (51)
we can multiply the matrices of Eq. 49 and obtain
[
n (n1m1 + n2m2)t
(n1m1 + n2m2) SW + n1m1mt1 + n2m2m
t
2
][
w0
w
]
=
[
0
n(m1 −m2)
]
.
This can be viewed as a pair of equations, the first of which can be solved for w0 in
terms of w:
w0 = −mtw, (52)
where m is the mean of all of the samples. Substituting this in the second equation
and performing a few algebraic manipulations, we obtain
[ 1
n
SW +
n1n2
n2
(m1 −m2)(m1 −m2)t
]
w = m1 −m2. (53)
Since the vector (m1−m2)(m1−m2)tw is in the direction of m1−m2 for any value
of w, we can write
n1n2
n2
(m1 −m2)(m1 −m2)tw = (1− α)(m1 −m2),
where α is some scalar. Then Eq. 53 yields
w = αnS−1W (m1 −m2), (54)
which, except for an unimportant scale factor, is identical to the solution for Fisher’s
linear discriminant. In addition, we obtain the threshold weight w0 and the following
decision rule: Decide ω1 if wt(x−m) > 0; otherwise decide ω2.
32 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
5.8.3 Asymptotic Approximation to an Optimal Discriminant
Another property of the MSE solution that recommends its use is that if b = 1n it
approaches a minimum mean-squared-error approximation to the Bayes discriminant
function
g0(x) = P (ω1|x)− P (ω2|x) (55)
in the limit as the number of samples approaches infinity. To demonstrate this fact,
we must assume that the samples are drawn independently, identically distributed
(i.i.d.) according to the probability law
p(x) = p(x|ω1)P (ω1) + p(x|ω2)P (ω2). (56)
In terms of the augmented vector y, the MSE solution yields the series expansion
g(x) = aty, where y = y(x). If we define the mean-squared approximation error by
“2 =
∫
[aty − g0(x)]2p(x) dx, (57)
then our goal is to show that “2 is minimized by the solution a = Y†1n.
The proof is simplified if we preserve the distinction between category ω1 and
category ω2 samples. In terms of the unnormalized data, the criterion function Js
becomes
Js(a) =
∑
y∈Y1
(aty − 1)2 +
∑
y∈Y2
(aty + 1)2
= n
[n1
n
1
n1
∑
y∈Y1
(aty − 1)2 + n2
n
1
n2
∑
y∈Y2
(aty + 1)2
]
. (58)
Thus, by the law of large numbers, as n approaches infinity (1/n)Js(a) approaches
J̄(a) = P (ω1)E1[(aty − 1)2] + P (ω2)E2[(aty + 1)2], (59)
with probability one, where
E1[(aty − 1)2] =
∫
(aty − 1)2p(x|ω1) dx
and
E2[(aty + 1)2] =
∫
(aty + 1)2p(x|ω2) dx.
Now, if we recognize from Eq. 55 that
g0(x) =
p(x, ω1)− p(x, ω2)
p(x)
we see that
5.8. MINIMUM SQUARED ERROR PROCEDURES 33
J̄(a) =
∫
(aty − 1)2p(x, ω1) dx +
∫
(aty + 1)2p(x, ω2) dx
=
∫
(aty)2p(x) dx− 2
∫
atyg0(x)p(x) dx + 1
=
∫
[aty − g0(x)]2p(x) dx
︸ ︷︷ ︸
�2
+
[
1−
∫
g20(x)p(x) dx
]
︸ ︷︷ ︸
indep. of a
. (60)
The second term in this sum is independent of the weight vector a. Hence, the a
that minimizes Js also minimizes “2 — the mean-squared-error between aty and g(x)
(Fig. 5.16). In Chap. ?? we shall see that analogous properties also holds for many
multilayer networks.
-4 -2 2 4
-4
-2
2
4
-4 -2 2 4
.5
1
-4 -2 2 4
.1
.2
.3
p(x|ω2)
p(x|ω1)
P(ω2|x)P(ω1|x)
g0(x)
g(x)
x
x
x
p(x|ω)
P(ω|x)
g(x)
Figure 5.16: The top figure shows two class-conditional densities, and the middle figure
the posteriors, assuming equal priors. Minimizing the MSE error also minimizes the
mean-squared-error between aty and the discriminant function g(x) (here a 7th-order
polynomial) measured over the data distribution, as shown at the bottom. Note that
the resulting g(x) best approximates g0(x) in the regions where the data points lie.
This result gives considerable insight into the MSE procedure. By approximat-
ing g0(x), the discriminant function aty gives direct information about the posterior
probabilities P (ω1|x) = (1 + g0)/2 and P (ω2|x) = (1 − g0)/2. The quality of the
approximation depends on the functions yi(x) and the number of terms in the expan-
sion aty. Unfortunately, the mean-square-error criterion places emphasis on points
34 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
where p(x) is larger, rather than on points near the decision surface g0(x) = 0. Thus,
the discriminant function that “best” approximates the Bayes discriminant does not
necessarily minimize the probability of error. Despite this property, the MSE solution
has interesting properties, and has received considerable attention in the literature.
We shall encounter the mean-square approximation of g0(x) again when we consider
stochastic approximation methods and multilayer neural networks.
5.8.4 The Widrow-Hoff Procedure
We remarked earlier that Js(a) = ‖Ya − b‖2 could be minimized by a gradient
descent procedure. Such an approach has two advantages over merely computing the
pseudoinverse: (1) it avoids the problems that arise when YtY is singular, and (2)
it avoids the need for working with large matrices. In addition, the computation
involved is effectively a feedback scheme which automatically copes with some of the
computational problems due to roundoff or truncation. Since ∇Js = 2Yt(Ya − b),
the obvious update rule is
a(1) arbitrary
a(k + 1) = a(k) + η(k)Yt(Yak − b).
}
In Problem 24 you are asked to show that if η(k) = η(1)/k, where η(1) is any positive
constant, then this rule generates a sequence of weight vectors that converges to a
limiting vector a satisfying
Yt(Ya− b) = 0.
Thus, the descent algorithm always yields a solution regardless of whether or not YtY
is singular.
While the d̂-by-d̂ matrix YtY is usually smaller than the d̂-by-n matrix Y†, the
storage requirements can be reduced still further by considering the samples sequen-
tially and using the Widrow-Hoff or LMS rule (least-mean-squared):LMS rule
a(1) arbitrary
a(k + 1) = a(k) + η(k)(bk − a(k)tyk)yk,
}
(61)
or in algorithm form:
Algorithm 10 (LMS)
1 begin initialize a,b, criterion θ, η(·), k = 0
2 do k ← k + 1
3 a← a + η(k)(bk − atyk)yk
4 until η(k)(bk − atyk)yk < θ
5 return a
6 end
At first glance this descent algorithm appears to be essentially the same as the re-
laxation rule. The primary difference is that the relaxation rule is an error-correction
rule, so that at(k)yk does not equal bk, and thus the corrections never cease. There-
fore, η(k) must decrease with k to obtain convergence, the choice η(k) = η(1)/k being
5.8. MINIMUM SQUARED ERROR PROCEDURES 35
common. Exact analysis of the behavior of the Widrow-Hoff rule in the deterministic
case is rather complicated, and merely indicates that the sequence of weight vectors
tends to converge to the desired solution. Instead of pursuing this topic further, we
shall turn to a very similar rule that arises from a stochastic descent procedure. We
note, however, that the solution need not give a separating vector, even if one exists,
as shown in Fig. 5.17 (Computer exercise 10).
y1
y2 separating hyperplane
LMS solution
Figure 5.17: The LMS algorithm need not converge to a separating hyperplane, even
if one exists. Since the LMS solution minimizes the sum of the squares of the distances
of the training points to the hyperplane, for this exmple the plane is rotated clockwise
compared to a separating hyperplane.
5.8.5 Stochastic Approximation Methods
All of the iterative descent procedures we have considered thus far have been described
in deterministic terms. We are given a particular set of samples, and we generate a
particular sequence of weight vectors. In this section we digress briefly to consider
an MSE procedure in which the samples are drawn randomly, resulting in a random
sequence of weight vectors. We will return in Chap. ?? to the theory of stochastic
approximation though here some of the main ideas will be presented without proof.
Suppose that samples are drawn independently by selecting a state of nature with
probability P (ωi) and then selecting an x according to the probability law p(x|ωi).
For each x we let θ be its label, with θ = +1 if x is labelled ω1 and θ = −1 if x
is labelled ω2. Then the data consist of an infinite sequence of independent pairs
(x, θ1), (x2, θ2), ..., (xk, θk), .... Even though the label variable θ is binary-valued it
can be thought of as a noisy version of the Bayes discriminant function g0(x). This
follows from the observation that
P (θ = 1|x) = P (ω1|x),
and
P (θ = −1|x) = P (ω2|x),
so that the conditional mean of θ is given by
Eθ|x[θ] =
∑
θ
θP (θ|x) = P (ω1|x)− P (ω2|x) = g0(x). (62)
36 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
Suppose that we wish to approximate g0(x) by the finite series expansion
g(x) = aty =
d̂∑
i=1
aiyi(x),
where both the basis functions yi(x) and the number of terms d̂ are known. Then we
can seek a weight vector â that minimizes the mean-squared approximation error
"2 = E [(aty − g0(x))2]. (63)
Minimization of "2 would appear to require knowledge of Bayes discriminant g0(x).
However, as one might have guessed from the analogous situation in Sect. 5.8.3, it
can be shown that the weight vector â that minimizes "2 also minimizes the criterion
function
Jm(a) = E [(aty − θ)2]. (64)
This should also be plausible from the fact that θ is essentially a noisy version of g0(x)
(Fig. ??). Since the gradient is
∇Jm = 2E [(aty − θ)y], (65)
we can obtain the closed-form solution
â = E [yyt]−1E [θy]. (66)
Thus, one way to use the samples is to estimate E [yyt] and E [θy], and use Eq. 66 to
obtain the MSE optimal linear discriminant. An alternative is to minimize Jm(a) by
a gradient descent procedure. Suppose that in place of the true gradient we substitute
the noisy version 2(atyk − θk)yk. This leads to the update rule
a(k + 1) = a(k) + η(θk − at(k)yk)yk, (67)
which is basically just the Widrow-Hoff rule. It can be shown (Problem ??) that if
E [yyt] is nonsingular and if the coefficients η(k) satisfy
lim
m→∞
m∑
k=1
η(k) = +∞ (68)
and
lim
m→∞
m∑
k=1
η2(k) <∞ (69)
then a(k) converges to â in mean square:
lim
k→∞
E [‖a(k)− â‖2] = 0. (70)
The reasons we need these conditions on η(k) are simple. The first condition keeps
the weight vector from converging so fast that a systematic error will remain forever
uncorrected. The second condition ensures that random fluctuations are eventually
suppressed. Both conditions are satisfied by the conventional choice η(k) = 1/k.
5.9. *THE HO-KASHYAP PROCEDURES 37
Unfortunately, this kind of programmed decrease of η(k), independent of the problem
at hand, often leads to very slow convergence.
Of course, this is neither the only nor the best descent algorithm for minimizing
Jm. For example, if we note that the matrix of second partial derivatives for Jm is
given by
D = 2E [yyt],
we see that Newton’s rule for minimizing Jm (Eq. 15) is
a(k + 1) = a(k) + E [yyt]−1E [(θ − aty)y].
A stochastic analog of this rule is
a(k + 1) = a(k) + Rk+1(θk − at(k)yk)yk. (71)
with
R−1k+1 = R
−1
k + yky
t
k, (72)
or, equivalently,∗
Rk+1 = Rk −
Rkyk(Rkyk)t
1 + ytkRkyk
. (73)
This rule also produces a sequence of weight vectors that converges to the optimal
solution in mean square. Its convergence is faster, but it requires more computation
per step (Computer exercise 8).
These gradient procedures can be viewed as methods for minimizing a criterion
function, or finding the zero of its gradient, in the presence of noise. In the statistical
literature, functions such as Jm and ∇Jm that have the form E [f(a,x)] are called
regression functions, and the iterative algorithms are called stochastic approximation regression
functionprocedures. Two well known ones are the Kiefer-Wolfowitz procedure for minimizing a
stochastic
approxi-
mation
regression function, and the Robbins-Monro procedure for finding a root of a regression
function. Often the easiest way to obtain a convergence proof for a particular descent
or approximation procedure is to show that it satisfies the convergence conditions for
these more general procedures. Unfortunately, an exposition of these methods in their
full generality would lead us rather far afield, and we must close this digression by
referring the interested reader to the literature.
5.9 The Ho-Kashyap Procedures
5.9.1 The Descent Procedure
The procedures we have considered thus far differ in several ways. The Perceptron
and relaxation procedures find separating vectors if the samples are linearly separable,
but do not converge on nonseparable problems. The MSE procedures yield a weight
vector whether the samples are linearly separable or not, but there is no guarantee
∗ This recursive formula for computing Rk, which is roughly (1/k)E[yyt]−1, cannot be used if Rk
is singular. The equivalence of Eq. 72 and Eq. 73 follows from Problem ?? of Chap. ??.
38 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
that this vector is a separating vector in the separable case (Fig. 5.17). If the margin
vector b is chosen arbitrarily, all we can say is that the MSE procedures minimize
‖Ya − b‖2. Now if the training samples happen to be linearly separable, then there
exists an â and a b̂ such that
Yâ = b̂ > 0,
where by b̂ > 0, we mean that every component of b̂ is positive. Clearly, were we
to take b = b̂ and apply the MSE procedure, we would obtain a separating vector.
Of course, we usually do not know b̂ beforehand. However, we shall now see how the
MSE procedure can be modified to obtain both a separating vector a and a margin
vector b. The underlying idea comes from the observation that if the samples are
separable, and if both a and b in the criterion function
Js(a,b) = ‖Ya− b‖2 (74)
are allowed to vary (subject to the constraint b > 0), then the minimum value of Js
is zero, and the a that achieves that minimum is a separating vector.
To minimize Js, we shall use a modified gradient descent procedure. The gradient
of Js with respect to a is given by
∇aJs = 2Yt(Ya− b), (75)
and the gradient of Js with respect to b is given by
∇bJs = −2(Ya− b). (76)
For any value of b, we can always take
a = Y†b, (77)
thereby obtaining ∇aJs = 0 and minimizing Js with respect to a in one step. We
are not so free to modify b, however, since we must respect the constraint b > 0,
and we must avoid a descent procedure that converges to b = 0. One way to prevent
b from converging to zero is to start with b > 0 and to refuse to reduce any of its
components. We can do this and still try to follow the negative gradient if we first set
all positive components of ∇bJs to zero. Thus, if we let |v| denote the vector whose
components are the magnitudes of the corresponding components of v, we are led to
consider an update rule for the margin of the form
b(k + 1) = b(k)− η 1
2
[∇bJs − |∇bJs|]. (78)
Using Eqs. 76 & 77, and being a bit more specific, we obtain the Ho-Kashyap rule for
minimizing Js(a,b):
b(1) > 0 but otherwise arbitrary
b(k + 1) = a(k) + 2η(k)e+(k),
}
(79)
where e(k) is the error vector
5.9. *THE HO-KASHYAP PROCEDURES 39
e(k) = Ya(k)− b(k), (80)
e+(k) is the positive part of the error vector
e+(k) =
1
2
(e(k) + |e(k)|), (81)
and
a(k) = Y†b(k), k = 1, 2, … (82)
Thus if we let bmin be a small convergence criterion and Abs[e] denote the positive
part of e, our algorithm is:
Algorithm 11 (Ho-Kashyap)
1 begin initialize a,b, η(·) < 1, criteria bmin, kmax
2 do k ← k + 1
3 e← Ya− b
4 e+ ← 1/2(e + Abs[e])
5 b← a + 2η(k)e+
6 a← Y†b
7 if Abs[e] ≤ bmin then return a,b and exit
8 until k = kmax
9 Print NO SOLUTION FOUND
10 end
Since the weight vector a(k) is completely determined by the margin vector b(k),
this is basically an algorithm for producing a sequence of margin vectors. The initial
vector b(1) is positive to begin with, and if η > 0, all subsequent vectors b(k) are
positive. We might worry that if none of the components of e(k) is positive, so that
b(k) stops changing, we might fail to find a solution. However, we shall see that in
that case either e(k) = 0 and we have a solution, or e(k) ≤ 0 and we have proof that
the samples are not linearly separable.
5.9.2 Convergence Proof
We shall now show that if the samples are linearly separable, and if 0 < η < 1, then
the Ho-Kashyap algorithm will yield a solution vector in a finite number of steps. To
make the algorithm terminate, we should add a terminating condition stating that
corrections cease once a solution vector is obtained or some large criterion number of
iterations have occurred. However, it is mathematically more convenient to let the
corrections continue and show that the error vector e(k) either becomes zero for some
finite k, or converges to zero as k goes to infinity.
It is clear that either e(k) = 0 for some k — say k0 — or there are no zero vectors
in the sequence e(1), e(2), ... In the first case, once a zero vector is obtained, no further
changes occur to a(k), b(k), or e(k), and Ya(k) = b(k) > 0 for all k ≥ k0. Thus, if
we happen to obtain a zero error vector, the algorithm automatically terminates with
a solution vector.
Suppose now that e(k) is never zero for finite k. To see that e(k) must nevertheless
converge to zero, we begin by asking whether or not we might possibly obtain an e(k)
with no positive components. This would be most unfortunate, since we would have
40 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
Ya(k) ≤ b(k), and since e+(k) would be zero, we would obtain no further changes
in a(k), b(k), or e(k). Fortunately, this can never happen if the samples are linearly
separable. A proof is simple, and is based on the fact that if YtYa(k) = Ytb, then
Yte(k) = 0. But if the samples are linearly separable, there exists an â and a b̂ > 0
such that
Yâ = b̂.
Thus,
et(k)Yâ = 0 = et(k)b̂,
and since all the components of b̂ are positive, either e(k) = 0 or at least one of the
components of e(k) must be positive. Since we have excluded the case e(k) = 0, it
follows that e+(k) cannot be zero for finite k.
The proof that the error vector always converges to zero exploits the fact that the
matrix YY† is symmetric, positive semidefinite, and satisfies
(YY†)t(YY†) = YY†. (83)
Although these results are true in general, for simplicity we demonstrate them only for
the case where YtY is nonsingular. In this case YY† = Y(YtY)−1Yt, and the sym-
metry is evident. Since YtY is positive definite, so is (YtY)−1; thus, bY(YtY)−1Ytb ≥
0 for any b, and YY† is at least positive semidefinite. Finally, Eq. 83 follows from
(YY†)t(YY†) = [Y(YtY)−1Yt][Y(YtY)−1Yt].
To see that e(k) must converge to zero, we eliminate a(k) between Eqs. 80 & 82
and obtain
e(k) = (YY† − I)b(k).
Then, using a contant learning rate and Eq. 79 we obtain the recursion relation
e(k + 1) = (YY† − I)(b(k) + 2ηe+(k))
= e(k) + 2η(YY† − I)e+(k), (84)
so that
1
4
‖e(k + 1)‖2 = 1
4
‖e(k)‖2 + ηet(k)(YY† − I)e+(k) + ‖η(YY† − I)e+(k)‖2.
Both the second and the third terms simplify considerably. Since et(k)Y = 0, the
second term becomes
ηet(k)(YY† − I)e+(k) = −ηet(k)e+t(k) = −η‖e+(k)‖2,
5.9. *THE HO-KASHYAP PROCEDURES 41
the nonzero components of e+(k) being the positive components of e(k). Since YY†
is symmetric and is equal to (YY†)t(YY†), the third term simplifies to
‖η(YY† − I)e+(k)‖2 = η2e+t(k)(YY† − I)t(YY† − I)e+(k)
= η2‖e+(k)‖2 − η2e+(k)YY†e+(k),
and thus we have
1
4
(‖e(k)‖2 − ‖e(k + 1)‖2) = η(1− η)‖e+(k)‖2 + η2e+t(k)YY†e+(k). (85)
Since e+(k) is nonzero by assumption, and since YY† is positive semidefinite,
‖e(k)‖2 > ‖e(k + 1)‖2 if 0 < η < 1. Thus the sequence ‖e(1)‖2, ‖e(2)‖2, ... is
monotonically decreasing and must converge to some limiting value ‖e‖2. But for
convergence to take place, e+(k) must converge to zero, so that all the positive com-
ponents of e(k) must converge to zero. Since et(k)b̂ = 0 for all k, it follows that all of
the components of e(k) must converge to zero. Thus, if 0 < η < 1 and if the samples
are linearly separable, a(k) will converge to a solution vector as k goes to infinity.
If we test the signs of the components of Ya(k) at each step and terminate the
algorithm when they are all positive, we will in fact obtain a separating vector in a
finite number of steps. This follows from the fact that Ya(k) = b(k)+ e(k), and that
the components of b(k) never decrease. Thus, if bmin is the smallest component of b(1)
and if e(k) converges to zero, then e(k) must enter the hypersphere ‖e(k)‖ = bmin after
a finite number of steps, at which point Ya(k) > 0. Although we ignored terminating
conditions to simplify the proof, such a terminating condition would always be used
in practice.
5.9.3 Nonseparable Behavior
If the convergence proof just given is examined to see how the assumption of sepa-
rability was employed, it will be seen that it was needed twice. First, the fact that
et(k)b̂ = 0 was used to show that either e(k) = 0 for some finite k, or e+(k) is never
zero and corrections go on forever. Second, this same constraint was used to show
that if e+(k) converges to zero, e(k) must also converge to zero.
If the samples are not linearly separable, it no longer follows that if e+(k) is zero
then e(k) must be zero. Indeed, on a nonseparable problem one may well obtain a
nonzero error vector having no positive components. If this occurs, the algorithm
automatically terminates and we have proof that the samples are not separable.
What happens if the patterns are not separable, but e+(k) is never zero? In this
case it still follows that
e(k + 1) = e(k) + 2η(YY† − I)e+(k) (86)
and
1
4
(‖e(k)‖2 − ‖e(k + 1)‖2) = η(1− η)‖e+(k)‖2 + η2e+t(k)YY†e+(k). (87)
Thus, the sequence ‖e(1)‖2, ‖e(2)‖2, … must still converge, though the limiting value
‖e‖2 cannot be zero. Since convergence requires that e+(k) = 0 for some finite k,
42 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
or e+(k) converges to zero while ‖e(k)‖ is bounded away from zero. Thus, the Ho-
Kashyap algorithm provides us with a separating vector in the separable case, and
with evidence of nonseparability in the nonseparable case. However, there is no bound
on the number of steps needed to disclose nonseparability.
5.9.4 Some Related Procedures
If we write Y† = (YtY)−1Yt and make use of the fact that Yte(k) = 0, we can
modify the Ho-Kashyap rule as follows
b(1) > 0 but otherwise arbitrary
a(1) = Y†b(1)
b(k + 1) = b(k) + η(e(k) + |e(k)|)
a(k + 1) = a(k) + ηY†|e(k)|,
(88)
where, as usual,
e(k) = Ya(k)− b(k). (89)
This then gives the algorithm for fixed learning rate:
Algorithm 12 (Modified Ho-Kashyap)
1 begin initialize a,b, η < 1, criterion bmin, kmax
2 do k ← k + 1
3 e← Ya− b
4 e+ ← 1/2(e + Abs[e])
5 b← b + 2η(k)(e + Abs[e])
6 a← Y†b
7 if Abs[e] ≤ bmin then return a,b and exit
8 until k = kmax
9 print NO SOLUTION FOUND
10 end
This algorithm differs from the Perceptron and relaxation algorithms for solving
linear inequalities in at least three ways: (1) it varies both the weight vector a and
the margin vector b, (2) it provides evidence of nonseparability, but (3) it requires
the computation of the pseudoinverse of Y. Even though this last computation need
be done only once, it can be time consuming, and it requires special treatment if YtY
is singular. An interesting alternative algorithm that resembles Eq. 88 but avoids the
need for computing Y† is
b(1) > 0 but otherwise arbitrary
a(1) = arbitrary
b(k + 1) = b(k) + (e(k) + |e(k)|)
a(k + 1) = a(k) + ηRYt|e(k)|
, (90)
where R is an arbitrary, constant, postive-definite d̂-by-d̂ matrix. We shall show that
if η is properly chosen, this algorithm also yields a solution vector in a finite number
of steps, provided that a solution exists. Furthermore, if no solution exists, the vector
Yt|e(k)| either vanishes, exposing the nonseparability, or converges to zero.
The proof is fairly straightforward. Whether the samples are linearly separable or
not, Eqs. 89 & 90 show that
5.9. *THE HO-KASHYAP PROCEDURES 43
e(k + 1) = Ya(k + 1)− b(k + 1)
= (ηYRYt − I)|e(k)|.
We can find, then, that the squared magnitude is
‖e(k + 1)‖2 = |e(k)|t(η2YRYtYRY − 2ηYRYt + I)|e(k)|,
and furthermore
‖e‖2 − ‖e(k + 1)‖2 = (Yt|e(k)|)tA(Yt|e(k)|), (91)
where
A = 2ηR− η2RYtR. (92)
Clearly, if η is positive but sufficiently small, A will be approximately 2ηR and hence
positive definite. Thus, if Yt|e(k)| �= 0 we will have ‖e(k)‖2 > ‖e(k + 1)‖2.
At this point we must distinguish between the separable and the nonseparable
case. In the separable case there exists an â and a b̂ > 0 satisfying Yâ = b̂. Thus, if
|e(k)| �= 0,
|e(k)|tYâ = |e(k)|tb̂ > 0,
so that Yt|e(k)| can not be zero unless e(k) is zero. Thus, the sequence ‖e(1)‖2, ‖e(2)‖2, …
is monotonically decreasing and must converge to some limiting value ‖e‖2. But for
convergence to take place, Yt|e(k)| must converge to zero, which implies that |e(k)|
and hence e(k) must converge to zero. Since e(k) starts out positive and never de-
creases, it follows that a(k) must converge to a separating vector. Moreover, by the
same argument used before, a solution must actually be obtained after a finite number
of steps.
In the nonseparable case, e(k) can neither be zero nor converge to zero. It may
happen that Yt|e(k)| = 0 at some step, which would provide proof of nonseparability.
However, it is also possible for the sequence of corrections to go on forever. In this
case, it again follows that the sequence ‖e(1)‖2, ‖e(2)‖2, … must converge to a limiting
value ‖e‖2 �= 0, and that Yt|e(k)| must converge to zero. Thus, we again obtain
evidence of nonseparability in the nonseparable case.
Before closing this discussion, let us look briefly at the question of choosing η and
R. The simplest choice for R is the identity matrix, in which case A = 2ηI− η2YtY.
This matrix will be positive definite, thereby assuring convergence, if 0 < η < 2/λmax,
where λmax is the largest eigenvalue of YtY. Since the trace of YtY is both the sum
of the eigenvalues of YtY and the sum of the squares of the elements of Y, one can
use the pessimistic bound d̂λmax ≤
∑
i
‖yi‖2 in selecting a value for η.
A more interesting approach is to change η at each step, selecting that value that
maximizes ‖e(k)‖2 − ‖e(k + 1)‖2. Equations 91 & 92 give
‖e(k)‖2 − ‖e(k + 1)‖2 = |e(k)|tY(2ηR− η2RYtYR)Yt|e(k)|. (93)
By differentiating with respect to η, we obtain the optimal value
44 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
η(k) =
|e(k)|tYRYt|e(k)|
|e(k)|tYRYtYRYt|e(k)|
, (94)
which, for R = I, simplifies to
η(k) =
‖Yt|e(k)| ‖2
‖YYt|e(k)| ‖2
. (95)
This same approach can also be used to select the matrix R. By replacing R in Eq. 93
by the symmetric matrix R + δR and neglecting second-order terms, we obtain
δ(‖e(k)‖2 − ‖e(k + 1)‖2) = |e(k)|Y[δRt(I− ηYtYR) + (I− ηRYtY)δR]Yt|e(k)|.
Thus, the decrease in the squared error vector is maximized by the choice
R =
1
η
(YtY)−1 (96)
and since ηRYt = Y†, the descent algorithm becomes virtually identical with the
original Ho-Kashyap algorithm.
5.10 Linear Programming Algorithms
5.10.1 Linear Programming
The Perceptron, relaxation and Ho-Kashyap procedures are basically gradient de-
scent procedures for solving simultaneous linear inequalities. Linear programming
techniques are procedures for maximizing or minimizing linear functions subject to
linear equality or inequality constraints. This at once suggests that one might be able
to solve linear inequalities by using them as the constraints in a suitable linear pro-
gramming problem. In this section we shall consider two of several ways that this can
be done. The reader need have no knowledge of linear programming to understand
these formulations, though such knowledge would certainly be useful in applying the
techniques.
A classical linear programming problem can be stated as follows: Find a vector
u = (u1, ..., um)t that minimizes the linear (scalar) objective functionobjective
function
z = αtu (97)
subject to the constraint
Au ≥ β, (98)
where α is an m-by-1 cost vector, β is an l-by-1 vector, and A is an l-by-m matrix.
The simplex algorithm is the classical iterative procedure for solving this problemsimplex
algorithm (Fig. 5.18). For technical reasons, it requires the imposition of one more constraint,
viz., u ≥ 0.
If we think of u as being the weight vector a, this constraint is unacceptable, since
in most cases the solution vector will have both positive and negative components.
However, suppose that we write
5.10. *LINEAR PROGRAMMING ALGORITHMS 45
u1
u2
u3
Figure 5.18: Surfaces of constant z = αtu are shown in gray, while constraints of the
form Au.β are shown in red. The simplex algorithm finds an extremum of z given
the constraints, i.e., where the gray plan intersects the red at a single point.
a ≡ a+ − a−, (99)
where
a+ ≡ 1
2
(|a|+ a) (100)
and
a− ≡ 1
2
(|a| − a). (101)
Then both a+ and a− are nonnegative, and by identifying the components of u with
the components of a+ and a−, for example, we can accept the constraint u ≥ 0.
5.10.2 The Linearly Separable Case
Suppose that we have a set of n samples y1, ...,yn and we want a weight vector a that
satisfies atyi ≥ bi > 0 for all i. How can we formulate this as a linear programming
problem? One approach is to introduce what is called an artificial variable τ ≥ 0 by
writing
atyi + τ ≥ bi.
If τ is sufficiently large, there is no problem in satisfying these constraints; for example,
they are satisfied if a = 0 and τ = maxi bi.∗ However, this hardly solves our original
problem. What we want is a solution with τ = 0, which is the smallest value τ can
have and still satisfy τ ≥ 0. Thus, we are led to consider the following problem:
Minimize τ over all values of τ and a that satisfy the conditions atyi ≥ bi and τ ≥ 0.
∗ In the terminology of linear programming, any solution satisfying the constraints is called a feasible
solution. A feasible solution for which the number of nonzero variables does not exceed the number
of constraints (not counting the simplex requirement for nonnegative variables) is called a basic
feasible solution. Thus, the solution a = 0 and τ = maxi bi is a basic feasible solution. Possession
of such a solution simplifies the application of the simplex algorithm.
46 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
If the answer is zero, the samples are linearly separable, and we have a solution. If the
answer is positive, there is no separating vector, but we have proof that the samples
are nonseparable.
Formally, our problem is to find a vector u that minimizes the objective function
z = αtu subject to the constraints Au ≥ β and u ≥ 0, where
A =
yt1 −yt1 1
yt2 −yt2 1
…
…
…
ytn −ytn 1
, u =
a
+
a−
τ
, α =
00
1
, β =
b1
b2
…
bn
.
Thus, the linear programming problem involves m = 2d̂ + 1 variables and l = n
constraints, plus the simplex algorithm constraints u ≥ 0. The simplex algorithm will
find the minimum value of the objective function z = αtu = τ in a finite number
of steps, and will exhibit a vector û yielding that value. If the samples are linearly
separable, the minimum value of τ will be zero, and a solution vector â can be obtained
from û. If the samples are not separable, the minimum value of τ will be positive.
The resulting û is usually not very useful as an approximate solution, but at least one
obtains proof of nonseparability.
5.10.3 Minimizing the Perceptron Criterion Function
In the vast majority of pattern classification applications we cannot assume that the
samples are linearly separable. In particular, when the patterns are not separable,
one still wants to obtain a weight vector that classifies as many samples correctly as
possible. Unfortunately, the number of errors is not a linear function of the compo-
nents of the weight vector, and its minimization is not a linear programming problem.
However, it turns out that the problem of minimizing the Perceptron criterion func-
tion can be posed as a problem in linear programming. Since minimization of this
criterion function yields a separating vector in the separable case and a reasonable
solution in the nonseparable case, this approach is quite attractive.
Recall from Sect. ?? that the basic Perceptron criterion function is given by
Jp(a) =
∑
y∈Y
(−aty), (102)
where Y(a) is the set of training samples misclassified by a. To avoid the useless
solution a = 0, we introduce a positive margin vector b and write
J ′p(a) =
∑
y∈Y′
(bi − aty), (103)
where yi ∈ Y ′ if atyi ≤ bi. Clearly, J ′p is a piecewise-linear function of a, not a
linear function, and linear programming techniques are not immediately applicable.
However, by introducing n artificial variables and their constraints we can construct
an equivalent linear objective function. Consider the problem of finding vectors a and
τ that minimize the linear function
a =
n∑
i=1
τi
5.10. *LINEAR PROGRAMMING ALGORITHMS 47
subject to the constraints
τk ≥ 0 and τi ≥ bi − atyi.
Of course for any fixed value of a, the minimum value of z is exactly equal to J ′p(a),
since under these constraints the best we can do is to take τi = max[0, bi − atyi]. If
we minimize z over t and a, we shall obtain the minimum possible value of J ′p(a).
Thus, we have converted the problem of minimizing J ′p(a) to one of minimizing a
linear function z subject to linear inequality constraints. Letting un denote an n-
dimensional unit vector, we obtain the following problem with m = 2d̂ + n variables
and l = n constraints: Minimize αtu subject to Au ≥ β and u ≥ 0, where
A =
yt1 −yt1 1 0 · · · 0
yt2 −yt2 0 1 · · · 0
…
…
…
…
. . .
…
ytn −ytn 0 0 · · · 1
, u =
a
+
a−
τ
, α =
00
1n
, β =
b1
b2
…
bn
.
The choice a = 0 and τi = bi provides a basic feasible solution to start the simplex
algorithm, and the simplex algorithm will provide an â minimizing J ′p(a) in a finite
number of steps.
We have shown two ways to formulate the problem of finding a linear discriminant
function as a problem in linear programming. There are other possible formulations,
the ones involving the so-called dual problem being of particular interest from a com-
putational standpoint. Generally speaking, methods such as the simplex method are
merely sophisticated gradient descent methods for extremizing linear functions sub-
ject to linear constraints. The coding of a linear programming algorithm is usually
more complicated than the coding of the simpler descent procedures we described ear-
lier, and these descent procedures generalize naturally to multilayer neural networks.
However, general purpose linear programming packages can often be used directly or
modified appropriately with relatively little effort. When this can be done, one can
secure the advantage of guaranteed convergence on both separable and nonseparable
problems.
The various algorithms for finding linear discriminant functions presented in this
chapter are summarized in Table 5.1. It is natural to ask which one is best, but none
uniformly dominates or is uniformly dominated by all others. The choice depends
upon such considerations as desired characteristics, ease of programming, the number
of samples, and the dimensionality of the samples. If a linear discriminant function
can yield a low error rate, any of these procedures, intelligently applied, can provide
good performance.
48 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
Table 5.1: Descent Procedures for Obtaining Linear Discriminant Functions
Name Criterion Algorithm Conditions Rema
Fixed
Increment
Jp =
∑
aty≤0
(−aty)
a(k + 1) = a(k) + yk
(at(k)yk ≤ 0)
—
Finite
linear
soluti
a(k) a
Variable
Increment
J ′p =
∑
aty≤0
−(aty − b)
a(k + 1) = a(k) + η(k)yk
(at(k)yk ≤ b)
η(k) ≥ 0∑
η(k)→∞
∑
η2(k)
(
∑
η(k))2
→ 0
Conve
separ
with a
Finite
0 < α
Relaxation Jr = 12
∑
aty≤b
(aty−b)2
‖y‖2
a(k + 1) = a(k) + η b−a
t(k)yk
‖yk‖2 y
k
(at(k)yk ≤ b)
0 < η < 2
Conve
separ
with a
finite
soluti
Widrow-Hoff
(LMS)
Js =
∑
i
(atyi − bi)2
a(k + 1) =
a(k) + η(k)(bk − at(k)yk)yk
η(k) > 0
η(k)→ 0
Tends
minim
Stochastic Jm = E
[
(aty − z)2
]
a(k + 1) =
a(k) + η(k)(zk − at(k)yk)yk
∑
η(k)→∞
∑
η2(k)→ L <∞
Involv
numb
drawn
verges
to a s
Approx.
a(k + 1) =
a(k) + R(k)(z(k)− a(k)tyk)yk R
−1(k + 1) = R−1(k) + ykytk
mizin
vides
imatio
discri
Pseudo-
inverse
Js = ‖Ya− b‖2 a = Y†b —
Class
specia
yield
discri
appro
Bayes
Ho-Kashyap Js = ‖Ya− b‖2
b(k + 1) = b(k) + η(e(k) + |e(k)|)
e(k) = Ya(k)− b(k)
a(k) = Y†b(k)
0 < η < 1
b(1) > 0
a(k) i
for ea
vergen
separ
but e
sampl
non-s
b(k + 1) = b(k) + η(e(k) + (|e(k)|)
a(k + 1) = a(k) + ηRYt|e(k)|
η(k) =
|e(k)|tYRYt|e(k)|
|e(k)|tYRYtYRYt|e(k)|
is optimum;
R sym., pos. def.;
b(1) > 0
Finite
linear
if Yt|
e(k) �=
are no
Linear
τ =max
atyi≤bi
[−(atyi − bi)] Simplex algorithm
atyi + τ ≥ bi
b ≥ 0
Finite
both
nonse
useful
if sep
Program-
ming J ′p =
n∑
i=1
τi
=
∑
atyi≤bi
−(atyi − bi)
Simplex algorithm
atyi + τ ≥ bi
b ≥ 0
Finite
both
nonse
useful
if sep
5.11. *SUPPORT VECTOR MACHINES 49
5.11 *Support Vector Machines
We have seen how to train linear machines with margins. Support Vector Machines
(SVMs) are motivated by many of the same considerations, but rely on preprocessing
the data to represent patterns in a high dimension — typically much higher than the
original feature space. With an appropriate nonlinear mapping ϕ() to a sufficiently
high dimension, data from two categories can always be separated by a hyperplane
(Problem 27). Here we assume each pattern xk has been transformed to yk = ϕ(xk);
we return to the choice of ϕ() below. For each of the n patterns, k = 1, 2, …, n, we let
zk = ±1, according to whether pattern k is in ω1 or ω2. A linear discriminant in an
augmented y space is
g(y) = aty, (104)
where both the weight vector and the transformed pattern vector are augmented (by
a0 = w0 and y0 = 1, respectively). Thus a separating hyperplane insures
zkg(yk) ≥ 1 k = 1, …, n, (105)
much as was shown in Fig. 5.8.
In Sect. ??, the margin was any positive distance from the decision hyperplane.
The goal in training a Support Vector Machine is to find the separating hyperplane
with the largest margin; we expect that the larger the margin, the better generalization
of the classifier. As illustrated in Fig. 5.2 the distance from any hyperplane to a
(transformed) pattern y is |g(y)|/||a||, and assuming that a positive margin b exists,
Eq. 105 implies
zkg(yk)
||a||
≥ b k = 1, …, n; (106)
the goal is to find the weight vector a that maximizes b. Of course, the solution
vector can be scaled arbitrarily and still preserve the hyperplane, and thus to insure
uniqueness we impose the constraint b ||a|| = 1; that is, we demand the solution to
Eqs. 104 & 105 also minimize ||a||2.
The support vectors are the (transformed) training patterns for which Eq. 105 rep- support
vectorresents an equality — that is, the support vectors are (equally) close to the hyperplane
(Fig. 5.19). The support vectors are the training samples that define the optimal sepa-
rating hyperplane and are the most difficult patterns to classify. Informally speaking,
they are the patterns most informative for the classification task.
If Ns denotes the total number of support vectors, then for n training patterns
the expected value of the generalization error rate is bounded, according to
En[error rate] ≤
En[Ns]
n
, (107)
where the expectation is over all training sets of size n drawn from the (stationary)
distributions describing the categories. This bound is independent of the dimension-
ality of the space of transformed vectors, determined by ϕ(). We will return to this
equation in Chap. ??, but for now we can understand this informally by means of
the leave one out bound. Suppose we have n points in the training set, and train a leave-one-
out boundSupport Vector Machine on n − 1 of them, and test on the single remaining point.
If that remaining point happens to be a support vector for the full n sample case,
then there will be an error; otherwise, there will not. Note that if we can find a
50 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
transformation ϕ() that well separates the data — so the expected number of support
vectors is small — then Eq. 107 shows that the expected error rate will be lower.
y1
y2
R 1
R 2
optimal hyperplane
m
ax
im
um
m
ar
gi
n
b
m
ax
im
um
m
ar
gi
n
b
Figure 5.19: Training a Support Vector Machine consists of finding the optimal hy-
perplane, i.e., the one with the maximum distance from the nearest training patterns.
The support vectors are those (nearest) patterns, a distance b from the hyperplane.
The three support vectors are shown in solid dots.
5.11.1 SVM training
We now turn to the problem of training an SVM. The first step is, of course, to choose
the nonlinear ϕ-functions that map the input to a higher dimensional space. Often
this choice will be informed by the designer’s knowledge of the problem domain. In
the absense of such information, one might choose to use polynomials, Gaussians or
yet other basis functions. The dimensionality of the mapped space can be arbitrarily
high (though in practice it may be limited by computational resources).
We begin by recasting the problem of minimizing the magnitude of the weight
vector constrained by the separation into an unconstrained problem by the method
of Lagrange undetermined multipliers. Thus from Eq. 106 and our goal of minimizing
||a||, we construct the functional
L(a, α) =
1
2
||a||2 −
n∑
k=1
αk[zka
tyk − 1]. (108)
and seek to minimize L() with respect to the weight vector a, and maximize it with
respect to the undetermined multipliers αk ≥ 0. The last term in Eq. 108 expresses
the goal of classifying the points correctly. It can be shown using the so-called Kuhn-
Tucker construction (Problem 30) (also associated with Karush whose 1939 thesis
addressed the same problem) that this optimization can be reformulated as maximiz-
ing
L(α) =
n∑
k=1
αi −
1
2
n∑
k,j
αkαjzkzjy
t
jyk, (109)
subject to the constraints
5.11. *SUPPORT VECTOR MACHINES 51
n∑
k=1
zkαk = 0 αk ≥ 0, k = 1, …, n, (110)
given the training data. While these equations can be solved using quadratic pro-
gramming, a number of alternate schemes have been devised (cf. Bibliography).
Example 2: SVM for the XOR problem
The exclusive-OR is the simplest problem that cannot be solved using a linear
discriminant operating directly on the features. The points k = 1, 3 at x = (1, 1)t
and (−1,−1)t are in category ω1 (red in the figure), while k = 2, 4 at x = (1,−1)t
and (−1, 1)t are in ω2 (black in the figure). Following the approach of Support Vector
Machines, we preprocess the features to map them to a higher dimension space where
they can be linearly separated. While many ϕ-functions could be used, here we use
the simplest expansion up to second order: 1,
√
2×1,
√
2×2,
√
2x1x2, x21 and x
2
2, where
the
√
2 is convenient for normalization.
We seek to maximize Eq. 109,
4∑
k=1
αk −
1
2
n∑
k,j
αkαjzkzjy
t
jyk
subject to the constraints (Eq. 110)
α1 − α2 + α3 − α4 = 0
0 ≤ αk k = 1, 2, 3, 4.
It is clear from the symmetry of the problem that α1 = α3 and that α2 = α4 at the
solution. While we could use iterative gradient descent as described in Sect. 5.9, for
this small problem we can use analytic techniques instead. The solution is a∗k = 1/8,
for k = 1, 2, 3, 4, and from the last term in Eq. 108 this implies that all four training
patterns are support vectors — an unusual case due to the highly symmetric nature
of the XOR problem.
The final discriminant function is g(x) = g(x1, x2) = x1x2, and the decision
hyperplane is defined by g = 0, which properly classifies all training patterns. The
margin is easily computed from the solution ||a|| and is found to be b = 1/||a|| =
√
2.
The figure at the right shows the margin projected into two dimensions of the five
dimensional transformed space. Problem 28 asks you to consider this margin as viewed
in other two-dimensional projected sub-spaces.
An important benefit of the Support Vector Machine approach is that the com-
plexity of the resulting classifier is characterized by the number of support vectors —
independent of the dimensionality of the transformed space. This
52 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
x
1
x 2
0 1 2-1-2
0
1
2
-1
-2
g = 1
g = 1
g
=
-1
g
=
-1
2 x
1
2
x
1
x
2
0 1 2-1-2
0
1
2
-1
-2
g = 1
g = -1
g = 0
√
√
g = 0 b
=
2
√
The XOR problem in the original x1 − x2 feature space is shown at the left; the two
red patterns are in category ω1 and the two black ones in ω2. These four training
patterns x are mapped to a six-dimensional space by 1,
√
2×1,
√
2×2,
√
2x1x2, x21 and
x22. In this space, the optimal hyperplane is found to be g(x1, x2) = x1x2 = 0 and the
margin is b =
√
2. A two-dimensional projection of this space is shown at the right.
The hyperplanes through the support vectors are
√
2x1x2 = ±1, and correspond to
the hyperbolas x1x2 = ±1 in the original feature space, as shown.
5.12 Multicategory Generalizations
5.12.1 Kesler’s Construction
There is no uniform way to extend all of the two-category procedures we have discussed
to the multicategory case. In Sect. 5.2.2 we defined a multicategory classifier called a
linear machine which classifies a pattern by computing c linear discriminant functions
gi(x) = w
tx + wi0 i = 1, …, c,
and assigning x to the category corresponding to the largest discriminant. This is
a natural generalization for the multiclass case, particularly in view of the results
of Chap. ?? for the multivariate normal problem. It can be extended simply to
generalized linear discriminant functions by letting y(x) be a d̂-dimensional vector of
functions of x, and by writing
gi(x) = a
t
iy i = 1, …, c, (111)
where again x is assigned to ωi if gi(x) > gj(x) for all j �= i.
The generalization of our procedures from a two-category linear classifier to a
multicategory linear machine is simplest in the linearly-separable case. Suppose that
we have a set of labelled samples y1,y2, …,yn, with n1 in the subset Y1 labelled ω1,
n2 in the subset Y2 labelled ω2,…, and nc in the subset Yc labelled ωc. We say that
this set is linearly separable if there exists a linear machine that classifies all of them
correctly. That is, if these samples are linearly separable, then there exists a set of
weight vectors â1, …, âc such that if yk ∈ Yi, then
âtiyk > â
t
jyk (112)
for all j �= i.
One of the pleasant things about this definition is that it is possible to manipulate
these inequalities and reduce the multicategory problem to the two-category case.
Suppose for the moment that y ∈ Y1, so that Eq. 112 becomes
5.12. MULTICATEGORY GENERALIZATIONS 53
âtiyk − âtjyk > 0, j = 2, …, c. (113)
This set of c − 1 inequalities can be thought of as requiring that the cd̂-dimensional
weight vector
α̂ =
a1
a2
…
ac
correctly classifies all c− 1 of the cd̂-dimensional sample sets
η12 =
y
−y
0
…
0
, η13 =
y
0
−y
…
0
, · · · , η1c =
y
0
0
…
−y
.
In other words, each η1j corresponds to “normalizing” the patterns in ω1 and ωj .
More generally, if y ∈ Yi, we construct (c− 1)cd̂-dimensional training samples ηij by
partitioning ηij into cd̂-dimensional subvectors, with the ith subvector being y, the
jth being −y, and all others being zero. Clearly, if α̂tηij > 0 for j �= i, then the
linear machine corresponding to the components of α̂ classifies y correctly.
This so-called Kesler construction multiplies the dimensionality of the data by c
and the number of samples by c − 1, which does not make its direct use attractive.
Its importance resides in the fact that it allows us to convert many multicategory
error-correction procedures to two-category procedures for the purpose of obtaining
a convergence proof.
5.12.2 Convergence of the Fixed-Increment Rule
We now use use Kesler’s construction to prove convergence for a generalization of the
fixed-increment rule for a linear machine. Suppose that we have a set of n linearly-
separable samples y1, …,yn, and we use them to form an infinite sequence in which
every sample appears infinitely often. Let Lk denote the linear machine whose weight
vectors are a1(k), …,ac(k). Starting with an arbitrary initial linear machine L1, we
want to use the sequence of samples to construct a sequence of linear machines that
converges to a solution machine, one that classifies all of the samples correctly. We
shall propose an error-correction rule in which weight changes are made if and only
if the present linear machine misclassifies a sample. Let yk denote the kth sample
requiring correction, and suppose that yk ∈ Yi. Since yk requires correction, there
must be at least one j �= i for which
ati(k)y
k ≤ aj(k)tyk. (114)
Then the fixed-increment rule for correcting Lk is
54 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
ai(k + 1) = ai(k) + yk
aj(k + 1) = aj(k)− yk
al(k + 1) = al(k), l �= i and l �= j.
(115)
That is, the weight vector for the desired category is incremented by the pattern, the
weight vector for the incorrectly chosen category is decremented, and all other weights
are left unchanged (Problem 33, Computer exercise 12).
We shall now show that this rule must lead to a solution machine after a finite
number of corrections. The proof is simple. For each linear machine Lk there corre-
sponds a weight vector
αk =
a1(k)
…
ac(k)
.
For each sample y ∈ Yi there are c − 1 samples ηij formed as described in Sect. ??.
In particular, corresponding to the vector yk satisfying Eq. 114 there is a vector
ηkij =
·
·
·
yk
·
·
·
−yk
·
·
·
← i
← j
satisfying
αt(k)ηkij ≤ 0.
Furthermore, the fixed-increment rule for correcting Lk is the fixed-increment rule for
correcting α(k), viz.,
α(k + 1) = α(k) + ηkij .
Thus, we have obtained a complete correspondence between the multicategory case
and the two-category case, in which the multicategory procedure produces a sequence
of samples η1,η2, …,ηk, … and a sequence of weight vectors α1,α2, …,αk, … By our
results for the the two-cateogry case, this latter sequence can not be infinite, but must
terminate in a solution vector. Hence, the sequence L1, L2, …, Lk, … must terminate
in a solution machine after a finite number of corrections.
This use of Kesler’s construction to establish equivalences between multicategory
and two-category procedures is a powerful theoretical tool. It can be used to extend
5.12. MULTICATEGORY GENERALIZATIONS 55
all of our results for the Perceptron and relaxation procedures to the multicategory
case, and applies as well to the error-correction rules for the method of potential
functions (Problem ??). Unfortunately, it is not as directly useful in generalizing the
MSE or the linear programming approaches.
5.12.3 Generalizations for MSE Procedures
Perhaps the simplest way to obtain a natural generalization of the MSE procedures
to the multiclass case is to consider the problem as a set of c two-class problems. The
ith problem is to obtain a weight vector ai that is minimum-squared-error solution to
the equations
atiy = 1 for all y ∈ Yi
atiy = −1 for all y /∈ Yi.
}
In view of the results of Sect. 5.8.3 the number of samples is very large we will obtain
a minimum mean-squared-error approximation to the Bayes discriminant function
P (ωi|x)− P (not ωi|x) = 2P (ωi|x)− 1.
This observation has two immediate consequences. First, it suggests a modification
in which we seek a weight vector ai that is a minimum-squared-error solution to the
equations
atiy = 1 for all y ∈ Yi
atiy = 0 for all y /∈ Yi
}
(116)
so that aty will be a minimum mean-squared-error approximation to P (ωi|x). Second,
it justifies the use of the resulting discriminant functions in a linear machine, in which
we assign y to ωi if atiy > a
t
jy for all j �= i.
The pseudoinverse solution to the multiclass MSE problem can be written in a
form analogous to the form for the two-class case. Let Y be the n-by-d̂ matrix of
training samples, which we assume to be partitioned as
Y =
Y1
Y2
…
Yc
, (117)
with the samples labelled ωi comprising the rows of Yi. Similarly, let A be the d̂-by-c
matrix of weight vectors
A = [a1 a2 · · · ac], (118)
and let B be the n-by-c matrix
B =
B1
B2
…
Bc
, (119)
56 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
where all of the elements of Bi are zero except for those in the ith column, which
are unity. Then the trace of the “squared” error matrix (YA − B)t × (YA − B) is
minimized by the solution∗
A = Y†B, (120)
where, as usual, Y† is the pseudoinverse of Y.
This result can be generalized in a theoretically interesting fashion. Let λij be
the loss incurred for deciding ωi when the true state of nature is ωj , and let the jth
submatrix of B be given by
Bj = −
λ1j λ2j · · · λcj
λ1j λ2j · · · λcj
…
…
λ1j λ2j · · · λcj
#
nj%
j = 1, …, c. (121)
Then, as the number of samples approaches infinity, the solution A = Y†B yields dis-
criminant functions atiy which provide a minimum-mean-square-error approximation
to the Bayes discriminant function
g0i = −
c∑
j=1
λijP (ωi|x). (122)
The proof of this is a direct extension of the proof given in Sect. 5.8.3 (Problem 34).
Summary
This chapter considers discriminant functions that are a linear function of a set of
parameters, generally called weights. In all two-category cases, such discriminants
lead to hyperplane decision boundaries, either in the feature space itself, or in a
space where the features have been mapped by a nonlinear function (general linear
discriminants).
In broad overview, techniques such as the Perceptron algorithm adjust the param-
eters to increase the inner product with patterns in category ω1 and decrease the inner
product with those in ω2. A very general approach is to form some criterion function
and perform gradient descent. Different creiterion functions have different strengths
and weaknesses related to computation and convergence, none uniformly dominates
the others. One can use linear algebra to solve for the weights (parameters) directly,
by means of pseudoinverse matrixes for small problems.
In Support Vector Machines, the input is mapped by a nonlinear function to a high-
dimensional space, and the optimal hyperplane found, the one that has the largest
margin. The support vectors are those (transformed) patterns that determine the
margin; they are informally the hardest patterns to classify, and the most informative
ones for designing the classifier. An upper bound on expected error rate of the classifier
depends linearly upon the expected number of support vectors.
For multi-category problems, the linear machines create decision boundaries con-
sisting of sections of such hyperplanes. One can prove convergence of multi-category
∗ If we let bi denote the ith column of B, the trace of (YA − B)t(YA − B) is equal to the sum
of the squared lengths of the error vectors Yai − bi. The solution A = Y†B not only minimizes
this sum, but it also minimizes each term in the sum.
5.12. BIBLIOGRAPHICAL AND HISTORICAL REMARKS 57
algorithms by first converting them to two-category algorithms and using the two-
category proofs. The simplex algorithm finds the optimimun of a linear function
subject to (inequality) constraints, and can be used for training linear classifiers.
Linear discriminants, while useful, are not sufficiently general for arbitrary chal-
lenging pattern recognition problems (such as those involving multi-modal densities)
unless an appropriate nonlinear mapping (ϕ function) can be found. In this chapter
we have not considered any principled approaches to choosing such functions, but
turn to that topic in Chap. ??.
Bibliographical and Historical Remarks
Because linear discriminant functions are so amenable to analysis, far more papers
have been written about them than the subject deserves. Historically, all of this work
begins with the classic paper by Ronald A. Fisher [4]. The application of linear dis-
criminant function to pattern classification was well described in [7], which posed the
problem of optimal (minimum-risk) linear discriminant, and proposed plausible gra-
dient descient procedures to determine a solution from samples. Unfortunately, little
can be said about such procedures without knowing the underlying distributions, and
even then the situation is analytically complex. The design of multicategory classifiers
using two-category procedures stems from [12]. Minsky and Papert’s Perceptrons
[11] was influential in pointing out the weaknesses of linear classifiers — weaknesses
that were overcome by the methods we shall study in Chap. ??. The Winnow algo-
rithms [8] in the error-free case and [9, 6] and subsequent work in the general case
have been useful in the computational learning community, as they allow one to derive
convergence bounds.
While this work was statistically oriented, many of the pattern recognition papers
that appeared in the late 1950s and early 1960s adopted other viewpoints. One
viewpoint was that of neural networks, in which individual neurons were modelled as
threshold elements, two-category linear machines — work that had its origins in the
famous paper by McCulloch and Pitts [10].
As linear machines have been applied to larger and larger data sets in higher and
higher dimensions, the computational burden of linear programming [2] has made this
approach less popular. Stochastic approximations, e.g, [15],
An early paper on the key ideas in Support Vector Machines is [1]. A more
extensive treatment, including complexity control, can be found in [14] — material
we shall visit in Chap. ??. A readable presentation of the method is [3], which provided
the inspiration behind our Example 2. The Kuhn-Tucker construction, used in the
SVM training method described in the text and explored in Problem 30, is from [5]
and used in [13]. The fundamental result is that exactly one of the following three
cases holds. 1) The original (primal) conditions have an optimal solution; in that case
the dual cases do too, and their objective values are equal, or 2) the primal conditions
are infeasible; in that case the dual is either unbounded or itself infeasible, or 3) the
primal conditions are unbounded; in that case the dual is infeasible.
Problems
⊕
Section 5.2
58 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
1. Consider a linear machine with discriminant functions gi(x) = wtx + wi0, i =
1, …, c. Show that the decision regions are convex by showing that if x1 ∈ Ri and
x2 ∈ Ri then λx1 + (1− λ)x2 ∈ Ri if 0 ≤ λ ≤ 1.
2. Figure 5.3 illustrates the two most popular methods for designing a c-category
classifier from linear boundary segments. Another method is to save the full
(
c
2
)
linear ωi/ωj boundaries, and classify any point by taking a vote based on all these
boundaries. Prove whether the resulting decision regions must be convex. If they need
not be convex, construct a non-pathological example yielding at least one non-convex
decision region.
3. Consider the hyperplane used for discriminant functions.
(a) Show that the distance from the hyperplane g(x) = wtx + w0 = 0 to the point
xa is |g(xa)|/‖w‖ by minimizing ‖x− xa‖2 subject to the constraint g(x) = 0.
(b) Show that the projection of xa onto the hyperplane is given by
xp = xa −
g(xa)
‖w‖2
w.
4. Consider the three-category linear machine with discriminant functions gi(x) =
wtix + wi0, i = 1, 2, 3.
(a) For the special case where x is two-dimensional and the threshold weights wi0
are zero, sketch the weight vectors with their tails at the origin, the three lines
joining their heads, and the decision boundaries.
(b) How does this sketch change when a constant vector c is added to each of the
three weight vectors?
5. In the multicategory case, a set of samples is said to be linearly separable if there
exists a linear machine that can classify them all correctly. If any samples labelled
ωi can be separated from all others by a single hyperplane, we shall say the samples
are totally linearly separable. Show that totally linearly separable samples must betotal
linear
separability
linearly separable, but that the converse need not be true. (Hint: For the converse,
consider a case in which a linear machine like the one in Problem 4 separates the
samples.)
6. A set of samples is said to be pairwise linearly separable if there exist c(c− 1)/2pairwise
linear
separability
hyperplanes Hij such that Hij separates samples labelled ωi from samples ωj . Show
that a pairwise-linearly-separable set of patterns may not be linearly separable.
7. Let {y1, …,yn} be a finite set of linearly separable training samples, and let a be
called a solution vector if atyi ≥ 0 for all i. Show that the minimum-length solution
vector is unique. (Hint: Consider the effect of averaging two solution vectors.)
8. The convex hull of a set of vectors xi, i = 1, . . . , n is the set of all vectors of theconvex
hull form
x =
n∑
i=1
αixi,
where the coefficients αi are nonnegative and sum to one. Given two sets of vectors,
show that either they are linearly separable or their convex hulls intersect. (Hint:
Suppose that both statements are true, and consider the classification of a point in
the intersection of the convex hulls.)
5.12. PROBLEMS 59
9. A classifier is said to be a piecewise linear machine if its discriminant functionspiecewise
linear
machine
have the form
gi(x) = max
j=1,…,ni
gij(x),
where
gij(x) = w
t
ijx + wij0,
i = 1, …, c
j = 1, …, ni.
(a) Indicate how a piecewise linear machine can be viewed in terms of a linear
machine for classifying subclasses of patterns.
(b) Show that the decision regions of a piecewise linear machine can be nonconvex
and even multiply connected.
(c) Sketch a plot of gij(x) for a one-dimensional example in which n1 = 2 and
n2 = 1 to illustrate your answer to part (b).
10. Let the d components of x be either 0 or 1. Suppose we assign x to ω1 if the
number of non-zero components of x is odd, and to ω2 otherwise. (This is called the
d-bit parity problem.)
(a) Show that this dichotomy is not linearly separable if d > 1.
(b) Show that this problem can be solved by a piecewise linear machine with d+ 1
weight vectors wij (see Problem 9). (Hint: Consider vectors of the form wij =
αij(1, 1, …, 1)t.)
⊕
Section 5.3
11. Consider the quadratic discriminant function (Eq. 4)
g(x) = w0 +
d∑
i=1
wixi +
d∑
i=1
d∑
j=1
wijxixj ,
and define the symmetric, nonsingular matrix W = [wij ]. Show that the basic
character of the decision boundary can be described in terms of the scaled matrix
W̄ = W/(wtW−1w − 4w0) as follows:
(a) If W̄ ∝ I (the identity matrix), then the decision boundary is a hypersphere.
(b) If W̄ is positive definite, then the decision boundary is a hyperellipsoid.
(c) If some eigenvalues of W̄ are positive and some negative, then the decision
boundary is a hyperhyperboloid.
(d) Suppose w =
52
-3
and W =
1 2 02 5 1
0 1 -3
. What is the character of
the solution?
60 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
(e) Repeat part (d) for w =
2-1
3
and W =
1 2 32 0 4
3 4 -5
.
⊕
Section 5.4
12. Derive Eq. 14, where J(·) depends on the iteration step k.
13. Consider the sum-of-squared-error criterion function (Eq. 43),
Js(a) =
n∑
i=1
(atyi − bi)2.
Let bi = b and consider the following six training points:
ω1 : (1, 5)t, (2, 9)t, (−5,−3)t
ω2 : (2,−3)t, (−1,−4)t, (0, 2)t
(a) Calculate the Hessian matrix for this problem.
(b) Assuming the quadratic criterion function calculate the optimal learning rate η.
⊕
Section 5.5
14. In the convergence proof for the Perceptron algorithm (Theorem 5.1) the scale
factor α was taken to be β2/γ.
(a) Using the notation of Sect. 5.5, show that if α is greater than β2/(2γ) the
maximum number of corrections is given by
k0 =
‖a1 − αa‖2
2αγ − β2
.
(b) If a1 = 0, what value of α minimizes k0?
15. Modify the convergence proof given in Sect. 5.5.2 (Theorem 5.1) to prove the
convergence of the following correction procedure: starting with an arbitrary initial
weight vector a1, correct a(k) according to
a(k + 1) = a(k) + η(k)yk,
if and only if at(k)yk fails to exceed the margin b, where η(k) is bounded by 0 < ηa ≤ η(k) ≤ ηb <∞. What happens if b is negative? 16. Let {y1, ...,yn} be a finite set of linearly separable samples in d dimensions. (a) Suggest an exhaustive procedure that will find a separating vector in a finite number of steps. (Hint: Consider weight vectors whose components are integer valued.) 5.12. PROBLEMS 61 (b) What is the computational complexity of your procedure? 17. Consider the criterion function Jq(a) = ∑ y∈Y(a) (aty − b)2 where Y(a) is the set of samples for which aty ≤ b. Suppose that y1 is the only sample in Y(a(k)). Show that ∇Jq(a(k)) = 2(at(k)y1 − b)y1 and that the matrix of second partial derivatives is given by D = 2y1yt1. Use this to show that when the optimal η(k) is used in Eq. ?? the gradient descent algorithm yields a(k + 1) = a(k) + b− aty1 ‖y1‖2 y1. 18. Given the conditions in Eqs. 28 – 30, show that a(k) in the variable increment descent rule indeed converges for atyi > b for all i.⊕
Section 5.6
19. Sketch a figure to illustrate the proof in Sec. 5.6.2. Be sure to take a general
case, and label all variables.
⊕
Section 5.7
⊕
Section 5.8
20. Show that the scale factor α in the MSE solution corresponding to Fisher’s linear
discriminant (Sect. ??) is given by
α =
[
1 +
n1n2
n
(m1 −m2)tS−1W (m1 −m2)
]−1
21. Generalize the results of Sect. 5.8.3 to show that the vector a that minimizes
the criterion function
J ′s(a) =
∑
y∈Y1
(aty − (λ21 − λ11))2 +
∑
y∈Y2
(aty − (λ12 − λ22))2
provides asymptotically a minimum-mean-squared-error approximation to the Bayes
discriminant function (λ21 − λ11)P (ω1|x)− (λ12 − λ22)P (ω2|x).
22. Consider the criterion function Jm(a) = E [(aty(x)− z)2] and the Bayes discrim-
inant function g0(x).
(a) Show that
Jm = E [(aty − g0)2]− 2E [(aty − g0)(z − g0)] + E [(z − g0)2].
(b) Use the fact that the conditional mean of z is g0(x) in showing that the â that
minimizes Jm also minimizes E [(aty − g0)2] .
62 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
23. A scalar analog of the relation R−1k+1 = R
−1
k + yky
t
k used in stochastic approxi-
mation is η−1(k + 1) = η−1(k) + y2k.
(a) Show that this has the closed form solution
η(k) =
η(1)
1 + η(1)
k−1∑
i=1
y2i
.
(b) Assume that η(1) > 0 and 0 < a ≤ y2i ≤ b <∞, and indicate why this sequence
of coefficients will satisfy
∑
η(k)→∞ and
∑
η(k)2 → L <∞.
24. Show that for the Widrow-Hoff or LMS rule that if η(k) = η(1)/k then the
sequence of weight vectors converges to a limiting vector a satisfying Y†(Ya−b) = 0
(Eq. 61).
⊕
Section 5.9
25. Consider the following six data points:
ω1 : (1, 2)t, (2,−4)t, (−3,−1)t
ω2 : (2, 4)t, (−1,−5)t, (5, 0)t
(a) Are they linearly separable?
(b) Using the approach in the text, assume R = I, the identity matrix, and calculate
the optimal learning rate η by Eq. 85.
⊕
Section 5.10
26. The linear programming problem formulated in Sect. 5.10.2 involved minimizing
a single artificial variable τ under the constraints atyi + τ > bi and τ ≥ 0. Show that
the resulting weight vector minimizes the criterion function
Jτ (a) = max
atyi≤bi
[bi − atyi].
⊕
Section 5.11
27. Discuss qualitatively why if samples from two categories are distinct (i.e., no
feature point is labelled by both categories), there always exists a nonlinear mapping
to a higher dimension that leaves the points linearly separable.
28. The figure in Example 2 shows the maximum margin for a Support Vector
Machine applied to the exclusive-OR problem mapped to a five-dimensional space.
That figure shows the training patterns and contours of the discriminant function, as
projected in the two-dimensional subspace defined by the features
√
2×1 and
√
2x1x2.
Ignore the constant feature, and consider the other four features. For each of the(
4
2
)
− 1 = 5 pairs of features other than the one shown in the Example, show the
5.12. PROBLEMS 63
patterns and the lines corresponding to the discriminant g = ±1. Are the margins in
your figures the same? Explain why or why not?
29. Consider a Support Vector Machine and the following training data from two
categories:
category x1 x2
ω1 1 1
ω1 2 2
ω1 2 0
ω2 0 0
ω2 1 0
ω2 0 1
(a) Plot these six training points, and construct by inspection the weight vector for
the optimal hyperplane, and the optimal margin.
(b) What are the support vectors?
(c) Construct the solution in the dual space by finding the Lagrange undetermined
multipliers, αi. Compare your result to that in part (a).
30. This problem asks you to follow the Kuhn-Tucker theorem to convert the con-
strained optimization problem in Support Vector Machines into a dual, unconstrained
one. For SVMs, the goal is to find the minimum length weight vector a subject to
the (classification) constraints
zka
tyk ≥ 1 k = 1, …, n,
where zk = ±1 indicates the target getegory of each of the n patterns yk. Note that
a and y are augmented (by a0 and y0 = 1, respectively).
(a) Consider the unconstrained optimization associated with SVMS:
L(a,α) =
1
2
||a||2 −
n∑
k=1
αk[zka
tyk − 1].
In the space determined by the components of a, and the n (scalar) undeter-
mined multipliers αk, the desired solution is a saddle point, rather than a global
maximum or minimum. Explain.
(b) Next eliminate the dependency of this (“primal”) functional upon a, i.e., refor-
mulated the optimization in a dual form, by the following steps. Note that at
the saddle point of the primal functional, we have
∂L(a∗,α∗)
∂a
= 0.
Solve for the partial derivatives and conclude
n∑
k=1
α∗kzk = 0 α
∗
k ≥ 0, k = 1, …n.
64 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
(c) Prove that at this inflection point, the optimal hyperplane is a linear combina-
tion of the training vectors:
a∗ =
n∑
k=1
α∗kzkyk.
(d) According to the Kuhn-Tucker theorem (cf. Bibliography), an undetermined
multiplier α∗k is non-zero only if the corresponding sample yk satisfies zka
tyk =
0. Show that this can be expressed as
α∗k[zka
∗tyk − 1] = 0, k = 1, …, n.
(The samples where α∗k are nonzero, i.e., zka
tyk = 1, are the support vectors.)
(e) Use the results from parts (b) – (c) to eliminate the weight vector in the func-
tional, and thereby construct the dual functional
L̃(a,α) =
1
2
||a||2 −
n∑
k=1
αkzka
tyk +
n∑
k=1
αk.
(f) Substitute the solution a∗ from part (c) to find the dual functional
L̃(α) = −1
2
n∑
j,k=1
αjαkzjzk(y
t
jyk) +
n∑
j=1
αj .
31. Repeat Example 2.⊕
Section 5.12
32. Suppose that for each two-dimensional training point yi in category ω1 there is
a corresponding (symmetric) point in ω2 at −yi.
(a) Prove that a separating hyperplane (should one exist) or LMS solution must go
through the origin.
(b) Consider such a symmetric, six-point problem with the following points:
ω1 : (1, 2)t, (2,−4)t,y
ω2 : (−1,−2)t, (−2, 4)t,−y
Find the matematical conditions on y such that the LMS solution for this prob-
lem not give a separating hyperplane.
(c) Generalize your result as follows. Suppose ω1 consists of y1 and y2 (known)
and the symmetric versions in ω2. What is the condition on y3 such that the
LMS solution does not separate the points.
5.12. COMPUTER EXERCISES 65
33. Write pseudocode for a fixed increment multicategory algorithm based on
Eq. 115. Discuss the strenths and weakness of such an implementation.
34. Generalize the discussion in Sect. 5.8.3 in order to prove that the solution
derived from Eq. 120 provieds a minimum-mean-square-error approximation to the
Bayes discriminant function given in Eq. 122.
Computer exercises
Several of the exercises use the data in the following table.
ω1 ω2 ω3 ω4
sample x1 x2 x1 x2 x1 x2 x1 x2
1 0.1 1.1 7.1 4.2 -3.0 -2.9 -2.0 -8.4
2 6.8 7.1 -1.4 -4.3 0.5 8.7 -8.9 0.2
3 -3.5 -4.1 4.5 0.0 2.9 2.1 -4.2 -7.7
4 2.0 2.7 6.3 1.6 -0.1 5.2 -8.5 -3.2
5 4.1 2.8 4.2 1.9 -4.0 2.2 -6.7 -4.0
6 3.1 5.0 1.4 -3.2 -1.3 3.7 -0.5 -9.2
7 -0.8 -1.3 2.4 -4.0 -3.4 6.2 -5.3 -6.7
8 0.9 1.2 2.5 -6.1 -4.1 3.4 -8.7 -6.4
9 5.0 6.4 8.4 3.7 -5.1 1.6 -7.1 -9.7
10 3.9 4.0 4.1 -2.2 1.9 5.1 -8.0 -6.3⊕
Section 5.4
1. Consider basic gradient descent (Algorithm 1) and Newton’s algorithm (Algo-
rithm 2) applied to the data in the tables.
(a) Apply both to the three-dimensional data in categories ω1 and ω3. For the
gradient descent use η(k) = 0.1. Plot the criterion function as function of the
iteration number.
(b) Estimate the total number of mathematical operations in the two algorithms.
(c) Plot the convergence time versus learning rate. What is the minimum learning
rate that fails to lead to convergence?
⊕
Section 5.5
2. Write a program to implement the Perceptron algorithm.
(a) Starting with a = 0, apply your program to the training data from ω1 and ω2.
Note the number of iterations required for convergence.
(b) Apply your program to ω3 and ω2. Again, note the number of iterations required
for convergence.
(c) Explain the difference between the iterations required in the two cases.
3. The Pocket algorithm uses the criterion of longest sequence of correctly classified Pocket
algorithmpoints, and can be used in conjunction a number of basic learning algorithms. For
instance, one use the Pocket algorithm in conjunction with the Perceptron algorithm
in a sort of ratchet scheme as follows. There are two sets of weights, one for the normal
66 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
Perceptron algorithm, and a separate one (not directly used for training) which is kept
“in your pocket.” Both are randomly chosen at the start. The “pocket” weights are
tested on the full data set to find the longest run of patterns properly classified. (At
the beginning, this run will be short.) The Perceptron weights are trained as usual,
but after every weight update (or after some finite number of such weight updates),
the Perceptron weight is tested on data points, randomly selected, to determine the
longest run of properly classified points. If this length is greater than the pocket
weights, the Perceptron weights replace the pocket weights, and perceptron training
continues. In this way, the poscket weights continually improve, classifying longer and
longer runs of randomly selected points.
(a) Write a pocket algorithm to be employed with Perceptron algorithm.
(b) Apply it to the data from ω1 and ω3. How often are the pocket weights updated?
4. Start with a randomly chosen a, Calculate β2 (Eq. 21 At the end of training
calculate γ (Eq. 22). Verify k0 (Eq. 25).
5. Show that the first xx points of categories ωx and ωxxx. Construct by hand
a nonlinear mapping of the feature space to make them linearly separable. Train a
Perceptron classifier on them.
6. Consider a version of the Balanced Winnow training algorithm (Algorithm 7).
Classification of test data is given by line 2. Compare the converge rate of Balanced
Winnow with the fixed-increment, single-sample Perceptron (Algorithm 4) on a prob-
lem with large number of redundant features, as follows.
(a) Generate a training set of 2000 100-dimensional patterns (1000 from each of two
categories) in which only the first ten features are informative, in the following
way. For patterns in category ω1, each of the first ten features are chosen ran-
domly and uniformly from the range +1 ≤ xi ≤ 2, for i = 1, …, 10. Conversely,
for patterns in ω2, each of the first ten features are chosen randomly and uni-
formly from the range −2 ≤ xi ≤ −1. All other features from both categories
are chosen from the range −2 ≤ xi ≤ +2.
(b) Construct by hand the obvious separating hyperplane.
(c) Adjust the learning rates so that your two algorithms have roughly the same
convergence rate on the full training set when only the first ten features are
considered. That is, assume each of the 2000 training patterns consists of just
the first ten features.
(d) Now apply your two algorithms to 2000 50-dimensional patterns, in which the
first ten features are informative and the remaining 40 are not. Plot the total
number of errors versus iteration.
(e) Now apply your two algorithms to the full training set of 2000 100-dimensional
patterns.
(f) Summarize your answers to parts (c) – (e).
⊕
Section 5.6
7. Consider relaxation methods.
5.12. COMPUTER EXERCISES 67
(a) Implement batch relaxation with margin (Algorithm 8), set b = 0.1 and a(1) = 0
and apply it to the data in ω1 and ω3. Plot the criterion function as a function
of the number of passes through the training set.
(b) Repeat for b = 0.5 and a(1) = 0. Explain qualitatively any difference you find
in the convergence rates.
(c) Modify your program to use single sample learning. Again, plot the criterion as
a function of the number of passes through the training set.
(d) Discuss any differences, being sure to consider the learning rate.
⊕
Section 5.8
8. Write a single-sample relaxation algorithm and use Eq. ?? for updating R. Apply
your program to the data in ω2 and ω3.⊕
Section 5.9
9. Implement the Ho-Kashyap algorithm (Algorithm 11) and apply to the data in
categories ω1 and ω3. Repeat for categories ω4 and ω2.⊕
Section 5.10
10. example where the LMS rule need not give the separating vector, even if one
exists⊕
Section 5.11
11. Support Vector Machine xxx. Apply it to the classification of ω3 and ω4.⊕
Section 5.12
12. Write a programto implement a multicategory generalization of basic single-
sample relaxation without margin (Algorithm ??).
(a) Apply it to the data in all four categories in the table.
(b) Use your algorithm in a two-category mode to form ωi/notωi boundaries for i =
1, 2, 3, 4. Find any regions whose categorization by your system is ambiguous.
68 CHAPTER 5. LINEAR DISCRIMINANT FUNCTIONS
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[6] Adam J. Grove, Nicholas Littlestone, and Dale Schuurmans. General convergence
results for linear discriminant updates. In Proceedings of the COLT 97, pages
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threshold learning using Winnow. In Proceedings of COLT 91, pages 147–156.
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[13] Gilbert Strang. Introduction to Applied Mathematics. Wellesley Cambridge Press,
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on Computers, C-17:861–872, 1968.
Index
batch training, 16
bias (offset), 4
bound
leave-one-out, 49
classifier
Gaussian, 7
linear
margin, 12
simplicity, 3
complexity
classifier, 51
convergence
Ho-Kashyap, 39
criterion function
minimizing, 3
second-order expansion, 13
curse of dimensionality, 9
decision boundary
hyperplane, 5
bias, 5
negative side, 5
positive side, 5
linear
convex, 6
decision surface
hyperellipsoid, 7
hyperhyperboloid, 7
hypersphere, 7
descent procedures
table, 48
discriminant
generalized linear, 7
homogeneous linear, 8
hyperquadratic, 7
linear
hyperplane, 4
multicategory, 6
optimal, 3
quadratic, 6
dual functional, 63
error
training, 3
problems, 3
estimation
nonparametric, 3
exclusive-OR, 51
fixed increment, 48
training
Algorithm, 18, 21
gap
Winnow algorithm, see Winnow al-
gorithm, gap
gradient descent, 13
Algorithm, 13
convergence, 17
fixed increment, 17, 18
Hessian
singular, 14
Hessian matrix, 13
Ho-Kashyap
bound, 43
convergence, 43
separability, 41
training
Algorithm, 39
Ho-Kasyap
nonseparability, 43
Kessler’s construction, 52
fixed increment, 53
Kiefer-Wolfowitz, 37
Kuhn-Tucker construction, 50
learning rate, 13
least-mean-square
Algorithm, 34
71
72 INDEX
leave-one-out bound, see bound, leave-
one-out
linear discriminant
dual, 47
linear machine
fixed increment, 53
linear programming
objective, 44
linear separability, 11, 16, 40
LMS, see least-mean-square, 48
LMs
training
Algorithm, 34
margin, see classifier,linear,margin
Modified Ho-Kashyap
training
Algorithm, 42
MSE algorithm
generalizations, 55
Newton
descent
Algorithm, 14
Newton’s algorithm, 14
Perceptron
Batch
Algorithm, 15
batch
Algorithm, 22
perceptron
criterion, 14
Perceptron algorithm
criterion, 46
phi function, 51
Pocket algorithm, 65
polynomial discriminant, 7
primal functional, 63
pseudoinverse, 48
quadratic programming, 51
relaxation
batch
Algorithm, 24
training
Algorithm, 25
relaxation rule
single-sample, 25
risk
sample, see error, training
simplex algorithm, 44, 48
constraint, 46
feasible solution, 45
nonseparability, 46
solution region, 12
stochastic descent, 48
support vector, 49
Support Vector Machines, 49–51
translation
axis, 7
variable increment, 48
vector
separating, see vector, solution
solution, 11
support, see support vector
weight, 4
weight space, 11
weight vector, 19
Winnow
Algorithm
balanced, 23
Winnow algorithm, 23
balanced, 23, 66
gap, 23
Contents
6 Multilayer Neural Networks 3
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
6.2 Feedforward operation and classification . . . . . . . . . . . . . . . . . 4
6.2.1 General feedforward operation . . . . . . . . . . . . . . . . . . 8
6.2.2 Expressive power of multilayer networks . . . . . . . . . . . . . 8
6.3 Backpropagation algorithm . . . . . . . . . . . . . . . . . . . . . . . . 10
6.3.1 Network learning . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6.3.2 Training protocols . . . . . . . . . . . . . . . . . . . . . . . . . 15
Algorithm 1: Stochastic Backpropagation . . . . . . . . . . . . . . . . . 15
Algorithm 2: Batch Backpropagation . . . . . . . . . . . . . . . . . . . 16
6.3.3 Learning curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.4 Error surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.4.1 Some small networks . . . . . . . . . . . . . . . . . . . . . . . . 17
6.4.2 XOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.4.3 Larger networks . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.4.4 How important are multiple minima? . . . . . . . . . . . . . . 20
6.5 Backpropagation as feature mapping . . . . . . . . . . . . . . . . . . . 21
6.5.1 Representations at the hidden layer — weights . . . . . . . . . 23
6.6 Backpropagation, Bayes theory and probability . . . . . . . . . . . . . 24
6.6.1 Bayes discriminants and neural networks . . . . . . . . . . . . . 25
6.6.2 Outputs as probabilities . . . . . . . . . . . . . . . . . . . . . . 26
6.7 *Related statistical techniques . . . . . . . . . . . . . . . . . . . . . . . 28
6.8 Practical techniques for backpropagation . . . . . . . . . . . . . . . . . 29
6.8.1 Transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.8.2 Parameters for the sigmoid . . . . . . . . . . . . . . . . . . . . 30
6.8.3 Scaling input . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.8.4 Target values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.8.5 Training with noise . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.8.6 Manufacturing data . . . . . . . . . . . . . . . . . . . . . . . . 32
6.8.7 Number of hidden units . . . . . . . . . . . . . . . . . . . . . . 33
6.8.8 Initializing weights . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.8.9 Learning rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.8.10 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Algorithm 3: Stochastic Backpropagation with momentum . . . . . . . 36
6.8.11 Weight decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.8.12 Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.8.13 On-line, stochastic or batch training? . . . . . . . . . . . . . . 38
6.8.14 Stopped training . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1
2 CONTENTS
6.8.15 How many hidden layers? . . . . . . . . . . . . . . . . . . . . . 39
6.8.16 Criterion function . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.9 *Second-order methods . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.9.1 Hessian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.9.2 Newton’s method . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.9.3 Quickprop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.9.4 Conjugate gradient descent . . . . . . . . . . . . . . . . . . . . 43
Example 1: Conjugate gradient descent . . . . . . . . . . . . . . . . . . 44
6.10 *Additional networks and training methods . . . . . . . . . . . . . . . 46
6.10.1 Radial basis function networks (RBF) . . . . . . . . . . . . . . 46
6.10.2 Special bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.10.3 Time delay neural networks (TDNN) . . . . . . . . . . . . . . . 47
6.10.4 Recurrent networks . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.10.5 Counterpropagation . . . . . . . . . . . . . . . . . . . . . . . . 49
6.10.6 Cascade-Correlation . . . . . . . . . . . . . . . . . . . . . . . . 50
Algorithm 4: Cascade-correlation . . . . . . . . . . . . . . . . . . . . . 50
6.10.7 Neocognitron . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.11 Regularization and complexity adjustment . . . . . . . . . . . . . . . . 51
6.11.1 Complexity measurement . . . . . . . . . . . . . . . . . . . . . 53
6.11.2 Wald statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Bibliographical and Historical Remarks . . . . . . . . . . . . . . . . . . . . 56
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Computer exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Chapter 6
Multilayer Neural Networks
6.1 Introduction
I n the previous chapter we saw a number of methods for training classifiers con-sisting of input units connected by modifiable weights to output units. The LMS
algorithm, in particular, provided a powerful gradient descent method for reducing
the error, even when the patterns are not linearly separable. Unfortunately, the class
of solutions that can be obtained from such networks — hyperplane discriminants
— while surprisingly good on a range or real-world problems, is simply not general
enough in demanding applications: there are many problems for which linear discrim-
inants are insufficient for minimum error.
With a clever choice of nonlinear ϕ functions, however, we can obtain arbitrary
decisions, in particular the one leading to minimum error. The central difficulty is,
naturally, choosing the appropriate nonlinear functions. One brute force approach
might be to choose a complete basis set (all polynomials, say) but this will not work;
such a classifier would have too many free parameters to be determined from a limited
number of training patterns (Chap. ??). Alternatively, we may have prior knowledge
relevant to the classification problem and this might guide our choice of nonlinearity.
In the absence of such information, up to now we have seen no principled or auto-
matic method for finding the nonlinearities. What we seek, then, is a way to learn
the nonlinearity at the same time as the linear discriminant. This is the approach
of multilayer neural networks (also called multilayer Perceptrons): the parameters
governing the nonlinear mapping are learned at the same time as those governing the
linear discriminant.
We shall revisit the limitations of the two-layer networks of the previous chapter,∗
and see how three-layer (and four-layer…) nets overcome those drawbacks — indeed
how such multilayer networks can, at least in principle, provide the optimal solution
to an arbitrary classification problem. There is nothing particularly magical about
multilayer neural networks; at base they implement linear discriminants, but in a space
where the inputs have been mapped nonlinearly. The key power provided by such
networks is that they admit fairly simple algorithms where the form of the nonlinearity
∗ Some authors describe such networks as single layer networks because they have only one layer of
modifiable weights, but we shall instead refer to them based on the number of layers of units.
3
4 CHAPTER 6. MULTILAYER NEURAL NETWORKS
can be learned from training data. The models are thus extremely powerful, have nice
theoretical properties, and apply well to a vast array of real-world applications.
One of the most popular methods for training such multilayer networks is based
on gradient descent in error — the backpropagation algorithm (or generalized deltabackpropagation
rule), a natural extension of the LMS algorithm. We shall study backpropagation
in depth, first of all because it is powerful, useful and relatively easy to understand,
but also because many other training methods can be seen as modifications of it.
The backpropagation training method is simple even for complex models (networks)
having hundreds or thousands of parameters. In part because of the intuitive graphical
representation and the simplicity of design of these models, practitioners can test
different models quickly and easily; neural networks are thus a sort of “poor person’s”
technique for doing statistical pattern recognition with complicated models. The
conceptual and algorithmic simplicity of backpropagation, along with its manifest
success on many real-world problems, help to explain why it is a mainstay in adaptive
pattern recognition.
While the basic theory of backpropagation is simple, a number of tricks — some
a bit subtle — are often used to improve performance and increase training speed.
Choices involving the scaling of input values and initial weights, desired output values,
and more can be made based on an analysis of networks and their function. We shall
also discuss alternate training schemes, for instance ones that are faster, or adjust
their complexity automatically in response to training data.
Network architecture or topology plays an important role for neural net classifi-
cation, and the optimal topology will depend upon the problem at hand. It is here
that another great benefit of networks becomes apparent: often knowledge of the
problem domain which might be of an informal or heuristic nature can be easily in-
corporated into network architectures through choices in the number of hidden layers,
units, feedback connections, and so on. Thus setting the topology of the network is
heuristic model selection. The practical ease in selecting models (network topologies)
and estimating parameters (training via backpropagation) enable classifier designers
to try out alternate models fairly simply.
A deep problem in the use of neural network techniques involves regularization,regular-
ization complexity adjustment, or model selection, that is, selecting (or adjusting) the com-
plexity of the network. Whereas the number of inputs and outputs is given by the
feature space and number of categories, the total number of weights or parameters in
the network is not — or at least not directly. If too many free parameters are used,
generalization will be poor; conversely if too few parameters are used, the training
data cannot be learned adequately. How shall we adjust the complexity to achieve
the best generalization? We shall explore a number of methods for complexity ad-
justment, and return in Chap. ?? to their theoretical foundations.
It is crucial to remember that neural networks do not exempt designers from
intimate knowledge of the data and problem domain. Networks provide a powerful
and speedy tool for building classifiers, and as with any tool or technique one gains
intuition and expertise through analysis and repeated experimentation over a broad
range of problems.
6.2 Feedforward operation and classification
Figure 6.1 shows a simple three-layer neural network. This one consists of an input
layer (having two input units), a hidden layer with (two hidden units)∗ and an outputhidden
layer
6.2. FEEDFORWARD OPERATION AND CLASSIFICATION 5
layer (a single unit), interconnected by modifiable weights, represented by links be-
tween layers. There is, furthermore, a single bias unit that is connected to each unit bias
other than the input units. The function of units is loosely based on properties of bio-
logical neurons, and hence they are sometimes called “neurons.” We are interested in neuron
the use of such networks for pattern recognition, where the input units represent the
components of a feature vector (to be learned or to be classified) and signals emitted
by output units will be discriminant functions used for classification.
∗ We call any units that are neither input nor output units “hidden” because their activations are
not directly “seen” by the external environment, i.e., the input or output.
6 CHAPTER 6. MULTILAYER NEURAL NETWORKS
We can clarify our notation and describe the feedforward (or classification or recall)recall
operation of such a network on what is perhaps the simplest nonlinear problem: the
exclusive-OR (XOR) problem (Fig. 6.1); a three-layer network can indeed solve this
problem whereas a linear machine operating directly on the features cannot.
Each two-dimensional input vector is presented to the input layer, and the output
of each input unit equals the corresponding component in the vector. Each hidden
unit performs the weighted sum of its inputs to form its (scalar) net activation ornet
activation simply net. That is, the net activation is the inner product of the inputs with the
weights at the hidden unit. For simplicity, we augment both the input vector (i.e.,
append a feature value x0 = 1) and the weight vector (i.e., append a value w0), and
can then write
netj =
d∑
i=1
xiwji + wj0 =
d∑
i=0
xiwji ≡ wtjx, (1)
where the subscript i indexes units on the input layer, j for the hidden; wji denotes
the input-to-hidden layer weights at the hidden unit j. In analogy with neurobiol-
ogy, such weights or connections are sometimes called “synapses” and the value ofsynapse
the connection the “synaptic weights.” Each hidden unit emits an output that is a
nonlinear function of its activation, f(net), i.e.,
yj = f(netj). (2)
The example shows a simple threshold or sign (read “signum”) function,
f(net) = Sgn(net) ≡
{
1 if net ≥ 0
−1 if net < 0, (3)
but as we shall see, other functions have more desirable properties and are hence
more commonly used. This f() is sometimes called the transfer function or merelytransfer
function “nonlinearity” of a unit, and serves as a ϕ function discussed in Chap. ??. We have
assumed the same nonlinearity is used at the various hidden and output units, though
this is not crucial.
Each output unit similarly computes its net activation based on the hidden unit
signals as
netk =
nH∑
j=1
yjwkj + wk0 =
nH∑
j=0
yjwkj = w
t
ky, (4)
where the subscript k indexes units in the output layer (one, in the figure) and nH
denotes the number of hidden units (two, in the figure). We have mathematically
treated the bias unit as equivalent to one of the hidden units whose output is always
y0 = 1. Each output unit then computes the nonlinear function of its net, emitting
zk = f(netk). (5)
where in the figure we assume that this nonlinearity is also a sign function. It is these
final output signals that represent the different discriminant functions. We would
typically have c such output units and the classification decision is to label the input
pattern with the label corresponding to the maximum yk = gk(x). In a two-category
case such as XOR, it is traditional to use a single output unit and label a pattern by
the sign of the output z.
6.2. FEEDFORWARD OPERATION AND CLASSIFICATION 7
bias
hidden j
output k
input i
111 1
.5
-1.5
.7
-.4-1
x1 x2
x1
x2
z=1
z= -1
z= -1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
-1
0
1
R2
R2
R1
y1 y2
z
zk
wkj
wji
Figure 6.1: The two-bit parity or exclusive-OR problem can be solved by a three-layer
network. At the bottom is the two-dimensional feature space x1 − x2, and the four
patterns to be classified. The three-layer network is shown in the middle. The input
units are linear and merely distribute their (feature) values through multiplicative
weights to the hidden units. The hidden and output units here are linear threshold
units, each of which forms the linear sum of its inputs times their associated weight,
and emits a +1 if this sum is greater than or equal to 0, and −1 otherwise, as shown
by the graphs. Positive (“excitatory”) weights are denoted by solid lines, negative
(“inhibitory”) weights by dashed lines; the weight magnitude is indicated by the
relative thickness, and is labeled. The single output unit sums the weighted signals
from the hidden units (and bias) and emits a +1 if that sum is greater than or equal
to 0 and a -1 otherwise. Within each unit we show a graph of its input-output or
transfer function — f(net) vs. net. This function is linear for the input units, a
constant for the bias, and a step or sign function elsewhere. We say that this network
has a 2-2-1 fully connected topology, describing the number of units (other than the
bias) in successive layers.
8 CHAPTER 6. MULTILAYER NEURAL NETWORKS
It is easy to verify that the three-layer network with the weight values listed indeed
solves the XOR problem. The hidden unit computing y1 acts like a Perceptron, and
computes the boundary x1 + x2 + 0.5 = 0; input vectors for which x1 + x2 + 0.5 ≥ 0
lead to y1 = 1, all other inputs lead to y1 = −1. Likewise the other hidden unit
computes the boundary x1 +x2−1.5 = 0. The final output unit emits z1 = +1 if and
only if both y1 and y2 have value +1. This gives to the appropriate nonlinear decision
region shown in the figure — the XOR problem is solved.
6.2.1 General feedforward operation
From the above example, it should be clear that nonlinear multilayer networks (i.e.,
ones with input units, hidden units and output units) have greater computational or
expressive power than similar networks that otherwise lack hidden units; that is, theyexpressive
power can implement more functions. Indeed, we shall see in Sect. 6.2.2 that given sufficient
number of hidden units of a general type any function can be so represented.
Clearly, we can generalize the above discussion to more inputs, other nonlineari-
ties, and arbitrary number of output units. For classification, we will have c output
units, one for each of the categories, and the signal from each output unit is the dis-
criminant function gk(x). We gather the results from Eqs. 1, 2, 4, & 5, to express
such discriminant functions as:
gk(x) ≡ zk = f
nH∑
j=1
wkj f
(
d∑
i=1
wjixi + wj0
)
+ wk0
. (6)
This, then, is the class of functions that can be implemented by a three-layer neural
network. An even broader generalization would allow transfer functions at the output
layer to differ from those in the hidden layer, or indeed even different functions at
each individual unit. We will have cause to use such networks later, but the attendant
notational complexities would cloud our presentation of the key ideas in learning in
networks.
6.2.2 Expressive power of multilayer networks
It is natural to ask if every decision can be implemented by such a three-layer network
(Eq. 6). The answer, due ultimately to Kolmogorov but refined by others, is “yes”
— any continuous function from input to output can be implemented in a three-layer
net, given sufficient number of hidden units nH , proper nonlinearities, and weights.
In particular, any posterior probabilities can be represented. In the c-category class-
ification case, we can merely apply a max[·] function to the set of network outputs
(just as we saw in Chap. ??) and thereby obtain any decision boundary.
Specifically, Kolmogorov proved that any continuous function g(x) defined on the
unit hypercube In (I = [0, 1] and n ≥ 2) can be represented in the form
g(x) =
2n+1∑
j=1
Ξj
(
d∑
i=1
ψij(xi)
)
(7)
for properly chosen functions Ξj and ψij . We can always scale the input region of
interest to lie in a hypercube, and thus this condition on the feature space is not
limiting. Equation 7 can be expressed in neural network terminology as follows: each
of 2n + 1 hidden units takes as input a sum of d nonlinear functions, one for each
6.2. FEEDFORWARD OPERATION AND CLASSIFICATION 9
input feature xi. Each hidden unit emits a nonlinear function Ξ of its total input; the
output unit merely emits the sum of the contributions of the hidden units.
Unfortunately, the relationship of Kolmogorov’s theorem to practical neural net-
works is a bit tenuous, for several reasons. In particular, the functions Ξj and ψij
are not the simple weighted sums passed through nonlinearities favored in neural net-
works. In fact those functions can be extremely complex; they are not smooth, and
indeed for subtle mathematical reasons they cannot be smooth. As we shall soon
see, smoothness is important for gradient descent learning. Most importantly, Kol-
mogorov’s Theorem tells us very little about how to find the nonlinear functions based
on data — the central problem in network based pattern recognition.
A more intuitive proof of the universal expressive power of three-layer nets is in-
spired by Fourier’s Theorem that any continuous function g(x) can be approximated
arbitrarily closely by a (possibly infinite) sum of harmonic functions (Problem 2). One
can imagine networks whose hidden units implement such harmonic functions. Proper
hidden-to-output weights related to the coefficients in a Fourier synthesis would then
enable the full network to implement the desired function. Informally speaking, we
need not build up harmonic functions for Fourier-like synthesis of a desired function.
Instead a sufficiently large number of “bumps” at different input locations, of different
amplitude and sign, can be put together to give our desired function. Such localized
bumps might be implemented in a number of ways, for instance by sigmoidal transfer
functions grouped appropriately (Fig. 6.2). The Fourier analogy and bump construc-
tions are conceptual tools, they do not explain the way networks in fact function. In
short, this is not how neural networks “work” — we never find that through train-
ing (Sect. 6.3) simple networks build a Fourier-like representation, or learn to group
sigmoids to get component bumps.
y1
y2
y4
y3
y3 y4y2y1
x1 x2
z1
z1
x1
x2
Figure 6.2: A 2-4-1 network (with bias) along with the response functions at different
units; each hidden and output unit has sigmoidal transfer function f(·). In the case
shown, the hidden unit outputs are paired in opposition thereby producing a “bump”
at the output unit. Given a sufficiently large number of hidden units, any continuous
function from input to output can be approximated arbitrarily well by such a network.
10 CHAPTER 6. MULTILAYER NEURAL NETWORKS
While we can be confident that a complete set of functions, such as all polynomi-
als, can represent any function it is nevertheless a fact that a single functional form
also suffices, so long as each component has appropriate variable parameters. In the
absence of information suggesting otherwise, we generally use a single functional form
for the transfer functions.
While these latter constructions show that any desired function can be imple-
mented by a three-layer network, they are not particularly practical because for most
problems we know ahead of time neither the number of hidden units required, nor
the proper weight values. Even if there were a constructive proof, it would be of little
use in pattern recognition since we do not know the desired function anyway — it
is related to the training patterns in a very complicated way. All in all, then, these
results on the expressive power of networks give us confidence we are on the right
track, but shed little practical light on the problems of designing and training neural
networks — their main benefit for pattern recognition (Fig. 6.3).
TwoRlayer
ThreeRlayer
x1 x2
x1
x2
...
x1 x2
fl
R1
R2
R1
R2
R2
R1
x2
x1
Figure 6.3: Whereas a two-layer network classifier can only implement a linear decision
boundary, given an adequate number of hidden units, three-, four- and higher-layer
networks can implement arbitrary decision boundaries. The decision regions need not
be convex, nor simply connected.
6.3 Backpropagation algorithm
We have just seen that any function from input to output can be implemented as a
three-layer neural network. We now turn to the crucial problem of setting the weights
based on training patterns and desired output.
6.3. BACKPROPAGATION ALGORITHM 11
Backpropagation is one of the simplest and most general methods for supervised
training of multilayer neural networks — it is the natural extension of the LMS al-
gorithm for linear systems we saw in Chap. ??. Other methods may be faster or
have other desirable properties, but few are more instructive. The LMS algorithm
worked for two-layer systems because we had an error (proportional to the square of
the difference between the actual output and the desired output) evaluated at the
output unit. Similarly, in a three-layer net it is a straightforward matter to find how
the output (and thus error) depends on the hidden-to-output layer weights. In fact
this dependency is the same as in the analogous two-layer case, and thus the learning
rule is the same.
But how should the input-to-hidden weights be learned, the ones governing the
nonlinear transformation of the input vectors? If the “proper” outputs for a hidden
unit were known for any pattern, the input-to-hidden weights could be adjusted to
approximate it. However, there is no explicit teacher to state what the hidden unit’s
output should be. This is called the credit assignment problem. The power of back- credit
assignmentpropagation is that it allows us to calculate an effective error for each hidden unit,
and thus derive a learning rule for the input-to-hidden weights.
Networks have two primary modes of operation: feedforward and learning. Feed-
forward operation, such as illustrated in our XOR example above, consists of present-
ing a pattern to the input units and passing the signals through the network in order
to yield outputs from the output units. Supervised learning consists of presenting
an input pattern as well as a desired, teaching or target pattern to the output layer target
patternand changing the network parameters (e.g., weights) in order to make the actual out-
put more similar to the target one. Figure 6.4 shows a three-layer network and the
notation we shall use.
6.3.1 Network learning
The basic approach in learning is to start with an untrained network, present an input
training pattern and determine the output. The error or criterion function is some
scalar function of the weights that is minimized when the network outputs match the
desired outputs. The weights are adjusted to reduce this measure of error. Here we
present the learning rule on a per pattern basis, and return to other protocols later.
We consider the training error on a pattern to be the sum over output units of the training
errorsquared difference between the desired output tk (given by a teacher) and the actual
output zk, much as we had in the LMS algorithm for two-layer nets:
J(w) ≡ 1/2
c∑
k=1
(tk − zk)2 = 1/2(t− z)2, (8)
where t and z are the target and the network output vectors of length c; w represents
all the weights in the network.
The backpropagation learning rule is based on gradient descent. The weights are
initialized with random values, and are changed in a direction that will reduce the
error:
∆w = −η ∂J
∂w
, (9)
or in component form
12 CHAPTER 6. MULTILAYER NEURAL NETWORKS
wkj
z1
wji
z2 zk zc... ...
... ...
... ...
... ...
x1 x2 xi xd... ...
outputRz
x1 x2 xi xd
y1 y2 yj ynH
t1 t2 tk tctargetRt
inputRx
output
hidden
input
Figure 6.4: A d-nH -c fully connected three-layer network and the notation we shall use
(bias not shown). During feedforward operation, a d-dimensional input pattern x is
presented to the input layer; each input unit then emits its corresponding component
xi. Each of the nH hidden units computes its net activation, netj , as the inner
product of the input layer signals with weights wji at the hidden unit. The hidden
unit emits yj = f(netj), where f(·) is the nonlinear transfer function, shown here as
a sigmoid. Each of the c output units functions in the same manner as the hidden
units do, computing netk as the inner product of the hidden unit signals and weights
at the output unit. The final signals emitted by the network, zk = f(netk) are used
as discriminant functions for classification. During network training, these output
signals are compared with a teaching or target vector t, and any difference is used in
training the weights throughout the network.
∆wmn = −η
∂J
∂wmn
, (10)
where η is the learning rate, and merely indicates the relative size of the changelearning
rate in weights. The power of Eqs. 9 & 10 is in their simplicity: they merely demand
that we take a step in weight space that lowers the criterion function. Because this
criterion can never be negative, moreover, this rule guarantees learning will stop
(except in pathological cases). This iterative algorithm requires taking a weight vector
at iteration m and updating it as:
w(m + 1) = w(m) + ∆w(m), (11)
where m indexes the particular pattern presentation (but see also Sect. 6.8).
6.3. BACKPROPAGATION ALGORITHM 13
We now turn to the problem of evaluating Eq. 10 for a three-layer net. Consider
first the hidden-to-output weights, wjk. Because the error is not explicitly dependent
upon wjk, we must use the chain rule for differentiation:
∂J
∂wkj
=
∂J
∂netk
∂netk
∂wkj
= δk
∂netk
∂wkj
, (12)
where the sensitivity of unit k is defined to be sensitivity
δk ≡ −∂J/∂netk, (13)
and describes how the overall error changes with the unit’s activation. We differentiate
Eq. 8 and find that for such an output unit δk is simply:
δk ≡ −
∂J
∂netk
= − ∂J
∂zk
∂zk
∂netk
= (tk − zk)f ′(netk). (14)
The last derivative in Eq. 12 is found using Eq. 4:
∂netk
∂wkj
= yj . (15)
Taken together, these results give the weight update (learning rule) for the hidden-
to-output weights:
∆wkj = ηδkyj = η(tk − zk)f ′(netk)yj . (16)
The learning rule for the input-to-hidden units is more subtle, indeed, it is the
crux of the solution to the credit assignment problem. From Eq. 10, and again using
the chain rule, we calculate
∂J
∂wji
=
∂J
∂yj
∂yj
∂netj
∂netj
∂wji
. (17)
The first term on the right hand side requires just a bit of care:
∂J
∂yj
=
∂
∂yj
[
1/2
c∑
k=1
(tk − zk)2
]
= −
c∑
k=1
(tk − zk)
∂zk
∂yj
= −
c∑
k=1
(tk − zk)
∂zk
∂netk
∂netk
∂yj
= −
c∑
k=1
(tk − zk)f ′(netk)wjk. (18)
For the second step above we had to use the chain rule yet again. The final sum over
output units in Eq. 18 expresses how the hidden unit output, yj , affects the error at
each output unit. In analogy with Eq. 13 we use Eq. 18 to define the sensitivity for
a hidden unit as:
δj ≡ f ′(netj)
c∑
k=1
wkjδk. (19)
14 CHAPTER 6. MULTILAYER NEURAL NETWORKS
Equation 19 is the core of the solution to the credit assigment problem: the sensitivity
at a hidden unit is simply the sum of the individual sensitivities at the output units
weighted by the hidden-to-output weights wjk, all multiplied by f ′(netj). Thus the
learning rule for the input-to-hidden weights is:
∆wji = ηxiδj = ηxif
′(netj)
c∑
k=1
wkjδk. (20)
Equations 16 & 20, together with training protocols such as described below, give
the backpropagation algorithm — or more specifically the “backpropagation of errors”
algorithm — so-called because during training an “error” (actually, the sensitivities
δk) must be propagated from the output layer back to the hidden layer in order to
perform the learning of the input-to-hidden weights by Eq. 20 (Fig. 6.5). At base then,
backpropagation is “just” gradient descent in layered models where the chain rule
through continuous functions allows the computation of derivatives of the criterion
function with respect to all model parameters (i.e., weights).
wkj
ω1
... ...δ1
ω2
δ2
ω3
δ3
ωk
δk
ωc
δc
δj
output
hidden
input
Figure 6.5: The sensitivity at a hidden unit is proportional to the weighted sum of the
sensitivities at the output units: δj = f ′(netj)
c∑
k=1
wkjδk. The output unit sensitivities
are thus propagated “back” to the hidden units.
These learning rules make intuitive sense. Consider first the rule for learning
weights at the output units (Eq. 16). The weight update at unit k should indeed be
proportional to (tk − zk) — if we get the desired output (zk = tk), then there should
be no weight change. For a typical sigmoidal f(·) we shall use most often, f ′(netk) is
always positive. Thus if yj and (tk − zk) are both positive, then the actual output is
too small and the weight must be increased; indeed, the proper sign is given by the
learning rule. Finally, the weight update should be proportional to the input value; if
yj = 0, then hidden unit j has no effect on the output (and hence the error), and thus
changing wji will not change the error on the pattern presented. A similar analysis
of Eq. 20 yields insight of the input-to-hidden weights (Problem 5).
Problem 7 asks you to show that the presence of the bias unit does not materially
affect the above results. Further, with moderate notational and bookkeeping effort
(Problem 11), the above learning algorithm can be generalized directly to feed-forward
networks in which
• input units are connected directly to output units (as well as to hidden units)
• there are more than three layers of units
• there are different nonlinearities for different layers
6.3. BACKPROPAGATION ALGORITHM 15
• each unit has its own nonlinearity
• each unit has a different learning rate.
It is a more subtle matter to perform incorporate learning into networks having con-
nections within a layer, or feedback connections from units in higher layers back to
those in lower layers. We shall consider such recurrent networks in Sect. ??.
6.3.2 Training protocols
In broad overview, supervised training consists in presenting to the network patterns
whose category label we know — the training set — finding the output of the net and training
setadjusting the weights so as to make the actual output more like the desired or teaching
signal. The three most useful training protocols are: stochastic, batch and on-line.
In stochastic training (or pattern training), patterns are chosen randomly from the stochastic
trainingtraining set, and the network weights are updated for each pattern presentation. This
method is called stochastic because the training data can be considered a random
variable. In batch training, all patterns are presented to the network before learning batch
training(weight update) takes place. In virtually every case we must make several passes
through the training data. In on-line training, each pattern is presented once and
on-line
protocol
only once; there is no use of memory for storing the patterns.∗
A fourth protocol is learning with queries where the output of the network is used
learning
with
queries
to select new training patterns. Such queries generally focus on points that are likely
to give the most information to the classifier, for instance those near category decision
boundaries (Chap. ??). While this protocol may be faster in many cases, its drawback
is that the training samples are no longer independent, identically distributed (i.i.d.),
being skewed instead toward sample boundaries. This, in turn, generally distorts the
effective distributions and may or may not improve recognition accuracy (Computer
exercise ??).
We describe the overall amount of pattern presentations by epoch — the number of epoch
presentations of the full training set. For other variables being constant, the number
of epochs is an indication of the relative amount of learning.† The basic stochastic
and batch protocols of backpropagation for n patterns are shown in the procedures
below.
Algorithm 1 (Stochastic backpropagation)
1 begin initialize network topology (# hidden units),w, criterion θ, η,m← 0
2 do m← m + 1
3 xm ← randomly chosen pattern
4 wij ← wij + ηδjxi; wjk ← wjk + ηδkyj
5 until ∇J(w) < θ
6 return w
7 end
In the on-line version of backpropagation, line 3 of Algorithm 1 is replaced by sequen-
tial selection of training patterns (Problem 9). Line 5 makes the algorithm end when
the change in the criterion function J(w) is smaller than some pre-set value θ. While
this is perhaps the simplest meaningful stopping criterion, others generally lead to stopping
criterion∗ Some on-line training algorithms are considered models of biological learning, where the organism
is exposed to the environment and cannot store all input patterns for multiple “presentations.”
† The notion of epoch does not apply to on-line training, where instead the number of pattern
presentations is a more appropriate measure.
16 CHAPTER 6. MULTILAYER NEURAL NETWORKS
better performance, as we shall discuss in Sect. 6.8.14.
In the batch version, all the training patterns are presented first and their corre-
sponding weight updates summed; only then are the actual weights in the network
updated. This process is iterated until some stopping criterion is met.
So far we have considered the error on a single pattern, but in fact we want to
consider an error defined over the entirety of patterns in the training set. With minor
infelicities in notation we can write this total training error as the sum over the errors
on n individual patterns:
J =
n∑
p=1
Jp. (21)
In stochastic training, a weight update may reduce the error on the single pattern
being presented, yet increase the error on the full training set. Given a large number
of such individual updates, however, the total error as given in Eq. 21 decreases.
Algorithm 2 (Batch backpropagation)
1 begin initialize network topology (# hidden units),w, criterion θ, η, r ← 0
2 do r ← r + 1 (increment epoch)
3 m← 0; ∆wij ← 0; ∆wjk ← 0
4 do m← m + 1
5 xm ← select pattern
6 ∆wij ← ∆wij + ηδjxi; ∆wjk ← ∆wjk + ηδkyj
7 until m = n
8 wij ← wij + ∆wij ; wjk ← wjk + ∆wjk
9 until ∇J(w) < θ
10 return w
11 end
In batch backpropagation, we need not select pattern randomly, since the weights
are updated only after all patterns have been presented once. We shall consider the
merits and drawbacks of each protocol in Sect. 6.8.
6.3.3 Learning curves
Because the weights are initialized with random values, error on the training set
is large; through learning the error becomes lower, as shown in a learning curve
(Fig. 6.6). The (per pattern) training error ultimately reaches an asymptotic value
which depends upon the Bayes error, the amount of training data and the expressive
power (e.g., the number of weights) in the network — the higher the Bayes error
and the fewer the number of such weights, the higher this asymptotic value is likely
to be (Chap. ??). Since batch backpropagation performs gradient descent in the
criterion function, these training error decreases monotonically. The average error on
an independent test set is virtually always higher than on the training set, and while
it generally decreases, it can increase or oscillate.
Figure 6.6 also shows the average error on a validation set — patterns not usedvalidation
error directly for gradient descent training, and thus indirectly representative of novel pat-
terns yet to be classified. The validation set can be used in a stopping criterion in
both batch and stochastic protocols; gradient descent training on the training set is
stopped when a minimum is reached in the validation error (e.g., near epoch 5 in
6.4. ERROR SURFACES 17
J/n
epochs
training
test
validation
1 2 3 4 5 6 7 8 9 10 11
Figure 6.6: A learning curve shows the criterion function as a function of the amount
of training, typically indicated by the number of epochs or presentations of the full
training set. We plot the average error per pattern, i.e., 1/n
∑n
p=1 Jp. The validation
error and the test (or generalization) error per pattern are virtually always higher
than the training error. In some protocols, training is stopped at the minimum of the
validation set.
the figure). We shall return in Chap. ?? to understand in greater depth why this
version of cross validation stopping criterion often leads to networks having improved cross
validationrecognition accuracy.
6.4 Error surfaces
Since backpropagation is based on gradient descent in a criterion function, we can gain
understanding and intuition about the algorithm by studying error surfaces themselves
— the function J(w). Of course, such an error surface depends upon the training and
classification task; nevertheless there are some general properties of error surfaces that
seem to hold over a broad range of real-world pattern recognition problems. One of
the issues that concerns us are local minima; if many local minima plague the error
landscape, then it is unlikely that the network will find the global minimum. Does this
necessarily lead to poor performance? Another issue is the presence of plateaus —
regions where the error varies only slightly as a function of weights. If such plateaus
are plentiful, we can expect training according to Algorithms 1 & 2 to be slow. Since
training typically begins with small weights, the error surface in the neighborhood of
w � 0 will determine the general direction of descent. What can we say about the
error in this region? Most interesting real-world problems are of high dimensionality.
Are there any general properties of high dimensional error functions?
We now explore these issues in some illustrative systems.
6.4.1 Some small networks
Consider the simplest three-layer nonlinear network, here solving a two-category prob-
lem in one dimension; this 1-1-1 sigmoidal network (and bias) is shown in Fig. 6.7.
The data shown are linearly separable, and the optimal decision boundary (a point
somewhat below x1 = 0) separates the two categories. During learning, the weights
descends to the global minimum, and the problem is solved.
18 CHAPTER 6. MULTILAYER NEURAL NETWORKS
-4 -3 -2 -1 43210
-100
0
100
-40
-20
0
20
40
0
0.25
0.5
0.75
1
-100
0
100
w0
w1
x1
x0,y0
w1
x1
y1
w2
z1
R1 R2x*
w0
J(w)
Figure 6.7: Six one-dimensional patterns (three in each of two classes) are to be
learned by a 1-1-1 network with sigmoidal hidden and output units (and bias). The
error surface as a function of w1 and w2 is also shown (for the case where the bias
weights have their final values). The network starts with random weights, and through
(stochastic) training descends to the global minimum in error, as shown by the trajec-
tory. Note especially that a low error solution exists, which in fact leads to a decision
boundary separating the training points into their two categories.
Here the error surface has a single (global) minimum, which yields the decision
point separating the patterns of the two categories. Different plateaus in the surface
correspond roughly to different numbers of patterns properly classified; the maximum
number of such misclassified patterns is three in this example. The plateau regions,
where weight change does not lead to a change in error, here correspond to sets of
weights that lead to roughly the same decision point in the input space. Thus as w1
increases and w2 becomes more negative, the surface shows that the error does not
change, a result that can be informally confirmed by looking at the network itself.
Now consider the same network applied to another, harder, one-dimensional prob-
lem — one that is not linearly separable (Fig. 6.8). First, note that overall the error
surface is slightly higher than in Fig. 6.7 because even the best solution attainable
with this network leads to one pattern being misclassified. As before, the different
plateaus in error correspond to different numbers of training patterns properly learned.
However, one must not confuse the (squared) error measure with classification error
(cf. Chap. ??, Fig. ??). For instance here there are two general ways to misclassify
exactly two patterns, but these have different errors. Incidentally, a 1-3-1 network
(but not a 1-2-1 network) can solve this problem (Computer exercise 3).
From these very simple examples, where the correspondences among weight val-
ues, decision boundary and error are manifest, we can see how the error of the global
minimum is lower when the problem can be solved and that there are plateaus corre-
sponding to sets of weights that lead to nearly the same decision boundary. Further-
more, the surface near w � 0 (the traditional region for starting learning) has high
6.4. ERROR SURFACES 19
-100
0
100
-40
-20
0
20
40
0
0.25
0.5
0.75
1
-100
0
100
-4 -3 -2 -1 43210
x1
R1 R2x*
x1
x0,y0
w1
y1
w2
z1
w0
w0
w1
J(w)
Figure 6.8: As in Fig. 6.7, except here the patterns are not linearly separable; the
error surface is slightly higher than in that figure.
error and happens in this case to have a large slope; if the starting point had differed
somewhat, the network would descend to the same final weight values.
6.4.2 XOR
A somewhat more complicated problem is the XOR problem we have already consid-
ered. Figure ?? shows several two-dimensional slices through the nine-dimensional
weight space of the 2-2-1 sigmoidal network (with bias). The slices shown include a
global minimum in the error.
Notice first that the error varies a bit more gradually as a function of a single
weight than does the error in the networks solving the problems in Figs. 6.7 & 6.8.
This is because in a large network any single weight has on average a smaller relative
contribution to the output. Ridges, valleys and a variety of other shapes can all
be seen in the surface. Several local minima in the high-dimensional weight space
exist, which here correspond to solutions that classify three (but not four) patterns.
Although it is hard to show it graphically, the error surface is invariant with respect
to certain discrete permutations. For instance, if the labels on the two hidden units
are exchanged (and the weight values changed appropriately), the shape of the error
surface is unaffected (Problem ??).
6.4.3 Larger networks
Alas, the intuition we gain from considering error surfaces for small networks gives only
hints of what is going on in large networks, and at times can be quite misleading. Fig-
ure 6.10 shows a network with many weights solving a complicated high-dimensional
two-category pattern classification problem. Here, the error varies quite gradually as
a single weight is changed though we can get troughs, valleys, canyons, and a host of
20 CHAPTER 6. MULTILAYER NEURAL NETWORKS
0
1
2
3
4
- 4
- 3
- 2
- 1
0
0
1
2
3
4
0
1
2
3
4
- 4
- 3
- 2
- 1
0
0
1
2
3
4
0
1
2
3
4
- 4
- 3
- 2
- 1
0
- 4
- 3
- 2
- 1
0
- 4
- 3
- 2
- 1
0
0
1
2
3
4
- 4
- 3
- 2
- 1
0
- 4
- 3
- 2
- 1
0
- 4
- 3
- 2
- 1
0
0
1
2
3
4
- 4
- 3
- 2
- 1
0
J
J
J J
w11
w12 w21
w22w02
w01w'0
w'1
w'2
w'1
w'2
w'1
w'0
w21
w11
w22
w12
Figure 6.9: Two-dimensional slices through the nine-dimensional error surface after
extensive training for a 2-2-1 network solving the XOR problem.
shapes.
Whereas in low dimensional spaces local minima can be plentiful, in high dimen-
sion, the problem of local minima is different: the high-dimensional space may afford
more ways (dimensions) for the system to “get around” a barrier or local maximum
during learning. In networks with many superfluous weights (i.e., more than are
needed to learn the training set), one is less likely to get into local minima. However,
networks with an unnecessarily large number of weights are undesirable because of
the dangers of overfitting, as we shall see in Sect. 6.11.
6.4.4 How important are multiple minima?
The possibility of the presence of multiple local minima is one reason that we resort to
iterative gradient descent — analytic methods are highly unlikely to find a single global
minimum, especially in high-dimensional weight spaces. In computational practice, we
do not want our network to be caught in a local minimum having high training error
since this usually indicates that key features of the problem have not been learned by
the network. In such cases it is traditional to re-initialize the weights and train again,
possibly also altering other parameters in the net (Sect. 6.8).
In many problems, convergence to a non-global minimum is acceptable, if the
error is nevertheless fairly low. Furthermore, common stopping criteria demand that
training terminate even before the minimum is reached and thus it is not essential
that the network be converging toward the global minimum for acceptable performance
(Sect. 6.8.14).
6.5. BACKPROPAGATION AS FEATURE MAPPING 21
Figure 6.10: A network with xxx weights trained on data from a complicated pattern
recognition problem xxx.
6.5 Backpropagation as feature mapping
Since the hidden-to-output layer leads to a linear discriminant, the novel computa-
tional power provided by multilayer neural nets can be attributed to the nonlinear
warping of the input to the representation at the hidden units. Let us consider this
transformation, again with the help of the XOR problem.
Figure 6.11 shows a three-layer net addressing the XOR problem. For any input
pattern in the x1−x2 space, we can show the corresponding output of the two hidden
units in the y1 − y2 space. With small initial weights, the net activation of each
hidden unit is small, and thus the linear portion of their transfer function is used.
Such a linear transformation from x to y leaves the patterns linearly inseparable
(Problem 1). However, as learning progresses and the input-to-hidden weights increase
in magnitude, the nonlinearities of the hidden units warp and distort the mapping
from input to the hidden unit space. The linear decision boundary at the end of
learning found by the hidden-to-output weights is shown by the straight dashed line;
the nonlinearly separable problem at the inputs is transformed into a linearly separable
at the hidden units.
We can illustrate such distortion in the three-bit parity problem, where the output
= +1 if the number of 1s in the input is odd, and -1 otherwise — a generalization
of the XOR or two-bit parity problem (Fig. 6.12). As before, early in learning the
hidden units operate in their linear range and thus the representation after the hid-
den units remains linearly inseparable — the patterns from the two categories lie at
alternating vertexes of a cube. After learning and the weights have become larger,
the nonlinearities of the hidden units are expressed and patterns have been moved
and can be linearly separable, as shown.
Figure 6.13 shows a two-dimensional two-category problem and the pattern rep-
resentations in a 2-2-1 and in a 2-3-1 network of sigmoidal hidden units. Note that
22 CHAPTER 6. MULTILAYER NEURAL NETWORKS
-1 1
x2
-1
1
x1
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
2
fin
al
d
ec
isi
on
b
ou
nd
ar
y
Epoch
1
15
30
45
60
y1
y2 x2x1
y2y1
bias
10 20 30 40 50 60
Epoch
0.5
1
1.5
2
J
Figure 6.11: A 2-2-1 backpropagation network (with bias) and the four patterns of the
XOR problem are shown at the top. The middle figure shows the outputs of the hidden
units for each of the four patterns; these outputs move across the y1− y2 space as the
full network learns. In this space, early in training (epoch 1) the two categories are
not linearly separable. As the input-to-hidden weights learn, the categories become
linearly separable. Also shown (by the dashed line) is the linear decision boundary
determined by the hidden-to-output weights at the end of learning — indeed the
patterns of the two classes are separated by this boundary. The bottom graph shows
the learning curves — the error on individual patterns and the total error as a function
of epoch. While the error on each individual pattern does not decrease monotonically,
the total training error does decrease monotonically.
6.5. BACKPROPAGATION AS FEATURE MAPPING 23
25 50 75 100 125 150
Epoch
1
2
3
4
Error
-1 0 1 2
-2
-1
0
1
2
10 20 30 40 50 60 70
Epoch
1
2
3
4
Error
-2
-1
0
1
2
y1
-2
-1
0
1
2
y2
-2
-1
0
1
2
y3
-2
-1
0
1
2
-2
-1
0
1
2
Figure 6.12: A 3-3-1 backpropagation network (plus bias) can indeed solve the three-
bit parity problem. The representation of the eight patterns at the hidden units
(y1−y2−y3 space) as the system learns and the (planar) decision boundary found by
the hidden-to-output weights at the end of learning. The patterns of the two classes
are separated by this plane. The learning curve shows the error on individual patterns
and the total error as a function of epoch.
in the two-hidden unit net, the categories are separated somewhat, but not enough
for error-free classification; the expressive power of the net is not sufficiently high.
In contrast, the three-hidden unit net can separate the patterns. In general, given
sufficiently many hidden units in a sigmoidal network, any set of different patterns
can be learned in this way.
6.5.1 Representations at the hidden layer — weights
In addition to focusing on the transformation of patterns, we can also consider the
representation of learned weights themselves. Since the hidden-to-output weights
merely leads to a linear discriminant, it is instead the input-to-hidden weights that
are most instructive. In particular, such weights at a single hidden unit describe the
input pattern that leads to maximum activation of that hidden unit, analogous to
a “matched filter.” Because the hidden unit transfer functions are nonlinear, the matched
filtercorrespondence with classical methods such as matched filters (and principal compo-
nents, Sect. ??) is not exact; nevertheless it is often convenient to think of the hidden
units as finding feature groupings useful for the linear classifier implemented by the
hidden-to-output layer weights.
Figure 6.14 shows the input-to-hidden weights (displayed as patterns) for a simple
task of character recognition. Note that one hidden unit seems “tuned” for a pair of
horizontal bars while the other to a single lower bar. Both of these feature groupings
are useful building blocks for the patterns presented. In complex, high-dimensional
problems, however, the pattern of learned weights may not appear to be simply related
to the features we suspect are appropriate for the task. This could be because we
may be mistaken about which are the true, relevant feature groupings; nonlinear
24 CHAPTER 6. MULTILAYER NEURAL NETWORKS
-1
0
1
-1
0
1
0
0.5
1
1.5
2
-1
0
1
1 2 3 4 5
1
2
3
4
5
x1
x2
y1
y2
y1
y2
y3
input
2-2-1 2-3-1
Figure 6.13: Seven patterns from a two-dimesional two-category nonlinearly separable
classification problem are shown at the bottom. The figure at the top left shows the
hidden unit representations of the patterns in a 2-2-1 sigmoidal network (with bias)
fully trained to the global error minimum; the linear boundary implemented by the
hidden-to-output weights is also shown. Note that the categories are almost linearly
separable in this y1 − y2 space, but one training point is misclassified. At the top
right is the analogous hidden unit representation for a fully trained 2-3-1 network
(with bias). Because of the higher dimension of the hidden layer representation, the
categories are now linearly separable; indeed the learned hidden-to-output weights
implement a plane that separates the categories.
interactions between features may be significant in a problem (and such interactions
are not manifest in the patterns of weights at a single hidden unit); or the network
may have too many weights (degrees of freedom), and thus the feature selectivity is
low.
It is generally much harder to represent the hidden-to-output layer weights in
terms of input features. Not only do the hidden units themselves already encode a
somewhat abstract pattern, there is moreover no natural ordering of the hidden units.
Together with the fact that the output of hidden units are nonlinearly related to the
inputs, this makes analyzing hidden-to-output weights somewhat problematic. Often
the best we can do is list the patterns of input weights for hidden units that have
strong connections to the output unit in question (Computer exercise 9).
6.6 Backpropagation, Bayes theory and probability
While multilayer neural networks may appear to be somewhat ad hoc, we now show
that when trained via backpropagation on a sum-squared error criterion they form a
least squares fit to the Bayes discriminant functions.
6.6. BACKPROPAGATION, BAYES THEORY AND PROBABILITY 25
sample training patterns
learned input-to-hidden weights
Figure 6.14: The top images represent patterns from a large training set used to train
a 64-2-3 sigmoidal network for classifying three characters. The bottom figures show
the input-to-hidden weights (represented as patterns) at the two hidden units after
training. Note that these learned weights indeed describe feature groupings useful for
the classification task. In large networks, such patterns of learned weights may be
difficult to interpret in this way.
6.6.1 Bayes discriminants and neural networks
As we saw in Chap. ?? Sect. ??, the LMS algorithm computed the approximation to
the Bayes discriminant function for two-layer nets. We now generalize this result in
two ways: to multiple categories and to nonlinear functions implemented by three-
layer neural networks. We use the network of Fig. 6.4 and let gk(x; w) be the output
of the kth output unit — the discriminant function corresponding to category ωk.
Recall first Bayes’ formula,
P (ωk|x) =
P (x|ωk)P (ωk)
c∑
i=1
P (x|ωi)P (ωi)
=
P (x, ωk)
P (x)
, (22)
and the Bayes decision for any pattern x: choose the category ωk having the largest
discriminant function gk(x) = P (ωk|x).
Suppose we train a network having c output units with a target signal according
to:
tk(x) =
{
1 if x ∈ ωk
0 otherwise.
(23)
(In practice, teaching values of ±1 are to be preferred, as we shall see in Sect. 6.8; we
use the values 0–1 in this derivation for computational simplicity.) The contribution
to the criterion function based on a single output unit k for finite number of training
samples x is:
J(w) =
∑
x
[gk(x; w)− tk]2 (24)
26 CHAPTER 6. MULTILAYER NEURAL NETWORKS
=
∑
x∈ωk
[gk(x; w)− 1]2 +
∑
x/∈ωk
[gk(x; w)− 0]2
= n
nk
n
1
nk
∑
x∈ωk
[gk(x; w)− 1]2 +
n− nk
n
1
n− nk
∑
x/∈ωk
[gk(x; w)− 0]2
,
where n is the total number of training patterns, nk of which are in ωk. In the limit
of infinite data we can use Bayes’ formula (Eq. 22) to express Eq. 24 as (Problem 17):
lim
n→∞
1
n
J(w) ≡ J̃(w) (25)
= P (ωk)
∫
[gk(x; w)− 1]2 p(x|ωk)dx + P (ωi �=k)
∫
g2k(x; w)p(x|ωi �=k)dx
=
∫
g2k(x; w)p(x)dx− 2
∫
gk(x; w)p(x, ωk)dx +
∫
p(x, ωk)dx
=
∫
[gk(x; w)− P (ωk|x)]2 p(x)dx +
∫
P (ωk|x)P (ωi �=k|x)p(x)dx
︸ ︷︷ ︸
independent of w
.
The backpropagation rule changes weights to minimize the left hand side of Eq. 25,
and thus it minimizes
∫
[gk(x; w)− P (ωk|x)]2p(x)dx. (26)
Since this is true for each category ωk (k = 1, 2, ..., c), backpropagation minimizes the
sum (Problem 22):
c∑
k=1
∫
[gk(x; w)− P (ωk|x)]2 p(x)dx. (27)
Thus in the limit of infinite data the outputs of the trained network will approximate
(in a least-squares sense) the true a posteriori probabilities, that is, the output units
represent the a posteriori probabilities,
gk(x; w) � P (ωk|x). (28)
Figure 6.15 illustrates the development of the learned outputs toward the Bayes dis-
criminants as the amount of training data and the expressive power of the net in-
creases.
We must be cautious in interpreting these results, however. A key assumption un-
derlying the argument is that the network can indeed represent the functions P (ωk|x);
with insufficient hidden units, this will not be true (Problem ??). Moreover, fitting
the discriminant function does not guarantee the optimal classification boundaries are
found, just as we saw in Chap. ??.
6.6.2 Outputs as probabilities
In the previous subsection we saw one way to make the c output units of a trained net
represent probabilities by training with 0–1 target values. While indeed given infinite
amounts of training data (and assuming the net can express the discriminants, does
6.6. BACKPROPAGATION, BAYES THEORY AND PROBABILITY 27
Figure 6.15: As a network is trained via backpropagation (under the assumptions
given in the text), its outputs more closely approximate posterior probabilities. The
figure shows the outputs of a 1-3-2 and a 1-8-2 sigmoidal network after backpropaga-
tion training with n = 10 and n = 1000 points from two categories. Note especially
the excellent agreement between the large net’s outputs and the Bayesian discriminant
functions in the regions of high p(x).
not fall into an undesirable local minimum, etc.), then the outputs will represent
probabilities. If, however, these conditions do not hold — in particular we have only
a finite amount of training data — then the outputs will not represent probabilities;
for instance there is no guarantee that they will sum to 1.0. In fact, if the sum of the
network outputs differs significantly from 1.0 within some range of the input space, it
is an indication that the network is not accurately modeling the posteriors. This, in
turn, may suggest changing the network topology, number of hidden units, or other
aspects of the net (Sect. 6.8).
One approach toward approximating probabilities is to choose the output unit
nonlinearity to be exponential rather than sigmoidal — f(netk) ∝ enetk — and for
each pattern normalize the outputs to sum to 1.0,
zk =
enetk
c∑
m=1
enetm
, (29)
and to train using 0–1 target signals. This is the softmax method — a smoothed or softmax
continuous version of a winner-take-all nonlinearity in which the maximum output is
winner-
take-all
transformed to 1.0, and all others reduced to 0.0. The softmax output finds theoretical
justification if for each category ωk the hidden unit representations y can be assumed
to come from an exponential distribution (Problem 20, Computer exercise 10).
A neural network classifier trained in this manner approximates the posterior
probabilities P (ωi|x), whether or not the data was sampled from unequal priors P (ωi).
If such a trained network is to be used on problems in which the priors have been
changed, it is a simple matter to rescale each network output, gi(x) = P (ωi|x) by the
ratio of such priors (Computer exercise 11).
28 CHAPTER 6. MULTILAYER NEURAL NETWORKS
6.7 *Related statistical techniques
While the graphical, topological representation of networks is useful and a guide to
intuition, we must not forget that the underlying mathematics of the feedforward
operation is governed by Eq. 6. A number of statistical methods bear similarities
to that equation. For instance, projection pursuit regression (or simply projectionprojection
pursuit pursuit) implements
z =
jmax∑
j=1
wjfj(v
t
jx + uj0) + w0. (30)
Here each vj and vj0 together define the projection of the input x onto one of jmax
different d-dimensional hyperplanes. These projections are transformed by nonlinear
functions fj(·) whose values are then linearly combined at the output; traditionally,
sigmoidal or Gaussian functions are used. The fj(·) have been called ridge functionsridge
function because for peaked fj(·), one obtains ridges in two dimensions. Equation 30 imple-
ments a mapping to a scalar function z; in a c-category classification problem there
would be c such outputs. In computational practice, the parameters are learned in
groups minimizing an LMS error, for instance first the components of v1 and v10, then
v2 and v20 up to vjmax and vjmax0; then the wj and w0, iterating until convergence.
Such models are related to the three-layer networks we have seen in that the vj
and vj0 are analogous to the input-to-hidden weights at a hidden unit and the effective
output unit is linear. The class of functions fj(·) at such hidden units are more general
and have more free parameters than do sigmoids. Moreover, such a model can have
an output much larger than 1.0, as might be needed in a general regression task. In
the classification tasks we have considered, a saturating output, such as a sigmoid is
more appropriate.
Another technique related to multilayer neural nets is generalized additive models,generalized
additive
model
which implement
z = f
(
d∑
i=1
fi(xi) + w0
)
, (31)
where again f(·) is often chosen to be a sigmoid, and the functions fi() operating on
the input features are nonlinear, and sometimes chosen to be sigmoidal. Such models
are trained by iteratively adjusting parameters of the component nonlinearities fi(·).
Indeed, the basic three-layer neural networks of Sect. 6.2 implement a special case of
general additive models (Problem 24), though the training methods differ.
An extremely flexible technique having many adjustable parameters is multivari-
ate adaptive regression splines (MARS). In this technique, localized spline functionsmultivariate
adaptive
regression
spline
(polynomials adjusted to insure continuous derivative) are used in the initial process-
ing. Here the output is the weighted sum of M products of splines:
z =
M∑
k=1
wk
rk∏
r=1
φkr(xq(k,r)) + w0, (32)
where the kth basis function is the product of rk one-dimensional spline functions φkr;
w0 is a scalar offset. The splines depend on the input values xq, such as the feature
component of an input, where the index is labeled q(k, r). Naturally, in a c-category
task, there would be one such output for each category.
6.8. PRACTICAL TECHNIQUES FOR BACKPROPAGATION 29
In broad overview, training in MARS begins by fitting the data with a spline
function along each feature dimension in turn. The spline that best fits the data (in
a sum squared error sense) is retained. This is the r = 1 term in Eq. 32. Next, each
of the other feature dimensions is considered, one by one. For each such dimension,
candidate splines are selected based on the data fit using the product of that spline
with the one previously selected, thereby giving the product r = 1 → 2. The best
such second spline is retained, thereby giving the r = 2 term. In this way, splines are
added incrementally up to some value rk, where some desired quality of fit is achieved.
The weights wk are learned using an LMS criterion.
For several reasons, multilayer neural nets have all but supplanted projection pur-
suit, MARS and earlier related techniques in practical pattern recognition research.
Backpropagation is simpler than learning in projection pursuit and MARS, especially
when the number of training patterns and the dimension is large; heuristic informa-
tion can be incorporated more simply into nets (Sect. 6.8.12); nets admit a variety of
simplification or regularization methods (Sec. 6.11) that have no direct counterpart
in those earlier methods. It is, moreover, usually simpler to refine a trained neural
net using additional training data than it is to modify classifiers based on projection
pursuit or MARS.
6.8 Practical techniques for improving backpropa-
gation
When creating a multilayer neural network classifier, the designer must make two ma-
jor types of decision: selection of the architecture and selection of parameters (though
the distinction is not always crisp or important). Our goal here is to give a princi-
pled basis for making such choices based on learning speed and optimal recognition
performance. In practice, while parameter adjustment is problem dependent several
rules of thumb emerge from an analysis of networks.
6.8.1 Transfer function
There are a number of desirable properties for f(·), but we must not lose sight of the
fact that backpropagation will work with virtually any transfer function, given that
a few simple conditions such as continuity of f and its derivative are met. In any
particular classification problem we may have a good reason for selecting a particular
transfer function. For instance, if we have prior information that the distributions
arise from a mixture of Gaussians, then Gaussian transfer functions are appropriate
(Sect. ??).
When not guided by such problem dependent information, what general proper-
ties might we seek in f(·)? First, of course, f(·) must be nonlinear — otherwise the
three-layer network provides no computational power above that of a two-layer net
(Problem 1). A second desirable property is that f(·) saturate, i.e., have some maxi-
mum and minimum output value. This will keep the weights and activations bounded,
and thus keep training time limited. (This property is less desirable in networks used
for regression, since there we may seek outputs values greater than any saturation
level selected before training.) A third property is continuity and smoothness, i.e.,
that f(·) and f ′(·) be defined throughout the range of their argument. Recall that
the fact that we could take a derivative of f(·) was crucial in the derivation of the
backpropagation learning rule. The rule would not, therefore, work with the threshold
30 CHAPTER 6. MULTILAYER NEURAL NETWORKS
or sign function of Eq. 3. Backpropagation can be made to work with piecewise linear
transfer functions, but with added complexity and few benefits.
Monotonicity is another convenient (but non-essential) property for f(·) — we
might wish the derivative have the same sign throughout the range of the argument,
e.g., f ′(·) ≥ 0. If f is not monotonic, additional (and undesirable) local extremum in
the error surface may become introduced (Computer Exercise ??). Non-monotonic
transfer functions such as radial basis functions can be used if proper care is taken
(Sect. 6.10.1). Another desirable property is linearity for small value of net, which will
enable the system to implement a linear model if adequate for low error. A property
that is might occasionally be of importance is computational simplicity — we seek a
function whose value and derivative can be easily computed.
We mention in passing that polynomial classifiers use transfer functions of thepolynomial
classifier form x1, x2, ..., xd, x21, x
2
2, ..., x
2
d, x1x2, ..., x1xd, and so forth — all terms up to some
limit; training is via gradient descent too. One drawback is that the outputs of the
hidden units (φ functions) can become extremely large even for realistic problems
(Problem 29, Computer exercise ??). Instead, standard neural networks employ the
same nonlinearity at each hidden unit.
One class of function that has all the above properties is the sigmoid such as asigmoid
hyperbolic tangent. The sigmoid is smooth, differentiable, nonlinear, and saturating.
It also admits a linear model if the network weights are small. A minor benefit is that
the derivative f ′(·) can be easily expressed in terms of f(·) itself (Problem 10). One
last benefit of the sigmoid is that it maximizes information transmission for features
that are normally distributed (Problem 25).
A hidden layer of sigmoidal units affords a distributed or global representationdistributed
representa-
tion
of the input. That is, any particular input x is likely to yield activity throughout
several hidden units. In contrast, if the hidden units have transfer functions that have
significant response only for inputs within a small range, then an input x generally
leads to fewer hidden units being active — a local representation. (Nearest neighborlocal
representa-
tion
classifiers employ local representations, of course.) It is often found in practice that
when there are few training points, distributed representations are superior because
more of the data influences the posteriors at any given input region (Computer exercise
14).
The sigmoid is the most widely used transfer function for the above reasons, and
in much of the following we shall employ sigmoids.
6.8.2 Parameters for the sigmoid
Given that we will use the sigmoidal form, there remain a number of parameters
to set. It is best to keep the function centered on zero and anti-symmetric, i.e.,
f(−net) = −f(net), rather than one whose value is always positive. Together with
the data preprocessing described in Sec. 6.8.3, anti-symmetric sigmoids speed learning
by eliminating the need to learn the mean values of the training data. Thus, sigmoid
functions of the form
f(net) = a tanh(b net) = a
[
1− eb net
1 + eb net
]
=
2a
1 + e−b net
− a (33)
work well. The overall range and slope are not important, since it is their relationship
to parameters such as the learning rate and magnitudes of the inputs and targets
that determine learning times (Problem 23). For convenience, though, we choose
a = 1.716 and b = 2/3 in Eq. 33 — values which insure f ′(0) � 1, that the linear
6.8. PRACTICAL TECHNIQUES FOR BACKPROPAGATION 31
range is −1 < net < +1, and that the extrema of the second derivative occur roughly
at net � ±2 (Fig. 6.16).
-4 -2 2 4
net
1
.1
f ''(net)
-4 -2 2 4
net
1.5
-1
-.5
.5
1
1.5
f(net)
4 -2 2 4
net
.2
.3
.4
.5
f '(net)
.1
0
Figure 6.16: A useful transfer function f(net) is an anti-symmetric sigmoid. For the
parameters given in the text, f(net) is nearly linear in the range −1 < net < +1 and
its second derivative, f ′′(net), has extrema near net � ±2.
6.8.3 Scaling input
Suppose we were using a two-input network to classify fish based on the features of
mass (measured in grams) and length (measured in meters). Such a representation
will have serious drawbacks for a neural network classifier: the numerical value of
the mass will be orders of magnitude larger than that for length. During training the
network will adjust weights from the “mass” input unit far more than for the “length”
input — indeed the error will hardly depend upon the tiny length values. If however,
the same physical information were presented but with mass measured in kilograms
and length in millimeters, the situation would be reversed. Naturally we do not want
our classifier to prefer one of these features over the other, since they differ solely in
the arbitrary representation. The difficulty arises even for features having the same
units but differing overall magnitude, of course, for instance if a fish’s length and its
fin thickness were both measured in millimeters.
In order to avoid such difficulties, the input patterns should be shifted so that the
average (over the training set) of each feature is zero. Moreover, the full data set
should then be scaled to have the same variance in each feature component — here
chosen to be 1.0 for reasons that will be clear in Sect. 6.8.8. That is, we standardize the standardize
32 CHAPTER 6. MULTILAYER NEURAL NETWORKS
training patterns. This data standardization is done once, before actually network
training, and thus represents a small one-time computational burden (Problem 27,
Computer exercise 15). Standardization can only be done for stochastic and batch
learning protocols, but not on-line protocols where the full data set is never available
at any one time.
6.8.4 Target values
For pattern recognition, we typically train with the pattern and its category label,
and thus we use a one-of-c representation for the target vector. Since the output units
saturate at ±1.716, we might naively feel that the target values should be those values;
however, that would present a difficulty. For any finite value of netk, the output would
be less than the saturation values, and thus there would be error. Full training would
never terminate as weights would become extremely large as netk would be driven to
± ∞.
This difficulty can be avoided by using teaching values of +1 for the target cat-
egory and -1 for the non-target categories. For instance, in a four-category prob-
lem if the pattern is in category ω3, the following target vector should be used:
t = (−1,−1,+1,−1). Of course, this target representation yields efficient learning for
categorization — the outputs here do not represent posterior probabilities (Sec. 6.6.2).
6.8.5 Training with noise
When the training set is small, one can generate virtual or surrogate training pat-
terns and use them as if they were normal training patterns sampled from the source
distributions. In the absence of problem-specific information, a natural assumption
is that such surrogate patterns should be made by adding d-dimensional Gaussian
noise to true training points. In particular, for the standardized inputs described in
Sect. 6.8.3, the variance of the added noise should be less than 1.0 (e.g., 0.1) and the
category label left unchanged. This method of training with noise can be used with
virtually every classification method, though it generally does not improve accuracy
for highly local classifiers such as ones based on the nearest neighbor (Problem 30).
6.8.6 Manufacturing data
If we have knowledge about the sources of variation among patterns (for instance due
to geometrical invariances), we can “manufacture” training data that conveys more
information than does the method of training with uncorrelated noise (Sec. 6.8.5).
For instance, in an optical character recognition problem, an input image may be pre-
sented rotated by various amounts. Hence during training we can take any particular
training pattern and rotate its image to “manufacture” a training point that may be
representative of a much larger training set. Likewise, we might scale a pattern, per-
form simple image processing to simulate a bold face character, and so on. If we have
information about the range of expected rotation angles, or the variation in thickness
of the character strokes, we should manufacture the data accordingly.
While this method bears formal equivalence to incorporating prior information in
a maximum likelihood approach, it is usually much simpler to implement, since we
need only the (forward) model for generating patterns. As with training with noise,
manufacturing data can be used with a wide range of pattern recognition methods.
A drawback is that the memory requirements may be large and overall training slow.
6.8. PRACTICAL TECHNIQUES FOR BACKPROPAGATION 33
6.8.7 Number of hidden units
While the number of input units and output units are dictated by the dimensionality
of the input vectors and the number of categories, respectively, the number of hidden
units is not simply related to such obvious properties of the classification problem.
The number of hidden units, nH , governs the expressive power of the net — and
thus the complexity of the decision boundary. If the patterns are well separated or
linearly separable, then few hidden units are needed; conversely, if the patterns are
drawn from complicated densities that are highly interspersed, then more hiddens are
needed. Thus without further information there is no foolproof method for setting
the number of hidden units before training.
Figure 6.17 shows the training and test error on a two-category classification prob-
lem for networks that differ solely in their number of hidden units. For large nH , the
training error can become small because such networks have high expressive power and
become tuned to the particular training data. Nevertheless, in this regime, the test
error is unacceptably high, an example of overfitting we shall study again in Chap. ??.
At the other extreme of too few hidden units, the net does not have enough free pa-
rameters to fit the training data well, and again the test error is high. We seek some
intermediate number of hidden units that will give low test error.
2 3 4 5 6 7 8 9
0.15
0.20
0.25
0.30
0.35
nH
J/n
test
train
9 13 17 21 25 29 33 37 total�number�fl
of�weights
0.40
Figure 6.17: The error per pattern for networks fully trained but differing in the
numbers of hidden units, nH . Each 2–nH–1 network (with bias) was trained with
90 two-dimensional patterns from each of two categories (sampled from a mixture of
three Gaussians); thus n = 180. The minimum of the test error occurs for networks in
the range 4 ≤ nH ≤ 5, i.e., the range of weights 17 to 21. This illustrates the rule of
thumb that choosing networks with roughly n/10 weights often gives low test error.
The number of hidden units determines the total number of weights in the net
— which we consider informally as the number of degrees of freedom — and thus
we should not have more weights than the total number of training points, n. A
convenient rule of thumb is to choose the number of hidden units such that the total
number of weights in the net is roughly n/10. This seems to work well over a range
of practical problems. A more principled method is to adjust the complexity of the
network in response to the training data, for instance start with a “large” number of
hiddens and prune or eliminate weights — techniques we shall study in Sect. ?? and
Chap. ??.
34 CHAPTER 6. MULTILAYER NEURAL NETWORKS
6.8.8 Initializing weights
Suppose we have fixed the network topology, and thus set the number of hidden
units. We now seek to set the initial weight values in order to have fast and uniform
learning, i.e., all weights reach their final equilibrium values at about the same time.uniform
learning One form of non-uniform learning occurs when category ωi is learned well before ωj .
In this undesirable case, the distribution of errors differs markedly from Bayes, and
the overall error rate is typically higher than necessary. (The data standarization
described above also helps to insure uniform learning.)
In setting weights in a given layer, we choose weights randomly from a single dis-
tribution to help insure uniform learning. Because data standardization gives positive
and negative values equally, on average, we want positive and negative weights as well;
thus we choose weights from a uniform distribution −w̃ < w < +w̃, for some w̃ yet
to be determined. If w̃ is chosen too small, the net activation of a hidden unit will be
small and the linear model will be implemented. Alternatively, if w̃ is too large, the
hidden unit may saturate even before learning begins. Hence we set w̃ such that the
net activation at a hidden unit is in the range −1 < netj < +1, since netj � ±1 are
the limits to its linear range (Fig. 6.16).
In order to calculate w̃, consider a hidden unit having a fan-in of d inputs. Suppose
too that all weights have the same value w̃. On average, then, the net activation from
d random variables of variance 1.0 from our standarized input through such weights
will be w̃
√
d. As mentioned, we would like this net activation to be roughly in the
range −1 < net < +1. This implies that w̃ = 1/
√
d and thus input weights should
be chosen in the range −1/
√
d < wji < +1/
√
d. The same argument holds for the
hidden-to-output weights, where the fan-in is nH ; hidden-to-output weights should
initialized with values chosen in the range −1/√nH < wkj < +1/
√
nH .
6.8.9 Learning rates
In principle, so long as the learning rate is small enough to assure convergence, its
value determines only the speed at which the network attains a minimum in the
criterion function J(w), not the final weight values themselves. In practice, however,
because networks are rarely fully trained to a training error minimum (Sect. 6.8.14),
the learning rate can affect the quality of the final network. If some weights converge
significantly earlier than others (non-uniform learning) then the network may notnonuniform
learning perform equally well throughout the full range of inputs, or equally well for patterns
in each category. Figure 6.18 shows the effect of different learning rates on convergence
in a single dimension.
The optimal learning rate is the one which leads to the local error minimum in one
learning step. A principled method of setting the learning rate comes from assuming
the criterion function can be reasonably approximated by a quadratic which thus gives
(Fig. 6.19):
∂2J
∂w2
∆w =
∂J
∂w
. (34)
The optimal rate is found directly to be
ηopt =
(
∂2J
∂w2
)−1
. (35)
6.8. PRACTICAL TECHNIQUES FOR BACKPROPAGATION 35
η=ηopt
w
J η<ηopt
w*
J J ηopt <η<ηopt J η>ηopt
w
w*
w
w*
w
w*
γ
Figure 6.18: Gradient descent in a one-dimensional quadratic criterion with different
learning rates. If η < ηopt, convergence is assured, but training can be needlessly
slow. If η = ηopt, a single learning step suffices to find the error minimum. If
ηopt < η < 2ηopt, the system will oscillate but nevertheless converge, but training is
needlessly slow. If η > 2ηopt, the system diverges.
Of course the maximum learning rate that will give convergence is ηmax = 2ηopt. It
should be noted that a learning rate η in the range ηopt < η < 2ηopt will lead to slower
convergence (Computer exercise 8).
J
w
w*
η=ηopt
w
w*
∆w
∂Jγ
∂w
Figure 6.19: If the criterion function is quadratic (above), its derivative is linear (be-
low). The optimal learning rate ηopt insures that the weight value yielding minimum
error, w∗ is found in a single learning step.
Thus, for rapid and uniform learning, we should calculate the second derivative of
the criterion function with respect to each weight and set the optimal learning rate
separately for each weight. We shall return in Sect. ?? to calculate second derivatives
in networks, and to alternate descent and training methods such as Quickprop that
give fast, uniform learning. For typical problems addressed with sigmoidal networks
and parameters discussed throughout this section, it is found that a learning rate
36 CHAPTER 6. MULTILAYER NEURAL NETWORKS
of η � 0.1 is often adequate as a first choice, and lowered if the criterion function
diverges, or raised if learning seems unduly slow.
6.8.10 Momentum
Error surfaces often have plateaus — regions in which the slope dJ(w)/dw is very
small — for instance because of “too many” weights. Momentum — loosely based
on the notion from physics that moving objects tend to keep moving unless acted
upon by outside forces — allows the network to learn more quickly when plateaus
in the error surface exist. The approach is to alter the learning rule in stochastic
backpropagation to include some fraction α of the previous weight update:
w(m + 1) = w(m) + ∆w(m)︸ ︷︷ ︸
gradient
descent
+α∆w(m− 1)︸ ︷︷ ︸
momentum
(36)
Of course, α must be less than 1.0 for stability; typical values are α � 0.9. It must
be stressed that momentum rarely changes the final solution, but merely allows it to
be found more rapidly. Momentum provides another benefit: effectively “averaging
out” stochastic variations in weight updates during stochastic learning and thereby
speeding learning, even far from error plateaus (Fig. 6.20).
w1
J(w)
sto
ch
as
tic
w.
m
om
en
tu
m
Figure 6.20: The incorporation of momentum into stochastic gradient descent by
Eq. 36 (white arrows) reduces the variation in overall gradient directions and speeds
learning, especially over plateaus in the error surface.
Algorithm 3 shows one way to incorporate momentum into gradient descent.
Algorithm 3 (Stochastic backpropagation with momentum)
1 begin initialize topology (# hidden units),w, criterion, α(< 1), θ, η,m← 0, bji ← 0, bkj ← 0
2 do m← m + 1
3 xm ← randomly chosen pattern
4 bji ← ηδjxi + αbji; bkj ← ηδkyj + αbkj
5 wji ← wji + bji; wkj ← wkj + bkj
6 until ∇J(w) < θ
6.8. PRACTICAL TECHNIQUES FOR BACKPROPAGATION 37
7 return w
8 end
6.8.11 Weight decay
One method of simplifying a network and avoiding overfitting is to impose a heuristic
that the weights should be small. There is no principled reason why such a method
of “weight decay” should always lead to improved network performance (indeed there
are occasional cases where it leads to degraded performance) but it is found in most
cases that it helps. The basic approach is to start with a network with “too many”
weights (or hidden units) and “decay” all weights during training. Small weights favor
models that are more nearly linear (Problems 1 & 41). One of the reasons weight
decay is so popular is its simplicity. After each weight update every weight is simply
“decayed” or shrunk according to:
wnew = wold(1− *), (37)
where 0 < * < 1. In this way, weights that are not needed for reducing the criterion
function become smaller and smaller, possibly to such a small value that they can be
eliminated altogether. Those weights that are needed to solve the problem cannot de-
cay indefinitely. In weight decay, then, the system achieves a balance between pattern
error (Eq. 60) and some measure of overall weight. It can be shown (Problem 43) that
the weight decay is equivalent to gradient descent in a new effective error or criterion
function:
Jef = J(w) +
2*
η
wtw. (38)
The second term on the right hand side of Eq. 38 preferentially penalizes a single large
weight. Another version of weight decay includes a decay parameter that depends
upon the value of the weight itself, and this tends to distribute the penalty throughout
the network:
*mr =
γη/2
(1 + w2mr)
2
. (39)
We shall discuss principled methods for setting *, and see how weight decay is an
instance of a more general regularization procedure in Chap. ??.
6.8.12 Hints
Often we have insufficient training data for adequate classification accuracy and we
would like to add information or constraints to improve the network. The approach
of learning with hints is to add output units for addressing an ancillary problem, one
related to the classification problem at hand. The expanded network is trained on the
classification problem of interest and the ancillary one, possibly simultaneously. For
instance, suppose we seek to train a network to classify c phonemes based on some
acoustic input. In a standard neural network we would have c output units. In learning
with hints, we might add two ancillary output units, one which represents vowels and
the other consonants. During training, the target vector must be lengthened to include
components for the hint outputs. During classification the hint units are not used;
they and their hidden-to-output weights can be discarded (Fig. 6.21).
38 CHAPTER 6. MULTILAYER NEURAL NETWORKS
ω1 ω2 ω3 ωc h1
...
h2
categories hints
output
hidden
input
Figure 6.21: In learning with hints, the output layer of a standard network having
c units (discriminant functions) is augmented with hint units. During training, the
target vectors are also augmented with signals for the hint units. In this way the
input-to-hidden weights learn improved feature groupings. During classification the
hint units are not used, and thus they and their hidden-to-output weights are removed
from the trained network.
The benefit provided by hints is in improved feature selection. So long as the hints
are related to the classification problem at hand, the feature groupings useful for the
hint task are likely to aid category learning. For instance, the feature groupings useful
for distinguishing vowel sounds from consonants in general are likely to be useful
for distinguishing the /b/ from /oo/ or the /g/ from /ii/ categories in particular.
Alternatively, one can train just the hint units in order to develop improved hidden
unit representations (Computer exercise 16).
Learning with hints illustrates another benefit of neural networks: hints are more
easily incorporated into neural networks than into classifiers based on other algo-
rithms, such as the nearest-neighbor or MARS.
6.8.13 On-line, stochastic or batch training?
Each of the three leading training protocols described in Sect. 6.3.2 has strengths and
drawbacks. On-line learning is to be used when the amount of training data is so
large, or that memory costs are so high, that storing the data is prohibitive. Most
practical neural network classification problems are addressed instead with batch or
stochastic protocols.
Batch learning is typically slower than stochastic learning. To see this, imag-
ine a training set of 50 patterns that consists of 10 copies each of five patterns
(x1,x2, ...,x5). In batch learning, the presentations of the duplicates of x1 provide as
much information as a single presentation of x1 in the stochastic case. For example,
suppose in the batch case the learning rate is set optimally. The same weight change
can be achieved with just a single presentation of each of the five different patterns in
the batch case (with learning rate correspondingly greater). Of course, true problems
do not have exact duplicates of individual patterns; nevertheless, true data sets are
generally highly redundant, and the above analysis holds.
For most applications — especially ones employing large redundant training sets
— stochastic training is hence to be preferred. Batch training admits some second-
6.8. PRACTICAL TECHNIQUES FOR BACKPROPAGATION 39
order techniques that cannot be easily incorporated into stochastic learning protocols
and in some problems should be preferred, as we shall see in Sect. ??.
6.8.14 Stopped training
In three-layer networks having many weights, excessive training can lead to poor
generalization, as the net implements a complex decision boundary “tuned” to the
specific training data rather than the general properties of the underlying distribu-
tions. In training the two-layer networks of Chap. ??, we could train as long as we like
without fear that it would degrade final recognition accuracy because the complexity
of the decision boundary is not changed — it is always simply a hyperplane. This
example shows that the general phenomenon should be called “overfitting,” and not
“overtraining.”
Because the network weights are initialized with small values, the units operate in
their linear range and the full network implements linear discriminants. As training
progresses, the nonlinearities of the units are expressed and the decision boundary
warps. Qualitatively speaking, stopping the training before gradient descent is com-
plete can help avoid overfitting. In practice, the elementary criterion of stopping when
the error function decreases less than some preset value (e.g., line ?? in Algorithm ??),
does not lead reliably to accurate classifiers as it is hard to know beforehand what an
appropriate threshold θ should be set. A far more effective method is to stop training
when the error on a separate validation set reaches a minimum (Fig. ??). We shall
explore the theory underlying this version of cross validation in Chap. ??. We note
in passing that weight decay is equivalent to a form of stopped training (Fig. 6.22).
w1
w2
learning
stopped
initial weights
Figure 6.22: When weights are initialized with small magnitudes, stopped training
is equivalent to a form of weight decay since the final weights are smaller than they
would be after extensive training.
6.8.15 How many hidden layers?
The backpropagation algorithm applies equally well to networks with three, four, or
more layers, so long as the units in such layers have differentiable transfer functions.
Since, as we have seen, three layers suffice to implement any arbitrary function, we
40 CHAPTER 6. MULTILAYER NEURAL NETWORKS
would need special problem conditions or requirements recommend the use of more
than three layers.
One possible such requirement is translation, rotation or other distortion invari-
ances. If the input layer represents the pixel image in an optical character recognition
problem, we generally want such a recognizer to be invariant with respect to such
transformations. It is easier for a three-layer net to accept small translations than to
accept large ones. In practice, then, networks with several hidden layers distribute
the invariance task throughout the net. Naturally, the weight initialization, learning
rate, data preprocessing arguments apply to these networks too. The Neocognitron
network architecture (Sec. 6.10.7) has many layers for just this reason (though it is
trained by a method somewhat different than backpropagation). It has been found
empirically that networks with multiple hidden layers are more prone to getting caught
in undesirable local minima.
In the absence of a problem-specific reason for multiple hidden layers, then, it is
simplest to proceed using just a single hidden layer.
6.8.16 Criterion function
The squared error criterion of Eq. 8 is the most common training criterion because
it is simple to compute, non-negative, and simplifies the proofs of some theorems.
Nevertheless, other training criteria occasionally have benefits. One popular alternate
is the cross entropy which for n patterns is of the form:
J(w)ce =
n∑
m=1
c∑
k=1
tmkln(tmk/zmk), (40)
where tmk and zmk are the target and the actual output of unit k for pattern m. Of
course, this criterion function requires both the teaching and the output values in the
range (0, 1).
Regularization and overfitting avoidance is generally achieved by penalizing com-
plexity of models or networks (Chap. ??). In regularization, the training error and the
complexity penalty should be of related functional forms. Thus if the pattern error is
the sum of squares, then a reasonable network penalty would be squared length of the
total weight vector (Eq. 38). Likewise, if the model penalty is some description length
(measured in bits), then a pattern error based on cross entropy would be appropriate
(Eq. 40).
Yet another criterion function is based on the Minkowski error:Minkowski
error
JMink(w) =
n∑
m=1
c∑
k=1
|zmk(x)− tmk(x)|R, (41)
much as we saw in Chap. ??. It is a straightforward matter to derive the backpropa-
gation rule for the this error (Problem ??). While in general the rule is a bit more
complex than for the (R = 2) sum squared error we have considered (since it includes
a Sgn[·] function), the Minkowski error for 1 ≤ R < 2 reduces the influence of long
tails in the distributions — tails that may be quite far from the category decision
boundaries. As such, the designer can adjust the “locality” of the classifier indirectly
through choice of R; the smaller the R, the more local the classifier.
Most of the heuristics described in this section can be used alone or in combination
with others. While they may interact in unexpected ways, all have found use in
6.9. *SECOND-ORDER METHODS 41
important pattern recognition problems and classifier designers should have experience
with all of them.
6.9 *Second-order methods
We have used a second-order analysis of the error in order to determine the optimal
learning rate. One can use second-order information more fully in other ways.
6.9.1 Hessian matrix
We derived the first-order derivatives of a sum-squared-error criterion function in
three-layer networks, summarized in Eqs. 16 & 20. We now turn to second-order
derivatives, which find use in rapid learning methods, as well as some pruning or
regularization algorithms. For our criterion function,
J(w) =
1
2
n∑
m=1
(tm − zm)2, (42)
where tm and zm are the target and output signals, and n the total number of training
patterns. The elements in the Hessian matrix are
∂2J(w)
∂wji∂wlk
=
n∑
m=1
∂J
∂wji
∂J
∂wlk
+
n∑
m=1
(z − t) ∂
2J
∂wji∂wlk
(43)
where we have used the subscripts to refer to any weight in the network — thus i, j, l
and k could all take on values that describe input-to-hidden weights, or that describe
hidden-to-output weights, or mixtures. Of course the Hessian matrix is symmetric.
The second term in Eq. 43 is often neglected as ; this approximation guarantees that
the resulting approximation is positive definite.
The second term is of order O(‖t− o‖); using Fisher’s method of scoring we set this
term to zero. This gives the expected value, a positive definite matrix thereby guar-
anteeing that gradient descent will progress. In this so-called Levenberg-Marquardt or Levenberg-
Marquardt
approxima-
tion
outer product approximation our Hessian reduces to:
The full exact calculation of the Hessian matrix for a three-layer network such as
we have considered is (Problem 31):
If the two weights are both in the hidden-to-output layer:
outout (44)
If the two weights are both in the input-to-hidden layer:
inin (45)
If the weights are in different layers:
inout (46)
42 CHAPTER 6. MULTILAYER NEURAL NETWORKS
6.9.2 Newton’s method
xxx
∆J(w) = J(w + ∆w)− J(w)
�
(
∂J(w)
∂w
)t
∆w +
1
2
∆wtH∆w, (47)
where H is the Hessian matrix. We differentiate Eq. 47 with respect to ∆ and find
that ∆J(w) is minimized for
(
∂J(w)
∂w
)
+ H∆w = 0, (48)
and thus the optimum change in weights can be expressed as
∆w = −H−1
(
∂J(w)
∂w
)
. (49)
Thus, if we have an estimate for the optimal weights w(m), we can get an improved
estimate using the weight change given by Eq. 49, i.e.,
w(m + 1) = w(m) + ∆w = w(m)−H−1(m)
(
∂J(w(m))
∂w
)
, (50)
Thus in this Newton’s algorithm, we iteratively recompute w.
Alas, the computation of the Hessian can be expensive, and there is no guarantee
that the Hessian is nonsingular.
xxx
6.9.3 Quickprop
The simplest method for using second-order information to increase training speed is
the Quickprop algorithm. In this method, the weights are assumed to be independent,
and the descent is optimized separately for each. The error surface is assumed to be
quadratic (i.e., a parabola) and the coefficients for the parabola are determined by
two successive evaluations of J(w) and dJ(w)/dw. The single weight w is then moved
to the computed minimum of the parabola (Fig. 6.23). It can be shown (Problem 34)
that this approach leads to the following weight update rule:
∆w(m + 1) =
dJ
dw
∣∣
m
dJ
dw
∣∣
m−1 −
dJ
dw
∣∣
m
∆w(m). (51)
If the third- and higher-order terms in the error are non-negligible, or if the assumption
of weight independence does not hold, then the computed error minimum will not
equal the true minimum, and further weight updates will be needed. When a number
of obvious heuristics are imposed — to reduce the effects of estimation error when
the surface is nearly flat, or the step actually increases the error — the method can
be significantly faster than standard backpropagation. Another benefit is that each
weight has, in effect, its own learning rate, and thus weights tend to converge at
roughly the same time, thereby reducing problems due to nonuniform learning.
6.9. *SECOND-ORDER METHODS 43
w
w*
J(w)
∆w(m)
dJ
dw
m-1
dJ
dw
m
∆w(m+1)
Figure 6.23: The quickprop weight update takes the error derivatives at two points
separated by a known amount, and by Eq. 51 makes its next weight value. If the
error can be fully expressed as a second-order function, then the weight update leads
to the weight (w∗) leading to minimum error.
6.9.4 Conjugate gradient descent
Another fast learning method is conjugate gradient descent, which employs a series
of line searches in weight or parameter space. One picks the first descent direction
(for instance, determined by the gradient) and moves along that direction until the
minimum in error is reached. The second descent direction is then computed: this
direction — the “conjugate direction” — is the one along which the gradient does not
change its direction, but merely its magnitude during the next descent. Descent along
this direction will not “spoil” the contribution from the previous descent iterations
(Fig. ??).
w
poor
�line
�sea
rch�
dire
ctio
n�fl
(gra
dien
t�dir
ectio
n�va
ries)
conjugate�direction�∆w()γ
(gradient�directionfl
does�not�vary)
fi
rs
t�
d
es
ce
n
t�
d
ir
ec
ti
o
n
�∆
w
(1
)
H�≠�I H�=�I
∆
w
(1
)
∆
w
(2)
w(1)
w(2)
w
w
w
Figure 6.24: Conjugate gradient descent in weight space employs a sequence of
line searches. If ∆w(1) is the first descent direction, the second direction obeys
∆wt(1)H∆w(2) = 0. Note especially that along this second descent, the gradient
changes only in magnitude, not direction; as such the second descent does not “spoil”
the contribution due to the previous line search. In the case where the Hessian is
diagonal (right), the directions of the line searches are orthogonal.
More specifically, if we let ∆w(m − 1) represent the direction of a line search on
44 CHAPTER 6. MULTILAYER NEURAL NETWORKS
step m− 1. (Note especially that this is not an overall magnitude of change, which is
determined by the line search). We demand that the subsequent direction, ∆w(m),
obey
∆wt(m− 1)H∆w(m) = 0, (52)
where H is the Hessian matrix. Pairs of descent directions that obey Eq. 52 are
called “conjugate.” If the Hessian is proportional to the identity matrix, then such
directions are orthogonal in weight space. Conjugate gradient requires batch training,
since the Hessian matrix is defined over the full training set.
The descent direction on iteration m is in the direction of the gradient plus a
component along the previous descent direction:
∆w(m) = −∇J(w(m)) + βm∆w(m− 1), (53)
and the relative proportions of these contributions is governed by β. This proportion
can be derived by insuring that the descent direction on iteration m does not spoil
that from direction m− 1, and indeed all earlier directions. It is generally calculated
in one of two ways. The first formula (Fletcher-Reeves) is
βm =
[∇J(w(m))]t ∇J(w(m))
[∇J(w(m− 1))]t ∇J(w(m− 1))
(54)
A slightly preferable formula (Polak-Ribiere) is more robust in non-quadratic error
functions is:
βm =
[∇J(w(m))]t [∇J(w(m))−∇J(w(m− 1))]
[∇J(w(m− 1))]t ∇J(w(m− 1))
. (55)
Equations 53 & 36 show that conjugate gradient descent algorithm is analogous
to calculating a “smart” momentum, where β plays the role of a momentum. If the
error function is quadratic, then the convergence of conjugate gradient descent is
guaranteed when the number of iterations equals the total number of weights.
Example 1: Conjugate gradient descent
Consider finding the miminimum of a simple quadratic criterion function centered
on the origin of weight space, J(w) = 1/2(.2w21 + w
2
2) = w
tHw, where by simple
differentiation the Hessian is found to be H =
(
.2 0
0 1
)
. We start descent descent at a
randomly selected position, which happens to be w(0) =
(−8
−4
)
, as shown in the figure.
The first descent direction is determined by a simple gradient, which is easily found to
be −∆J(w(0)) = −
(
.4w1(0)
2w2(0)
)
=
(
3.2
8
)
. In typical complex problems in high dimensions,
the minimum along this direction is found using a line search, in this simple case the
minimum can be found be calculus. We let s represent the distance along the first
descent direction, and find its value for the minimum of J(w) according to:
d
ds
[[(
−8
−4
)
+ s
(
3.2
8
)]t(
.2 0
0 1
) [(
−8
−4
)
+ s
(
3.2
8
)]]
= 0
which has solution s = 0.562. Therefore the minimum along this direction is
6.9. *SECOND-ORDER METHODS 45
w(1) = w(0) + 0.562(−∆J(w(0)))
=
(
−8
−4
)
+ 0.562
(
3.2
8
)
=
(
−6.202
0.496
)
.
Now we turn to the use of conjugate gradients for the next descent. The simple
gradient evaluated at w(1) is
−∆J(w(1)) = −
(
.4w1(1)
2w2(1)
)
=
(
2.48
−0.99
)
.
(It is easy to verify that this direction, shown as a black arrow in the figure, does not
point toward the global minimum at w =
(
0
0
)
.) We use the Fletcher-Reeves formula
(Eq. 54) to construct the conjugate gradient direction:
β1 =
[∆J(w(1))]t∆J(w(1))
[∆J(w(0))]t∆J(w(0))
=
(−2.48 .99)
(−2.48
.99
)
(−3.2 8)
(−3.2
8
) = 7.13
74
= 0.096.
Incidentally, for this quadratic error surface, the Polak-Ribiere formula (Eq. 55) would
give the same value. Thus the conjugate descent direction is
∆w(1) = −∆J(w(1)) + β1
(
3.2
8
)
=
(
2.788
−.223
)
.
-7.5 -5 -2.5 0 2.5 5 7.5
-7.5
-5
-2.5
0
2.5
5
7.5
w2
w1
w(0)
w(1)
-∆
J(
w
(0
)) -∆J(w(1))
Conjugate gradient descent in a quadratic error landscape, shown in contour plot,
starts at a random point w(0) and descends by a sequence of line searches. The first
direction is given by the standard gradient and terminates at a minimum of the error
— the point w(1). Standard gradient descent from w(1) would be along the black
vector, “spoiling” some of the gains made by the first descent; it would, furthermore,
miss the global minimum. Instead, the conjugate gradient (red vector) does not spoil
the gains from the first descent, and properly passes through the global error minimum
at w =
(
0
0
)
.
46 CHAPTER 6. MULTILAYER NEURAL NETWORKS
As above, rather than perform a traditional line search, we use calculus to find the
error minimum along this second descent direction:
d
ds
[
[w(1) + s∆w(1)]tH [w(1) + s∆w(1)]
]
=
d
ds
[[(
−6.202
0.496
)
+ s
(
2.788
−.223
)]t(
.2 0
0 1
) [(
−6.202
0.496
)
+ s
(
2.788
−.223
)]]
= 0
which has solution s = 2.231. This yields the next minimum to be
w(2) = w(1) + s∆w(1) =
(
−6.202
0.496
)
+ 2.231
(
2.788
−.223
)
=
(
0
0
)
.
Indeed, the conjugate gradient search finds the global minimum in this quadratic
error function in two search steps — the number of dimensions of the space.
6.10 *Additional networks and training methods
The elementary method of gradient descent used by backpropagation can be slow,
even with straightforward improvements. We now consider some alternate networks
and training methods.
6.10.1 Radial basis function networks (RBF)
We have already considered several classifiers, such as Parzen windows, that employ
densities estimated by localized basis functions such as Gaussians. In light of our
discussion of gradient descent and backpropagation in particular, we now turn to a
different method for training such networks. A radial basis function network with
linear output unit implements
zk(x) =
nH∑
j=0
wkjφj(x). (56)
where we have included a j = 0 bias unit. If we define a vector φ whose components
are the hidden unit outputs, and a matrix W whose entries are the hidden-to-output
weights, then Eq. 56 can be rewritten as: z(x) = Wφ. Minimizing the criterion
function
J(w) =
1
2
n∑
m=1
(y(xm; w)− tm)2 (57)
is formally equivalent to the linear problem we saw in Chap. ??. We let T be the
matrix consisting of target vectors and Φ the matrix whose columns are the vectors
φ, then the solution weights obey
ΦtΦWt = ΦtT, (58)
and the solution can be written directly: Wt = Φ†T. Recall that Φ† is the pseu-
doinverse of Φ. One of the benefits of such radial basis function or RBF networks
6.10. *ADDITIONAL NETWORKS AND TRAINING METHODS 47
with linear output units is that the solution requires merely such standard linear tech-
niques. Nevertheless, inverting large matrices can be computationally expensive, and
thus the above method is generally confined to problems of moderate size.
If the output units are nonlinear, that is, if the network implements
zk(x) = f
nH∑
j=0
wkjφj(x)
(59)
rather than Eq. 56, then standard backpropagation can be used. One need merely
take derivatives of the localized transfer functions. For classification problems it is
traditional to use a sigmoid for the output units in order to keep the output values
restricted to a fixed range. Some of the computational simplification afforded by
sigmoidal at the hidden units functions is absent, but this presents no conceptual
difficulties (Problem ??).
6.10.2 Special bases
Occasionally we may have special information about the functional form of the dis-
tributions underlying categories and then it makes sense to use corresponding hidden
unit transfer functions. In this way, fewer parameters need to be learned for a given
quality of fit to the data. This is an example of increasing the bias of our model, and
thereby reducing the variance in the solution, a crucial topic we shall consider again
in Chap. ??. For instance, if we know that each underlying distribution comes from
a mixture of two Gaussians, naturally we would use Gaussian transfer functions and
use a learning rule that set the parameters (such as the mean and covariance).
6.10.3 Time delay neural networks (TDNN)
One can also incorporate prior knowledge into the network architecture itself. For
instance, if we demand that our classifier be insensitive to translations of the pattern,
we can effectively replicate the recognizer at all such translations. This is the approach
taken in time delay neural networks (or TDNNs)
Figure 6.25 shows a typical TDNN architecture; while the architecture consists
of input, hidden and output layers, much as we have seen before, there is a crucial
difference. Each hidden unit accepts input from a restricted (spatial) range of posi-
tions in the input layer. Hidden units at “delayed” locations (i.e., shifted to the right)
accept inputs from the input layer that are similarly shifted. Training proceeds as in
standard backpropagation, but with the added constraint that corresponding weights
are forced to have the same value — an example of weight sharing. Thus, the weights weight
sharinglearned do not depend upon the position of the pattern (so long as the full pattern
lies in the domain of the input layer).
The feedforward operation of the network (during recognition) is the same as in
standard three-layer networks, but because of the weight sharing, the final output
does not depend upon the position of the input. The network gets its name from the
fact that it was developed for, and finds greatest use in speech and other temporal
phenomena, where the shift corresponds to delays in time. Such weight sharing can
be extended to translations in an orthogonal spatial dimensions, and has been used
in optical character recognition systems, where the location of an image in the input
space is not precisely known.
48 CHAPTER 6. MULTILAYER NEURAL NETWORKS
in
pu
t
hi
dd
en
ou
tp
ut
time
Figure 6.25: A time delay neural network (TDNN) uses weight sharing to insure that
patterns are recognized regardless of shift in one dimension; in practice, this dimension
generally corresponds to time. In this example, there are five input units at each time
step. Because we hypothesize that the input patterns are of four time steps or less
in duration, each of the hidden units at a given time step accepts inputs from only
4× 5 = 20 input units, as highlighted in gray. An analogous translation constraint is
also imposed between the hidden and output layer units.
6.10.4 Recurrent networks
Up to now we have considered only networks which use feedforward flow of information
during classification; the only feedback flow was of error signals during training. Now
we turn to feedback or recurrent networks. In their most general form, these have
found greatest use in time series prediction, but we consider here just one specific
type of recurrent net that has had some success in static classification tasks.
Figure 6.26 illustrates such an architecture, one in which the output unit values
are fed back and duplicated as auxiliary inputs, augmenting the traditional feature
values. During classification, a static pattern x is presented to the input units, the
feedforward flow computed, and the outputs fed back as auxiliary inputs. This, in
turn, leads to a different set of hidden unit activations, new output activations, and
so on. Ultimately, the activations stabilize, and the final output values are used for
classification. As such, this recurrent architecture, if “unfolded” in time, is equivalent
to the static network shown at the right of the figure, where it must be emphasized that
many sets of weights are constrained to be the same (weight sharing), as indicated.
This unfolded representation shows that recurrent networks can be trained via
standard backpropagation, but with the weight sharing constraint imposed, as in
TDNNs.
6.10. *ADDITIONAL NETWORKS AND TRAINING METHODS 49
x
y
z
x
y(1)
z(1)
y(2)
z(2)
x x
y(3)
z(3)
wji
wji
wji
wkj
wkj
wkj
Figure 6.26: The form of recurrent network most useful for static classification has
the architecture shown at the bottom, with the recurrent connections in red. It is
functionally equivalent to a static network with many hidden layers and extensive
weight sharing, as shown above. Note that the input is replicated.
6.10.5 Counterpropagation
Occasionally, one wants a rapid prototype of a network, yet one that has expressive
power greater than a mere two-layer network. Figure 6.27 shows a three-layer net,
which consists of familiar input, hidden and output layers.∗ When one is learning the
weights for a pattern in category ωi,
In this way, the hidden units create a Voronoi tesselation (cf. Chap. ??), and the
hidden-to-output weights pool information from such centers of Voronoi cells. The
processing at the hidden units is competitive learning (Chap. ??).
The speedup in counterpropagation is that only the weights from the single most
active hidden unit are adjusted during a pattern presentation. While this can yield
suboptimal recognition accuracy, counterpropagation can be orders of magnitude
faster than full backpropagation. As such, it can be useful during preliminary data
exploration. Finally, the learned weights often provide an excellent starting point for
refinement by subsequent full training via backpropagation.
∗ It is called “counterpropagation” for an earlier implementation that employed five layers with
signals that passed bottom-up as well as top-down.
50 CHAPTER 6. MULTILAYER NEURAL NETWORKS
Figure 6.27: The simplest version of a counterpropagation network consists of three
layers. During training, an input is presented and the most active hidden unit is
determined. The only weights that are modified are the input-to-hidden weights
leading to this most active hidden unit and the single hidden-to-output weight leading
to the proper category. Weights can be trained using an LMS criterion.
6.10.6 Cascade-Correlation
The central notion underlying the training of networks by cascade-correlation is quite
simple. We begin with a two-layer network and train to minimum of an LMS error. If
the resulting training error is low enough, training is stopped. In the more common
case in which the error is not low enough, we fix the weights but add a single hid-
den unit, fully connected from inputs and to output units. Then these new weights
are trained using an LMS criterion. If the resulting error is not sufficiently low, yet
another hidden unit is added, fully connected from the input layer and to the output
layer. Further, the output of each previous hidden unit is multiplied by a fixed weight
of -1 and presented to the new hidden unit. (This prevents the new hidden unit from
learning function already represented by the previous hidden units.) Then the new
weights are trained via an LMS criterion. Thus training proceeds by alternatively
training weights, then (if needed) adding a new hidden unit, training the new modi-
fiable weights, and so on. In this way the network grows to a size that depends upon
the problem at hand (Fig. 6.28).
The benefit is that often faster than strict backprop since fewer weights are up-
dated at any time (Computer exercise 18).
Algorithm 4 (Cascade-correlation)
1 begin initialize a, criterion θ, η, k ← 0
2 do m← m + 1
3 wki ← wki − η∇J(w)
4 until ∇J(w) � θ
5 if J(w > θ then add hidden unit else exit
6 do m← m + 1
7 wji ← wji − η∇J(w); wkj ← wkj − η∇J(w)
8 until ∇J(w) � θ
9 return w
10 end
6.11. REGULARIZATION AND COMPLEXITY ADJUSTMENT 51
ω1 ω2
1
2
3
ωc
x0
x1
xd
xi
z1 z1 zk zc
wki
wji
wkj
-1
-1
-1
Figure 6.28: The training of a multilayer network via cascade-correlation begins with
the input later fully connected to the output layer (black). Such weights, wki are
trained using an LMS criterion, as discussed in Chap. ??. If the resulting training
error is not sufficiently low, a first hidden unit (labeled 1, in red) is introduced, fully
interconnected from the input layer and to the output layer. These new red weights are
trained, while the previous (black) ones are held fixed. If the resulting training error
is still not sufficiently low, a second hidden unit (labeled 2) is likewise introduced,
fully interconnected; it also receives a the output from each previous hidden unit,
multiplied by -1. Training proceeds in this way, training successive hidden units until
the training error is acceptably low.
6.10.7 Neocognitron
The cognitron and its descendent, the Neocognitron, address the problem of recogni-
tion of characters in pixel input. The networks are noteworthy not for the learning
method, but instead for their reliance on a large number of layers for translation, scale
and rotation invariance.
The first layer consists of hand tuned feature detectors, such as vertical, horizon-
tal and diagonal line detectors. Subsequent layers consist of slightly more complex
features, such as Ts or Xx, and so forth — weighted groupings of the outputs of
units at earlier layers. The total number of weights in such a network is enormous
(Problem 35).
6.11 Regularization and complexity adjustment
Whereas the number of inputs and outputs of a backpropagation network are deter-
mined by the problem itself, we do not know a priori the number of hidden units,
or weights. If we have too many degrees of freedom, we will have overfitting. This
will depend upon the number of training patterns and the complexity of the problem
52 CHAPTER 6. MULTILAYER NEURAL NETWORKS
0 1 2 3 4 5 6 7 8 9
19 x 19 x 12
21 x 21 x 8
21 x 21 x 38
13 x 13 x 19
13 x 13 x 35
7 x 7 x 23
13 x 3 x 11
1 x 1 x 10
input
output
19 x 19 x 1
Figure 6.29: The neocognitron consists of a 19× 19 pixel input layer, seven interme-
diate layers, and an output layer consisting of 10 units, one for each digit. The earlier
layers consist of relatively fixed feature detectors (as shown); units in successively
layer respond to a spatial range of units in the previous layer. In this way, shift,
rotation and scale invariance is distributed throughout the network. The network is
trained one-layer at a time by a large number of patterns.
itself.
We could try different numbers of hidden units, apply knowledge of the problem
domain or add other constraints. The error is the sum of an error over patterns (such
as we have used before) plus a regularization term, which expresses constraints or
desirable properties of solutions:
J = Jpat + λJreg. (60)
The parameter λ is adjusted to impose the regularization more or less strongly.
Because a desirable constraint is simpler networks (i.e., simpler models), regular-
ization is often used to adjust complexity, as in weight decay.
6.11. REGULARIZATION AND COMPLEXITY ADJUSTMENT 53
6.11.1 Complexity measurement
xxx
6.11.2 Wald statistics
The fundamental theory of generalization favors simplicity. For a given level of per-
formance on observed data, models with fewer parameters can be expected to perform
better on test data. For instance weight decay leads to simpler decision boundaries
(closer to linear). Likewise, training via cascade-correlation adds weights only as
needed.
The fundamental idea in Wald statistics is that we can estimate the importance
of a parameter in a model, such as a weight, by how much the training error increases
if that parameter is eliminated. To this end the Optimal Brain Damage method
(OBD) seeks to delete weights by keeping the training error as small as possible.
OBS extended OBD to include the off-diagonal terms in the network’s Hessian, which
were shown to be significant and important for pruning in classical and benchmark
problems.
OBD and Optimal Brain Surgeon (OBS) share the same basic approach of training
a network to (local) minimum in error at weight w∗, and then pruning a weight that
leads to the smallest increase in the training error. The predicted functional increase
in the error for a change in full weight vector δw is:
δJ =
(
∂J
∂w
)T
· δw
︸ ︷︷ ︸
0
+
1
2
δwT · ∂
2J
∂w2︸ ︷︷ ︸
≡H
·δw + O(‖δw‖3)︸ ︷︷ ︸
0
, (61)
where H is the Hessian matrix. The first term vanishes because we are at a local
minimum in error; we ignore third- and higher-order terms. The general solution for
minimizing this function given the constraint of deleting one weight is (Problem ??):
δw = − wq
[H−1]qq
H−1 · uq and Lq =
1
2
w2q
[H−1]qq
. (62)
Here, uq is the unit vector along the qth direction in weight space and Lq is the
saliency of weight q — an estimate of the increase in training error if weight q is
pruned and the other weights updated by the left equation in Eq. 62 (Problem 42).
We define Xk ≡ ∂g(x
m; w)
∂w
and ak ≡ ∂
2d(tm,zm)
∂z2
, and can easily show that the
recursion for computing the inverse Hessian becomes:
H−1m+1 = H
−1
m −
H−1m ·Xm+1 ·XTm+1 ·H−1m
P
ak
+ XTm+1 ·H
−1
m ·Xm+1
,
H−10 = α
−1I (63)
H−1n = H
−1 , (64)
where α is a small parameter — effectively a weight decay constant (Problem 38).
Note how different error measures d(t, z) scale the gradient vectors Xk forming the
Hessian (Eq. ??). For the squared error d(t, z) = (t − z)2, we have ak = 1, and all
gradient vectors are weighted equally.
Problem: repeat for cross-entropy (Problem 36).
54 CHAPTER 6. MULTILAYER NEURAL NETWORKS
J(w)
w
0 w*
true error
quadratic
approximationa
pp
ro
x.
to
s
al
ie
nc
y
sa
li
en
cy
Figure 6.30: The saliency of a parameter, such as a weight, is the increase in the
training error when that weight is set to zero. One can approximate the saliency by
expanding the true error around a local minimum, w∗, and setting the weight to zero.
In this example the approximated saliency is smaller than the true saliency; this is
typically, but not always the case.
w*
0
mag
O
B
DOB
S
w2
w1
J(w)
Figure 6.31: In the second-order approximation to the criterion function, optimal
brain damage assumes the Hessian matrix is diagonal, while Optimal Brain Surgeon
uses the full Hessian matrix.
Summary
Multilayer nonlinear neural networks — nets with two or more layers of modifiable
weights — trained by gradient descent methods such as backpropagation perform a
maximum likelihood estimation of the weight values (parameters) in the model defined
by the network topology. One of the great benefits of learning in such networks is the
simplicity of the learning algorithm, the ease in model selection, and the incorporation
of heuristic constraints by means such as weight decay. Discrete pruning algorithms
such as Optimal Brain Surgeon and Optimal Brain Damage correspond to priors
favoring few weights, and can help avoid overfitting.
6.11. SUMMARY 55
Alternate networks and training algorithms have benefits. For instance radial basis
functions are most useful when the data clusters. Cascade-correlation and counter-
propagation are generally faster than backpropagation.
Complexity adjustment: weight decay, Wald statistic, which for networks is opti-
mal brain damage and optimal brain surgeon, which use the second-order approxima-
tion to the true saliency as a pruning criterion.
56 CHAPTER 6. MULTILAYER NEURAL NETWORKS
Bibliographical and Historical Remarks
McCulloch and Pitts provided the first principled mathematical and logical treatment
of the behavior of networks of simple neurons [49]. This pioneering work addressed
non-recurrent as well as recurrent nets (those possessing “circles,” in their termi-
nology), but not learning. Its concentration on all-or-none or threshold function of
neurons indirectly delayed the consideration of continuous valued neurons that would
later dominate the field. These authors later wrote an extremely important paper
on featural mapping (cf. Chap. ??), invariances, and learning in nervous systems and
thereby advanced the conceptual development of pattern recognition significantly [56].
Rosenblatt’s work on the (two-layer) Perceptron (cf. Chap. ??) [61, 62] was some
of the earliest to address learning, and was the first to include rigorous proofs about
convergence. A number of stochastic methods, including Pandemonium [66, 67], were
developed for training networks with several layers of processors, though in keeping
with the preoccupation with threshold functions, such processors generally computed
logical functions (AND or OR), rather than some continuous functions favored in later
neural network research. The limitations of networks implementing linear discrimi-
nants — linear machines — were well known in the 1950s and 1960s and discussed by
both their promoters [62, cf., Chapter xx, “Summary of Three-Layer Series-Coupled
Systems: Capabilities and Deficiencies”] and their detractors [51, cf., Chapter 5,
“ψCONNECTED: A Geometric Property with Unbounded Order”].
A popular early method was to design by hand three-layer networks with fixed
input-to-hidden weights, and then train the hidden-to-output weight [80, for a review].
Much of the difficulty in finding learning algorithms for all layers in a multilayer neural
network came from the prevalent use of linear threshold units. Since these do not have
useful derivatives throughout their entire range, the current approach of applying the
chain rule for derivatives and the resulting “backpropagation of errors” did not gain
more adherents earlier.
The development of backpropagation was gradual, with several steps, not all of
which were appreciated or used at the time. The earliest application of adaptive
methods that would ultimately become backpropagation came from the field of con-
trol. Kalman filtering from electrical engineering [38, 28] used an analog error (dif-
ference between predicted and measured output) for adjusting gain parameters in
predictors. Bryson, Denham and Dreyfus showed how Lagrangian methods could
train multilayer networks for control, as described in [6]. We saw in the last chapter
the work of Widrow, Hoff and their colleagues [81, 82] in using analog signals and
the LMS training criterion applied to pattern recognition in two-layer networks. Wer-
bos [77][78, Chapter 2], too, discussed a method for calculating the derivatives of a
function based on a sequence of samples (as in a time series), which, if interpreted
carefully carried the key ideas of backpropagation. Parker’s early “Learning logic”
[53, 54], developed independently, showed how layers of linear units could be learned
by a sufficient number of input-output pairs. This work lacked simulations on repre-
sentative or challenging problems (such as XOR) and was not appreciated adequately.
Le Cun independently developed a learning algorithm for three-layer networks [9, in
French] in which target values are propagated, rather than derivatives; the resulting
learning algorithm is equivalent to standard backpropagation, as pointed out shortly
thereafter [10].
Without question, the paper by Rumelhart, Hinton and Williams [64], later ex-
panded into a full and readable chapter [65], brought the backpropagation method to
the attention of the widest audience. These authors clearly appreciated the power of
6.11. BIBLIOGRAPHICAL AND HISTORICAL REMARKS 57
the method, demonstrated it on key tasks (such as the exclusive OR), and applied it
to pattern recognition more generally. An enormous number of papers and books of
applications — from speech production and perception, optical character recognition,
data mining, finance, game playing and much more — continues unabated. One novel
class of for such networks includes generalization for production [20, 21]. One view of
the history of backpropagation is [78]; two collections of key papers in the history of
neural processing more generally, including many in pattern recognition, are [3, 2].
Clear elementary papers on neural networks can be found in [46, 36], and several
good textbooks, which differ from the current one in their emphasis on neural networks
over other pattern recognition techniques, can be recommended [4, 60, 29, 27]. An
extensive treatment of the mathematical aspects of networks, much of which is beyond
that needed for mastering the use of networks for pattern classification, can be found
in [19]. There is continued exploration of the strong links between networks and more
standard statistical methods; White presents and overview [79], and books such as
[8, 68] explore a number of close relationships. The important relation of multilayer
Perceptrons to Bayesian methods and probability estimation can be found in [23,
59, 43, 5, 13, 63, 52].posterior probability!and backpropagation Original papers on
projection pursuit and MARS, can be found in [15] and [34], respectively, and a good
overview in [60].
Shortly after its wide dissemination, the backpropagation algorithm was criti-
cized for its lack of biological plausibility; in particular, Grossberg [22] discussed the
non-local nature of the algorithm, i.e., that synaptic weight values were transported
without physical means. Somewhat later Stork devised a local implementation of
backpropagation was [71, 45], and pointed out that it was nevertheless highly implau-
sible as a biological model.
The discussions and debates over the relevance of Kolmogorov’s Theorem [39] to
neural networks, e.g. [18, 40, 41, 33, 37, 12, 42], have centered on the expressive
power. The proof of the univerasal expressive power of three-layer nets based on
bumps and Fourier ideas appears in [31]. The expressive power of networks having
non-traditional transfer functions was explored in [72, 73] and elsewhere. The fact
that three-layer networks can have local minima in the criterion function was explored
in [50] and some of the properties of error surfaces illustrated in [35].
The Levenberg-Marquardt approximation and deeper analysis of second-order
methods can be found in [44, 48, 58, 24]. Three-layer networks trained via cascade-
correlation have been shown to perform well compared to standard three-layer nets
trained via backpropagation [14]. Our presentation of counterpropagation networks
focussed on just three of the five layers in a full such network [30]. Although there
was little from a learning theory new presented in Fukushima’s Neocognitron [16, 17],
its use of many layers and mixture of hand-crafted feature detectors and learning
groupings showed how networks could address shift, rotation and scale invariance.
Simple method of weight decay was introduced in [32], and gained greater accep-
tance due to the work of Weigend and others [76]. The method of hints was introduced
in [1]. While the Wald test [74, 75] has been used in traditional statistical research
[69], its application to multilayer network pruning began with the work of Le Cun
et al’s Optimal Brain Damage method [11], later extended to include non-diagonal
Hessian matrices [24, 25, 26], including some speedup methods [70]. A good review
of the computation and use of second order derivatives in networks can be found in
[7] and of pruning algorithms in [58].
58 CHAPTER 6. MULTILAYER NEURAL NETWORKS
Problems
⊕
Section 6.2
1. Show that if the transfer function of the hidden units is linear, a three-layer
network is equivalent to a two-layer one. Explain why, therefore, that a three-layer
network with linear hidden units cannot solve a non-linearly separable problem such
as XOR or n-bit parity.
2. Fourier’s Theorem can be used to show that a three-layer neural net with sigmoidal
hidden units can approximate to arbitrary accuracy any posterior function. Consider
two-dimensional input and a single output, z(x1, x2). Recall that Fourier’s Theorem
states that, given weak restrictions, any such functions can be written as a possibly
infinite sum of cosine functions, as
z(x1, x2) ≈
∑
f1
∑
f2
Af1f2cos(f1x1) cos(f2x2),
with coefficients Af1f2 .
(a) Use the trigonometric identity
cosα cosβ =
1
2
cos(α + β) +
1
2
cos(α− β)
to write z(x1, x2) as a linear combination of terms cos(f1x1 + f2x2) and
cos(f1x1 − f2x2).
(b) Show that cos(x) or indeed any continuous function f(x) can be approximated
to any accuracy by a linear combination of sign functions as:
f(x) ≈ f(x0) +
N∑
i=0
[f(xi+1)− f(xi)]
[
1 + Sgn(x− xi)
2
]
,
where the xi are sequential values of x; the smaller xi+1 − xi, the better the
approximation.
(c) Put your results together to show that z(x1, x2) can be expressed as a linear
combination of step functions or sign functions whose arguments are themselves
linear combinations of the input variables x1 and x2. Explain, in turn, why
this implies that a three-layer network with sigmoidal hidden units and a linear
output unit can implement any function that can be expressed by a Fourier
series.
(d) Does your construction guarantee that the derivative df(x)/dx can be well ap-
proximated too?
⊕
Section 6.3
3. Consider an d− nH − c network trained with n patterns for me epochs.
(a) What is the space complexity in this problem? (Consider both the storage of
network parameters as well as the storage of patterns, but not the program
itself.)
6.11. PROBLEMS 59
(b) Suppose the network is trained in stochastic mode. What is the time complex-
ity? Since this is dominated by the number of multiply-accumulations, use this
as a measure of the time complexity.
(c) Suppose the network is trained in batch mode. What is the time complexity?
4. Prove that the formula for the sensitivity δ for a hidden unit in a three-layer net
(Eq. 20) generalizes to a hidden unit in a four- (or higher-) layer network, where the
sensitivity is the weighted sum of sensitivities of units in the next higher layer.
5. Explain in words why the backpropagation rule for training input-to-hidden
weights makes intuitive sense by considering the dependency upon each of the terms
in Eq. 20.
6. One might reason that the the dependence of the backpropagation learning rules
(Eq. ??) should be roughly inversely related to f ′(net); i.e., that weight change should
be large where the output does not vary. In fact, of course, the learning rule is linear
in f ′(net). What, therefore, is wrong with the above view?
7. Show that the learning rule described in Eqs. 16 & 20 works for bias, where
x0 = y0 = 1 is treated as another input and hidden unit.
8. Consider a standard three-layer backpropagation net with d input units, nH
hidden units, c output units, and bias.
(a) How many weights are in the net?
(b) Consider the symmetry in the value of the weights. In particular, show that if
the sign if flipped on every weight, the network function is unaltered.
(c) Consider now the hidden unit exchange symmetry. There are no labels on
the hidden units, and thus they can be exchanged (along with corresponding
weights) and leave network function unaffected. Prove that the number of such
equivalent labellings — the exchange symmetry factor — is thus nH2nH . Eval-
uate this factor for the case nH = 10.
9. Using the style of procedure, write the procedure for on-line version of backpropa-
gation training, being careful to distinguish it from stochastic and batch procedures.
10. Express the derivative of a sigmoid in terms of the sigmoid itself in the following
two cases (for positive constants a and b):
(a) A sigmoid that is purely positive: f(net) = 1
1+ea net
.
(b) An anti-symmetric sigmoid: f(net) = atanh(b net).
11. Generalize the backpropagation to four layers, and individual (smooth, differ-
entiable) transfer functions at each unit. In particular, let xi, yj , vl and zk denote
the activations on units in successive layers of a four-layer fully connected network,
trained with target values tk. Let f1i be the transfer function of unit i in the first
layer, f2j in the second layer, and so on. Write a program, with greater detail than
that of Algorithm 1, showing the calculation of sensitivities, weight update, etc. for
the general four-layer network.⊕
Section 6.4
12. Use Eq. ?? to show why the input-to-hidden weights must be different from each
other (e.g., random) or else learning cannot proceed well (cf. Computer Exercise 2).
60 CHAPTER 6. MULTILAYER NEURAL NETWORKS
13. Show that proper preprocessing of the data will lead to faster convergence, at
least in a simple network 2-1 (two-layer) network with bias. Suppose the training
data come from two Gaussians, p(x|ω1) ∼ N(−.5, 1) and p(x|ω2) ∼ N(+.5, 1). Let
the teaching values for the two categories be t = ±1.
(a) Write the error as a sum over the n patterns of a function of the weights, inputs,
etc.
(b) Differentiate twice with respect to the weights to get the Hessian H. Express
your answer in words as well.
(c) Consider two data sets drawn from p(x|ωi) ∼ N(µi, I) for i = 1, 2 and I is the
2× 2 identity matrix. Calculate your Hessian in terms of µi.
(d) Calculate the maximum and minimum eigenvalues of the Hessian in terms of
the components of µi.
(e) Suppose µ1 = (1, 0)
t and µ2 = (0, 1)
t. Calculate the ratio of the eigenvalues,
and hence a measure of the convergence time.
(f) Now standardize your data, by subtracting means and scaling to have unit
covariances in each of the two dimensions. That is, find two new distributions
that have overall zero mean and the same covariance. Check your answer by
calculating the ratio of the maximum to minimum eigenvalues.
(g) If T denotes the total training time in the unprocessed data, express the time
required for the preprocessed data (cf. Computer exercise 13).
14. Consider the derivation of the bounds on the convergence time for gradient
descent. Complete the steps leading to Eq. ?? as follows:
(a) Express the error to second order in new coordinates w̃ that are parallel to the
principal axes of the Hessian.
(b) Write an equation analogous to that of Eq. ?? in the transformed space. Use Λ
as the diagonal matrix of eigenvalues of the Hessian.
(c) Inspect your result and use Eq. ?? to state a criterion for convergence in terms
of λmax, the maximum eigenvalue of the Hessian.
15. Assume that the criterion function J(w) is well described to second order by a
Hessian matrix H.
(a) Show that convergence of learning is assured if the learning rate obeys η < 2/λmax, where λmax is the largest eigenvalue of H. (b) Show that the learning time is thus dependent upon the ratio of the largest to the smallest non-negligible eigenvalue of H. (c) Explain why “standardizing” the training data can therefore reduce learning time. 6.11. PROBLEMS 61 ⊕ Section 6.5 16. Problem on feature mapping. xx ⊕ Section 6.6 17. Fill in the steps in the derivation leading to Eq. 25. 18. Consider Eq. 27, and confirm that one of the solutions to the minimum squared error condition yields outputs that are indeed posterior probabilities. Do this as follows: (a) To find the minimum of J̃(w), calculate its derivative ∂J̃(w)/∂w; this will consist of the sum of two integrals. Set ∂J̃(w)/∂w = 0 and solve to obtain the natural solution. (b) Apply Bayes’ rule and the normalization P (ωk|x) + P (ωi �=k|x) = 1 to prove that the outputs zk = gk(x; w) are indeed equal to the posterior probabilities P (ωk|x). 19. In the derivation that backpropagation finds a least squares fit to the posterior probabilities, it was implicitly assumed that the network could indeed represent the true underlying distribution. Explain where in the derivation this was assumed, and what in the subsequent steps may not hold if that assumption is violated. 20. Show that the softmax output (Eq. 29) indeed approximates posterior probabil- ities if the hidden unit outputs, y, belong to the family of exponential distributions as: p(y|ωk) = exp[A(w̃k) + B(y, φ) + w̃tky] for nH -dimensional vectors w̃k and y, and scalar φ and scalar functions A(·) and B(·, ·). Proceed as follows: (a) Given p(y|ωk), use Bayes’ Theorem to write the posterior probability P (ωk|y). (b) Interpret the parameters A(·), w̃k, B(·, ·) and φ in light of your results. 21. Consider a three-layer network for classification with output units employing softmax (Eq. 29), trained with 0− 1 signals. (a) Derive the learning rule if the criterion function (per pattern) is sum squared error, i.e., J(w) = 1 2 c∑ k=1 (tk − zk)2. (b) Repeat for the criterion function is cross-entropy, i.e., Jce(w) = c∑ k=1 tkln tk zk . 62 CHAPTER 6. MULTILAYER NEURAL NETWORKS 22. Clearly if the discriminant functions gk1(x; w) and gk2(x; w) were independent, the derivation of Eq. 26 would follow from Eq. 27. Show that the derivation is never- theless valid despite the fact that these functions are implemented in part using the same input-to-hidden weights.⊕ Section 6.7 23. Show that the slope of the sigmoid and the learning rates together determine the learning time. (a) That is, show that if the slope of the sigmoid is increased by a factor of γ, and the learning rate decreased by a factor 1/γ, that the total learning time remains the same. (b) Must the input be rescaled for this relationship to hold? 24. Show that the basic three-layer neural networks of Sect. 6.2 are special cases of general additive models by describing in detail the correspondences between Eqs. 6 & 31. 25. Show that the sigmoidal transfer function acts to transmit the maximum infor- mation if its inputs are distributed normally. Recall that the entropy (a measure of information) is defined as H = ∫ p(y)ln[p(y)]dy. (a) Consider a continuous input variable x drawn from the density p(x) ∼ N(0, σ2). What is entropy for this distribution? (b) Suppose samples x are passed through an antisymmetric sigmoidal function to give y = f(x), where the zero crossing of the sigmoid occurs at the peak of the Gaussian input, and the effective width of the linear region equal to the range −σ < x < +σ. What are the values of a and b in Eq. 33 insures this? (c) Calculate the entropy of the output distribution p(y). (d) Suppose instead that the transfer function were a Dirac delta function δ(x− 0). What is the entropy of the resulting output distribution p(y)? (e) Summarize your results of (c) and (d) in words. ⊕ Section 6.8 26. Consider the sigmoidal transfer function: f(net) = a tanh(b net) = a [ 1− eb net 1 + eb net ] = 2a 1 + e−b net − a. (a) Show that its derivative f ′(net) can be written simply in terms of f(net) itself. (b) What are f(net), f ′(net) and f ′′(net) at net = −∞? 0? +∞? (c) For which value of net is the second derivative f ′′(net) extremal? 27. Consider the computational burden for standardizing data, as described in the text. 6.11. PROBLEMS 63 (a) What is the computational complexity of standardizing a training set of n d- dimensional patterns? (b) Estimate the computational complexity of training. Use the heuristic for choos- ing the size of the network (i.e., number of weights) described in Sect. 6.8.7. Assume that the number of training epochs is nd. (c) Use your results from (a) and (b) to express the computational burden of stan- dardizing as a ratio. (Assume unknown constants are 1.) 28. Derive the gradient descent learning rule for a three-layer network with linear in- put units and sigmoidal hidden and output units for the Minkowski xxx and arbitrary R. Confirm that your answer reduces to Eqs. 16 & 20 for R = 2. 29. Training rule for polynomial classifier. Show that terms can become extremely large for realistic values of the input. 30. Train in noise. show improves bp under realistic assumptions; not so nearest neighbor ⊕ Section 6.9 31. Derive the exact expression for the full Hessian matrix for a sum squared error criterion in a three-layer network, as given in Eqs. 44 – 46. 32. Repeat Problem 31 but for a cross entropy error criterion. 33. Calculate a Hessian, see if it shrinks any vector. (Convergence assured.) 34. Derive Eq. 51 from the discussion in the text. ⊕ Section 6.10 35. What is the space complexity of the Neocognitron network of Fig. 6.29? If we used the heuristic of Sec. 6.8.7, how many training patterns would be needed? (In practice, since many weights are hand set in the form of feature detectors, fewer training patterns are needed.) 36. Derive the central equations for OBD and OBS in a three-layer sigmoidal network for a cross-entropy error. ⊕ Section 6.11 37. Consider a general constant matrix K and variable vector parameter x. (a) Write in summation notation with components explicit, and derive the formula for the derivative: d dx [xtKx] = (K + Kt)x. (b) Show simply that for the case where K is symmetric (as for instance the Hessian matrix H = Ht), we have: d dx [xtHx] = 2Hx as was used in Eq. ?? and in the derivation of the Optimal Brain Surgeon method. 64 CHAPTER 6. MULTILAYER NEURAL NETWORKS 38. Show that the constant α in the OBS derivation (Eq. ??) is equivalent to a weight decay. 39. (a) Find the space and the time computational complexities for one step in the nominal OBS method. (b) Find the space and the time computational complexities for pruning the first weight in OBS. What is it for pruning subsequent weights, if one uses Shur’s decomposition method? (c) Find the space and the time computational complexities for one step of OBD (without retraining). 40. Weight decay is equivalent to doing gradient descent on an error that has a “complexity” term. (a) Show that in the weight decay rule wnewij = w old ij (1− *) amounts to performing gradient descent in the error function Jef = J(w) + 2%η w tw (Eq.38). (b) Express γ in terms of the weight decay constant * and learning rate η. (c) Likewise, show that if wnewmr = w old mr(1 − *mr) where *mr = 1/(1 + w2mr)2, that the new effective error function is Jef = J(w) + γ ∑ mr w2mr/(1 +w 2 mr). Find γ in terms of η and *mr. (d) Consider a network with a wide range of magnitudes for weights. Describe qualitatively how the two different weight decay methods affect the network. 41. Show that the weight decay rule of Eq. 37 is equivalent to a prior on models that favors small weights. ⊕ Section ?? 42. (a) Fill in the steps between Eq. ?? and ?? for the saliency. (b) Find the saliency in OBD, where one assumes Hij = 0 for i �= j. ⊕ Section ?? 43. Prove that the weight decay rule of Eq. 37 leads to the Jreg of Eq. 38. Computer exercises Several exercises will make use of the following three-dimensional data sampled from three categories, denoted ωi. (CHANGE NUMBERS xxx) 6.11. COMPUTER EXERCISES 65 ω1 ω2 ω3 sample x1 x2 x3 x1 x2 x3 x1 x2 x3 1 0.28 1.31 -6.2 0.011 1.03 -0.21 1.36 2.17 0.14 2 0.07 0.58 -0.78 1.27 1.28 0.08 1.41 1.45 -0.38 3 1.54 2.01 -1.63 0.13 3.12 0.16 1.22 0.99 0.69 4 -0.44 1.18 -4.32 -0.21 1.23 -0.11 2.46 2.19 1.31 5 -0.81 0.21 5.73 -2.18 1.39 -0.19 0.68 0.79 0.87 6 1.52 3.16 2.77 0.34 1.96 -0.16 2.51 3.22 1.35 7 2.20 2.42 -0.19 -1.38 0.94 0.45 0.60 2.44 0.92 8 0.91 1.94 6.21 -0.12 0.82 0.17 0.64 0.13 0.97 9 0.65 1.93 4.38 -1.44 2.31 0.14 0.85 0.58 0.99 10 -0.26 0.82 -0.96 0.26 1.94 0.08 0.66 0.51 0.88 ⊕ Section 6.2 1. Consider a 2-2-1 network with bias, where the transfer function at the hidden units and the output unit is a sigmoid yj = a tanh[b netj ] for a = 1.716 and b = 2/3. Suppose the matrices describing the input-to-hidden weights (wji for j = 1, 2 and i = 0, 1, 2) and the hidden-to-output weights (wkj for k = 1 and j = 0, 1, 2) are, respectively, xx xxxx xx xx xx and xxxx xx . The network is to be used to classify patterns into one of two categories, based on the sign of the output unit signal. Shade a two-dimensional input space x1 − x2 (−5 ≤ x1, x2 ≤ +5) black or white according to the category given by the network. Repeat with xx xxxx xx xx xx and xxxx xx . xxx ⊕ Section 6.3 2. Create a 3-1-1 sigmoidal network with bias to be trained to classify patterns from ω1 and ω2 in the table above. Use stochastic backpropagation to (Algorithm 1) with learning rate η = 0.1 and sigmoid as described in Eq. 33 in Sect. 6.8.2. (a) Initialize all weights randomly in the range −1 ≤ w ≤ +1. Plot a learning curve — the training error as a function of epoch. (b) Now repeat (a) but with weights initialized to be the same throughout each level. In particular, let all input-to-hidden weights be initialized with wji = 0.5 and all hidden-to-output weights with wkj = −0.5. (c) Explain the source of the differences between your learning curves (cf. Prob- lem 12). 66 CHAPTER 6. MULTILAYER NEURAL NETWORKS 3. Consider the nonlinearly separable categorization problem shown in Fig. 6.8. (a) Train a 1-3-1 sigmoidal network with bias by means of batch backpropagation (Algorithm 2) to solve it. (b) Display your decision boundary by classifying points along separated by ∆x � 0.1. (c) Repeat with a 1-2-1 network. (d) Inspect the decision boundary for your 1-3-1 network (or construct by hand an optimal one) and explain why no 1-2-1 network with sigmoidal hidden units can achieve it. 4. Write a backpropagation program for a 2-2-1 network with bias to solve the XOR problem (see Fig. 6.1). Show the input-to-hidden weights and analyze the function of each hidden unit. 5. Write a basic backpropagation program for a 3-3-1 network with bias to solve the three-bit parity problem, i.e., return a +1 if the number of input units that are high is even, and -1 if odd. (a) Show the input-to-hidden weights and analyze the function of each hidden unit. (b) Retrain several times from a new random point until you get a local (but not global) minimum. Analyze the function of the hidden units now. (c) How many patterns are properly classified for your local minimum? 6. Write a stochastic backpropagation program for a 2− nH − 1 network with bias to classify points chosen randomly in the range −1 ≤ x1, x2 ≤ +1 with P (ω1) = P (ω2) = 0.5. Train using 40 points (20 from each category). Train with nH = 1, 2, 3 and 4 hidden units. Plot your minimum training error as a function of nH . How many hidden units are needed to implement your particular random function? 7. Train a 2-4-1 network having a different transfer function at each hidden unit on a random problem.⊕ Section 6.4 8. Measure H, show that convergence is slower for ηopt < η < 2ηopt.⊕ Section 6.5 9. Train net and show that the hidden⊕ Section 6.6 10. Three-layer with softmax outputs. 11. Train with one set of priors; test with other priors.⊕ Section 6.7 12. Consider several gradient descent methods applied to a criterion function in one dimension: simple gradient descent with learning rate η, optimized descent, Newton’s method, and Quickprop. Consider first the criterion function J(w) = w2 which of course has minimum J = 0 at w = 0. In all cases, start the descent at w(0) = 1. For definiteness, we consider convergence to be complete when 6.11. COMPUTER EXERCISES 67 (a) Plot the number of steps until convergence as a function of η for η = 0.01, 0.03, 0.1, 0.3, 1, 3. (b) Calculate the optimum learning rate ηopt by Eq. 35, and confirm that this value is correct from your graph in (??). (c) ⊕ Section 6.8 13. Demonstrate that preprocessing data can lead to significant reduction in time of learning. Consider a single linear output unit for a two-category classification task, with teaching values tω1 = + 1, tω2 = − 1, with squared error criterion. (a) Write a program to train the three weights based on training samples. (b) Generate 20 samples from each of two categories P (ω1) = P (ω2) = .5 and p(x|ωi) ∼ N(µi), I, where I is the 2× 2 identity matrix and µ1 = (??, ??)t and µ2 = (??, ??) t. (c) Find the optimal learning rate empirically by trying a few values. (d) Train to minimum error. Why is there no danger of overtraining in this case? (e) Why can we be sure that it is at least possible that this network can achieve the minimum (Bayes) error? (f) Generate 100 test samples, 50 from each category, and find the error rate. (g) Now preprocess the data by subtracting off the mean and scaling standard deviation in each dimension. (h) Repeat the above, and find the optimal learning rate. (i) Find the error rate on the (transformed) test set. (j) Verify that the accuracy is virtually the same in the two cases (any differences can be attributed to stochastic effects). 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Index augmented vector, 6 backpropagation, 4, 10–15 Algorithm, 50 stochastic, 15, 16 stochastic with momentum, 36 and neurobiology, 6 Bayes Theory, 24 biological plausibility, 57 feature mapping, 21 Bayes’ formula, 26 bias unit, 5 bump network response, 9 cascade-correlation, 50, 57 chain rule, see derivative, chain rule classifier polynomial, 30 conjugate gradient, 43 counterpropagation, 57 credit assignment problem, 11, 13 criterion stopping, see stopping criterion cross entropy, see entropy, cross cross validation, 17 degree of freedom network, 33 derivative chain rule, 13 discriminant hyperplane, 3 linear, 3 discriminant function neural network, 5, 8 distributed representation, see represen- tation, distributed entropy, 62 cross, 40 epoch, 15 error Minkowski, 40 training, 11 error surface, 17–20 plateau, 17 expressive power network, 8 filter matched, see matched filter Fletcher-Reeves equation, 44 Fourier’s Theorem, 9, 58 free parameter, 4 function harmonic, 9 generalized additive model, 28 generalized delta rule, see backpropa- gation global error minimum, 18 gradient descent, see also backpropa- gation hidden layer, 4 hidden unit exchange symmetry, see sym- metry, hidden unit exchange hidden unit weights representation, 23 hidden units number of, 33 hint, 37 hyperbolic tangent, 30 hypercube Kolmogorov Theorem, 8 independent, identically distributed, 15 information, 62 inner product in networks, 6 input scaling, see standarization, 31 75 76 INDEX input layer, 4 Kalman filtering, 56 Kolmogorov’s Theorem, 8, 57 Lagrangian method network training, see network train- ing, Lagrangian method learning curve, 16 nonuniform, 34, 42 rate, 12 uniform, 34 with queries, 15 learning logic, 56 learning rate optimal, 34 least mean square algorithm, 11, 25 Levenberg-Marquardt approximation, 41 linear machine, 6, 56 linear threshold unit, 6, 56 LMS Algorithm, 3, see least mean square, algorithm LMS error projection pursuit, 28 local minimum high dimensional, 20 local representation, see representation, local logical function, 56 LTU, see linear threshold unit MARS, see multivariate adaptive re- gression matched filter, 23 model section heuristic, 4 momentum, 36 multivariate adaptive regression, 28 net, see net activation net activation, 6 network expressive power, 57 network pruning, 57 network topology, 4 neural network feedforward operation, 11 learning, 11 time delay, 47 neuron, see unit nonuniform learning, see learning, nonuni- form OBD, see Optimal Brain Damage OBS, see Optimal Brain Surgeon Optimal Brain Damage, 53, 57 Optimal Brain Surgeon, 53, 57 outer product approximation, see Levenberg- Marquardt approximation Pandemonium, 56 parameter adjustment, 29 parity three-bit, 21 pattern desired, see pattern, target target, 11 teaching, see pattern, target pattern training, see training protocol, stochastic Perceptron, 8, 56 phi function, 3, see transfer function, 6 Polak-Ribiere equation, 44 polynomial classifier, see classifier, poly- nomial posterior probability and backpropagation, 26 principal components, 23 prior knowledge, 3 projection pursuit, 28 pseudoinverse, 46 query learning, see learning, with queries recall, see classification regularization, 51–52 network, 4 representation distributed, 30 global, see representation, distributed local, 30 ridge function, 28 ridge regression, 28 sensitivity unit, 13 sigmoid parameter, 30 sign function, 6 INDEX 77 softmax, 27 spline function, 28 standardize, 31, 34 standardized data, 60 stopping criterion, 15, 20 symmetry hidden unit exchange, 59 synapse, 6 tanh, see hyperbolic tangent target value, 32 target propagation, 56 TDNN, see neural network, time delay threshold function, 6 time delay neural network, see neural network, time delay topology network, see network topology training procedure stochastic, 15 training protocol batch, 15 learning with queries, 15 on-line, 15 stochastic, 15 transfer function, 6, 29 continuity, 29 linearity, 30 piecewise linear, 30 uniform learning, see learning, uniform unit, 5 hidden, 51 validation error, 16 vector augmented, see augmented vector target, 11 Wald statistic, 53 Wald test, 57 weight excitatory, 6 inhibitory, 6 initialization, 34 sharing, 47 weight decay, 37, 57 winner-take-all, 27 XOR problem error surface, 19 network, 6 Contents 7 Stochastic Methods 3 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7.2 Stochastic search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7.2.1 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . . 5 7.2.2 The Boltzmann factor . . . . . . . . . . . . . . . . . . . . . . . 5 Algorithm 1: Simulated annealing . . . . . . . . . . . . . . . . . . . . . 8 7.2.3 Deterministic simulated annealing . . . . . . . . . . . . . . . . 9 Algorithm 2: Deterministic annealing . . . . . . . . . . . . . . . . . . . 11 7.3 Boltzmann learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 7.3.1 Stochastic Boltzmann learning of visible states . . . . . . . . . 13 7.3.2 Missing features and category constraints . . . . . . . . . . . . 17 7.3.3 Deterministic Boltzmann learning . . . . . . . . . . . . . . . . 20 Algorithm 3: Deterministic Boltzmann learning . . . . . . . . . . . . . 20 7.3.4 Initialization and setting parameters . . . . . . . . . . . . . . . 20 7.4 *Boltzmann networks and graphical models . . . . . . . . . . . . . . . 23 7.5 *Evolutionary methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.5.1 Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 27 Algorithm 4: Basic Genetic algorithm . . . . . . . . . . . . . . . . . . 27 7.5.2 Further heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.5.3 Why do they work? . . . . . . . . . . . . . . . . . . . . . . . . 31 7.6 *Genetic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Bibliographical and Historical Remarks . . . . . . . . . . . . . . . . . . . . 35 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Computer exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1 2 CONTENTS Chapter 7 Stochastic Methods 7.1 Introduction Learning plays a central role in the construction of pattern classifiers. As we haveseen, the general approach is to specify a model having one or more parameters and then estimate their values from training data. When the models are fairly simple and of low dimension, we can use analytic methods such as computing derivatives and performing gradient descent to find optimal model parameters. If the models are somewhat more complicated, we may calculate local derivatives and use gradient methods, as in neural networks and some maximum-likelihood problems. In most high-dimensional and complicated models, there are multiple maxima and we must use a variety of tricks — such as performing the search multiple times from different starting conditions — to have any confidence that an acceptable local maximum has been found. These methods become increasingly unsatisfactory as the models become more complex. A naive approach — exhaustive search through solution space — rapidly gets out of hand and is completely impractical for real-world problems. The more complicated the model, the less the prior knowledge, and the less the training data, the more we must rely on sophisticated search for finding acceptable model parameters. In this chapter we consider stochastic methods for finding parameters, where randomness plays a crucial role in search and learning. The general approach is to bias the search toward regions where we expect the solution to be and allow randomness — somehow — to help find good parameters, even in very complicated models. We shall consider two general classes of such methods. The first, exemplified by Boltzmann learning, is based on concepts and techniques from physics, specifically statistical mechanics. The second, exemplified by genetic algorithms, is based on concepts from biology, specifically the mathematical theory of evolution. The former class has a highly developed and rigorous theory and many successes in pattern recog- nition; hence it will command most of our effort. The latter class is more heuristic yet affords flexibility and can be attractive when adequate computational resources are available. We shall generally illustrate these techniques in cases that are simple, and which might also be addressed with standard gradient procedures; nevertheless we emphasize that these stochastic methods may be preferable in complex problems. 3 4 CHAPTER 7. STOCHASTIC METHODS The methods have high computational burden, and would be of little use without computers. 7.2 Stochastic search We begin by discussing an important and general quadratic optimization problem. Analytic approaches do not scale well to large problems, however, and thus we focus here on methods of search through different candidate solutions. We then consider a form of stochastic search that finds use in learning for pattern recognition. Suppose we have a large number of variables si, i = 1, . . . , N where each variable can take one of two discrete values, for simplicity chosen to be ±1. The optimization problem is this: find the values of the si so as to minimize the cost or energyenergy E = −1 2 N∑ i,j=1 wijsisj , (1) where the wij can be positive or negative and are problem dependent. We require the self-feedback terms to vanish, i.e., wii = 0, since non-zero wii merely add an unimportant constant to E, independent of the si. This optimization problem can be visualized in terms of a network of nodes, where bi-directional links or interconnec- tions correspond to the weights wij = wji. (It is very simple to prove that we can always replace a non-symmetric connection matrix by its symmetric part, as asked in Problem 2. We avoid non-symmetric matrices because they unnecessarily complicate the dynamics described in Sect. 7.2.1.) Figure 7.1 shows such a network, where nodes are labeled input, output, and hidden, though for the moment we shall ignore such distinctions. This network suggests a physical analogy which in turn will guide our choice of solution method. Imagine the network represents N physical magnets, each of which can have its north pole pointing up (si = +1) or pointing down (si = −1). The wij are functions of the physical separations between the magnets. Each pair of magnets has an associated interaction energy which depends upon their state, separation and other physical properties: Eij = −1/2 wijsisj . The energy of the full system is the sum of all these interaction energies, as given in Eq. 1. The optimization task is to find the configuration of states of the magnets with the most stable configuration, the one with lowest energy. This general optimization problem appears in a wide range of applications, in many of which the weights do not have a physical interpretation.∗ As mentioned, we shall be particularly interested in its application to learning methods. Except for very small problems or few connections, it is infeasible to solve directly for the N values si that give the minimum energy — the space has 2N possible configurations (Problem 4). It is tempting to propose a greedy algorithm to search for the optimum configuration: Begin by randomly assigning a state to each node. Next consider each node in turn and calculate the energy with it in the si = +1 state and then in the si = −1 state, and choose the one giving the lower energy. (Naturally, this decision needs to be based on only those nodes connected to node i with non-zero weight wij .) Alas, this greedy search is rarely successful since the system usually gets caught in local energy minima or never converges (Computer exercise 1). Another method is required. ∗ Similar generalized energy functions, called Lyapunov functions or objective functions, can be used for finding optimum states in other problem domains as well. 7.2. STOCHASTIC SEARCH 5 h id d en wij -1 +1 -1 -1 -1 +1 -1 +1+1 -1 +1 -1 -1 +1 -1 +1 -1i j αβ visible visible vi si b le Figure 7.1: The class of optimization problems of Eq. 1 can be viewed in terms of a network of nodes or units, each of which can be in the si = +1 or si = −1 state. Every pair of nodes i and j is connected by bi-directional weights wij ; if a weight between two nodes is zero then no connection is drawn. (Because the networks we shall discuss can have an arbitrary interconnection, there is no notion of layers as in multilayer neural networks.) The optimization problem is to find a configuration (i.e., assignment of all si) that minimizes the energy described by Eq. 1. The state of the full network is indexed by an integer γ, and since here there are 17 binary nodes, γ is bounded 0 ≤ γ < 217. The state of the visible nodes and hidden nodes are indexed by α and β, respectively and in this case are bounded 0 ≤ α ≤ 210 and 0 ≤ β < 27. 7.2.1 Simulated annealing In physics, the method for allowing a system such as many magnets or atoms in an al- loy to find a low-energy configuration is based on annealing. In physical annealing the annealing system is heated, thereby conferring randomness to each component (magnet). As a result, each variable can temporarily assume a value that is energetically unfavorable and the full system explores configurations that have high energy. Annealing proceeds by gradually lowering the temperature of the system — ultimately toward zero and thus no randomness — so as to allow the system to relax into a low-energy config- uration. Such annealing is effective because even at moderately high temperatures, the system slightly favors regions in the configuration space that are overall lower in energy, and hence are more likely to contain the global minimum. As the temperature is lowered, the system has increased probability of finding the optimum configuration. This method is successful in a wide range of energy functions or energy “landscapes,” though there are pathological cases such as the “golf course” landscape in Fig. 7.2 where it is unlikely to succeed. Fortunately, the problems in learning we shall consider rarely involve such pathological functions. 7.2.2 The Boltzmann factor The statistical properties of large number of interacting physical components at a temperature T , such as molecules in a gas or magnetic atoms in a solid, have been thoroughly analyzed. A key result, which relies on a few very natural assumptions, is 6 CHAPTER 7. STOCHASTIC METHODS x1 x1 x2 x2 E E Figure 7.2: The energy function or energy “landscape” on the left is meant to suggest the types of optimization problems addressed by simulated annealing. The method uses randomness, governed by a control parameter or “temperature” T to avoid getting stuck in local energy minima and thus to find the global minimum, like a small ball rolling in the landscape as it is shaken. The pathological “golf course” landscape at the right is, generally speaking, not amenable to solution via simulated annealing because the region of lowest energy is so small and is surrounded by energetically unfavorable configurations. The configuration space of the problems we shall address are discrete and thus the continuous x1 − x2 space shown here is a bit misleading. the following: the probability the system is in a (discrete) configuration indexed by γ having energy Eγ is given by P (γ) = e−Eγ/T Z(T ) , (2) where Z is a normalization constant. The numerator is the Boltzmann factor and theBoltzmann factor denominator the partition function, the sum over all possible configurations partition function Z(T ) = ∑ γ′ e−Eγ′/T , (3) which guarantees Eq. 2 represents a true probability.∗ The number of configurations is very high, 2N , and in physical systems Z can be calculated only in simple cases. Fortunately, we need not calculate the partition function, as we shall see. Because of the fundamental importance of the Boltzmann factor in our discus- sions, it pays to take a slight detour to understand it, at least in an informal way. Consider a different, but nontheless related system: one consisting of a large number of non-interacting magnets, that is, without interconnecting weights, in a uniform external magnetic field. If a magnet is pointing up, si = +1 (in the same direction as the field), it contributes a small positive energy to the total system; if the magnet is pointing down, a small negative energy. The total energy of the collection is thus proportional to the total number of magnets pointing up. The probability the system has a particular total energy is related to the number of configurations that have that energy. Consider the highest energy configuration, with all magnets pointing up. There is only ( N N ) = 1 configuration that has this energy. The next to highest ∗ In the Boltzmann factor for physical systems there is a “Boltzmann constant” which converts a temperature into an energy; we can ignore this factor by scaling the temperature in our simulations. 7.2. STOCHASTIC SEARCH 7 energy comes with just a single magnet pointing down; there are ( N 1 ) = N such con- figurations. The next lower energy configurations have two magnets pointing down; there are ( N 2 ) = N(N − 1)/2 of these configurations, and so on. The number of states declines exponentially with increasing energy. Because of the statistical independence of the magnets, for large N the probability of finding the state in energy E also decays exponentially (Problem 7). In sum, then, the exponential form of the Boltz- mann factor in Eq. 2 is due to the exponential decrease in the number of accessible configurations with increasing energy. Further, at high temperature there is, roughly speaking, more energy available and thus an increased probability of higher-energy states. This describes qualitatively the dependence of the probability upon T in the Boltzmann factor — at high T , the probability is distributed roughly evenly among all configurations while at low T , it is concentrated at the lowest-energy configurations. If we move from the collection of independent magnets to the case of magnets interconnected by weights, the situation is a bit more complicated. Now the energy associated with a magnet pointing up or down depends upon the state of others. Nonetheless, in the case of large N , the number of configurations decays exponentially with the energy of the configuration, as described by the Boltzmann factor of Eq. 2. Simulated annealing algorithm The above discussion and the physical analogy suggest the following simulated an- nealing method for finding the optimum configuration to our general optimization problem. Start with randomized states throughout the network, si(1), and select a high initial “temperature” T (1). (Of course in the simulation T is merely a control parameter which will control the randomness; it is not a true physical temperature.) Next, choose a node i randomly. Suppose its state is si = +1. Calculate the system energy in this configuration, Ea; next recalculate the energy, Eb, for a candidate new state si = − 1. If this candidate state has a lower energy, accept this change in state. If however the energy is higher, accept this change with a probability equal to e−∆Eab/T , (4) where ∆Eab = Eb−Ea. This occasional acceptance of a state that is energetically less favorable is crucial to the success of simulated annealing, and is in marked distinc- tion to naive gradient descent and the greedy approach mentioned above. The key benefit is that it allows the system to jump out of unacceptable local energy minima. For example, at very high temperatures, every configuration has a Boltzmann factor e−E/T ≈ e0 roughly equal. After normalization by the partition function, then, every configuration is roughly equally likely. This implies every node is equally likely to be in either of its two states (Problem 6). The algorithm continues polling (selecting and testing) the nodes randomly several polling times and setting their states in this way. Next lower the temperature and repeat the polling. Now, according to Eq. 4, there will be a slightly smaller probability that a candidate higher energy state will be accepted. Next the algorithm polls all the nodes until each node has been visited several times. Then the temperature is lowered further, the polling repeated, and so forth. At very low temperatures, the probability that an energetically less favorable state will be accepted is small, and thus the search becomes more like a greedy algorithm. Simulated annealing terminates when the temperature is very low (near zero). If this cooling has been sufficiently slow, the system then has a high probability of being in a low energy state — hopefully the global energy minimum. 8 CHAPTER 7. STOCHASTIC METHODS Because it is the difference in energies between the two states that determines the acceptance probabilities, we need only consider nodes connected to the one being polled — all the units not connected to the polled unit are in the same state and contribute the same total amount to the full energy. We let Ni denote the set of nodes connected with non-zero weights to node i; in a fully connected net would include the complete set of N − 1 remaining nodes. Further, we let Rand[0, 1) denote a randomly selected positive real number less than 1. With this notation, then, the randomized or stochastic simulated annealing algorithm is: Algorithm 1 (Stochastic simulated annealing) 1 begin initialize T (k), kmax, si(1), wij for i, j = 1, . . . , N 2 k ← 0 3 do k ← k + 1 4 do select node i randomly; suppose its state is si 5 Ea ← −1/2 Ni∑ j wijsisj 6 Eb ← −Ea 7 if Eb < Ea 8 then si ← −si 9 else if e−(Eb−Ea)/T (k) > Rand[0, 1)
10 then si ← −si
11 until all nodes polled several times
12 until k = kmax or stopping criterion met
13 return E, si, for i = 1, . . . , N
14 end
Because units are polled one at a time, the algorithm is occasionally called sequential
simulated annealing. Note that in line 5, we define Ea based only on those units
connected to the polled one — a slightly different convention than in Eq. 1. Changing
the usage in this way has no effect, since in line 9 it is the difference in energies that
determines transition probabilities.
There are several aspects of the algorithm that must be considered carefully, in
particular the starting temperature, ending temperature, the rate at which the tem-
perature is decreased and the stopping criterion. This function is called the cooling
schedule or more frequently the annealing schedule, T (k), where k is an iteration in-annealing
schedule dex. We demand T (1) to be sufficiently high that all configurations have roughly equal
probability. This demands the temperature be larger than the maximum difference
in energy between any configurations. Such a high temperature allows the system to
move to any configuration which may be needed, since the random initial configura-
tion may be far from the optimal. The decrease in temperature must be both gradual
and slow enough that the system can move to any part of the state space before being
trapped in an unacceptable local minimum, points we shall consider below. At the
very least, annealing must allow N/2 transitions, since a global optimum never differs
from any configuration by more than this number of steps. (In practice, annealing
can require polling several orders of magnitude more times than this number.) The
final temperature must be low enough (or equivalently kmax must be large enough or
a stopping criterion must be good enough) that there is a negligible probability that
if the system is in a global minimum it will move out.
Figure 7.3 shows that early in the annealling process when the temperature is
high, the system explores a wide range of configurations. Later, as the temperature
7.2. STOCHASTIC SEARCH 9
is lowered, only states “close” to the global minimum are tested. Throughout the
process, each transition corresponds to the change in state of a single unit.
A typical choice of annealing schedule is T (k + 1) = cT (k) with 0 < c < 1. If
computational resources are of no concern, a high initial temperature, large c < 1,
and large kmax are most desirable. Values in the range 0.8 < c < 0.99 have been
found to work well in many real-world problems. In practice the algorithm is slow,
requiring many iterations and many passes through all the nodes, though for all but
the smallest problems it is still faster than exhaustive search (Problem 5). We shall
revisit the issue of parameter setting in the context of learning in Sect. 7.3.4.
While Fig. 7.3 displayed a single trajectory through the configuration space, a more
relevant property is the probability of being in a configuration as the system is annealed
gradually. Figure 7.4 shows such probability distributions at four temperatures. Note
especially that at the final, low temperature the probability is concentrated at the
global minima, as desired. While this figure shows that for positive temperature all
states have a non-zero probability of being visited, we must recognize that only a
small fraction of configurations are in fact visited in any anneal. In short, in the
vast majority of large problems, annealing does not require that all configurations be
explored, and hence it is more efficient than exhaustive search.
7.2.3 Deterministic simulated annealing
Stochastic simulated annealing is slow, in part because of the discrete nature of the
search through the space of all configurations, i.e., an N -dimensional hypercube. Each
trajectory is along a single edge, thereby missing full gradient information that would
be provided by analog state values in the “interior” of the hypercube. An alternate,
faster method is to allow each node to take on analog values during search; at the
end of the search the values are forced to be si = ±1, as required by the optimization
problem. Such a deterministic simulated annealing algorithm also follows from the
physical analogy. Consider a single node (magnet) i connected to several others; each
exerts a force tending to point node i up or down. In deterministic annealing we sum
the forces and give a continuous value for si. If there is a large “positive” force, then
si ≈ +1; if a large negative force, then si ≈ −1. In the general case si will lie between
these limits.
The value of si also depends upon the temperature. At high T (large randomness)
even a large upward force will not be enough to insure si = +1, whereas at low
temperature it will. We let li =
∑
j
wijsj be the force exerted on node i, the updated
value is:
si = f(li, T ) = tanh[li/T ], (5)
where there is an implied scaling of the force and temperature in the response function response
functionf(·, ·) (Fig. 7.5). In broad overview, deterministic annealing consists in setting an
annealing schedule and then at each temperature finding an equilibrium analog value
for every si. This analog value is merely the expected value of the discrete si in a
system at temperature T (Problem 8). At low temperatures (i.e., at the end of the
anneal), each variable will assume an extreme value ±1, as can be seen in the low-T
curve in Fig. 7.5.
It is instructive to consider the energy landscape for the continuous case. Differ-
entiation of Eq. 1 shows that the energy is linear in each variable when others held
fixed, as can be seen in Fig. 7.6 — there are no local minima along any “cut” parallel
10 CHAPTER 7. STOCHASTIC METHODS
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E
(k)
k k
E
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begin
end
γ
Figure 7.3: Stochastic simulated annealing (Algorithm 1) uses randomness, governed
by a control parameter or “temperature” T (k) to search through a discrete space
for a minimum of an energy function. In this example there are N = 6 variables;
the 26 = 64 configurations are shown at the bottom along as a column of + and -.
The plot of the associated energy of each configuration given by Eq. 1 for randomly
chosen weights. Every transition corresponds to the change of just a single si. (The
configurations have been arranged so that adjacent ones differ by the state of just
a single node; nevertheless most transitions corresponding to a single node appear
far apart in this ordering.) Because the system energy is invariant with respect
to a global interchange si ↔ −si, there are two “global” minima. The graph at
the upper left shows the annealing schedule — the decreasing temperature versus
iteration number k. The middle portion shows the configuration versus iteration
number generated by Algorithm 1. The trajectory through the configuration space
is colored red for transitions that increase the energy; late in the annealing such
energetically unfavorable (red) transitions are rarer. The graph at the right shows
the full energy E(k), which decreases to the global minimum.
7.2. STOCHASTIC SEARCH 11
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Figure 7.4: An estimate of the probability P (γ) of being in a configuration denoted
by γ is shown for four temperatures during a slow anneal. (These estimates, based on
a large number of runs, are nearly the theoretical values e−Eγ/T ) Early, at high T ,
each configuration is roughly equal in probability while late, at low T , the probability
is strongly concentrated at the global minima. The expected value of the energy, E [E]
(i.e., averaged at temperature T ), decreases gradually during the anneal.
to any axis. Note too that there are no stable local energy minima within the volume
of the space; the energy minima always occur at the “corners,” i.e., extreme si = ±1
for all i, as required by the optimization problem.
This search method is sometimes called mean-field annealing because each node
responds to the average or mean of the forces (fields) due to the nodes connected to
it. In essence the method approximates the effects of all other magnets while ignoring
their mutual interactions and their response to the magnet in question, node i. Such
annealing is also called deterministic because in principle we could deterministically
solve the simultaneous equations governing the si as the temperature is lowered. The
algorithm has a natural parallel mode of implementation, for instance where each value
si is updated simultaneously and deterministically as the temperature is lowered. In
and inherently serial simulation, however, the nodes are updated one at a time. Even
though the nodes might be polled pseudo randomly, the algorithm is in principle
deterministic — there need be no inherent randomness in the searchn. If we let si(1)
denote the initial state of unit i, the algorithm is:
Algorithm 2 (Deterministic simulated annealing)
1 begin initialize T (k), wij , si(1), i, j = 1, . . . N
2 k ← 0
3 do k ← k + 1
12 CHAPTER 7. STOCHASTIC METHODS
-4 -2 2 4
-1
-0.5
0.5
1
.01
.1
1
5
li = Σ wij sj
j
si = f(li,T)
Figure 7.5: In deterministic annealing, each node can take on a continuous value
−1 ≤ si ≤ +1, which equals the expected value of a binary node in the system at
temperature T . In other words, the analog value si replaces the expectation of the
discrete variable, E [si]. We let li denote a force exerted by the nodes connected to si.
The larger this force, the closer the analog si is to +1; the more negative this force,
the closer to −1. The temperature T (marked in red) also affects si. If T is large,
there is a great deal of randomness and even a large force will not insure si ≈ +1.
At low temperature, there is little or no randomness and even a small positive force
insures that si = +1. Thus at the end of an anneal, each node has value si = +1 or
si = −1.
4 Select node i randomly
5 li ←
Ni∑
j
wijsj
6 si ← f(li, T (k))
7 until k = kmax or convergence criterion met
8 return E, si, i = 1, . . . , N
9 end
In practice, deterministic and stochastic annealing give very similar solutions. In
large real-world problems deterministic annealing is faster, sometimes by two or three
orders of magnitude.
Simulated annealing can also be applied to other classes of optimization problem,
for instance, finding the minimum in
∑
ijk
wijksisjsk. We will not consider such higher-
order problems, though they can be the basis of learning methods as well.
7.3 Boltzmann learning
For pattern recognition, we will use a network such as that in Fig. 7.1, where the input
units accept binary feature information and the output units represent the categories,
generally in the familiar 1-of-c representation (Fig. 7.7). During classification the
input units are held fixed or clamped to the feature values of the input pattern; theclamp
remaining units are annealed to find the lowest energy, most probable configuration.
The category information is then read from the final values of the output units.
Of course, accurate recognition requires proper weights, and thus we now turn to a
7.3. BOLTZMANN LEARNING 13
1
-2
-1
0
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
E
-2
-1
0
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
E
-2
-1
0
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
E
s1
s2
s1
s2
s1
s3 = -1 s3 = 0 s3 = +1
Figure 7.6: If the state variables si can assume analog values (as in mean-field anneal-
ing), the energy in Eq. 1 is a general quadratic form having minima at the extreme val-
ues si = ±1. In this case N = 3 nodes are fully interconnected with arbitrary weights
wij . While the total energy function is three-dimensional, we show two-dimensional
surfaces for each of three values of s3. The energy is linear in each variable so long as
the other variables are held fixed. Further, the energy is invariant with respect to the
interchange of all variables si ↔ −si. In particular, here the global minimum occurs
as s1 = −1, s2 = +1 and s3 = −1 as well as the symmetric configuration s1 = +1,
s2 = −1 and s3 = +1.
method for learning weights from training patterns. There are two closely related
approaches to such learning, one based on stochastic and the other on deterministic
simulated annealing.
7.3.1 Stochastic Boltzmann learning of visible states
Before we turn to our central concern — learning categories from training patterns
— consider an alternate learning problem where we have a set of desired probabilities
for all the visible units, Q(α) (given by a training set), and seek weights so that the
actual probability P (α), achieved in random simulations, matches these probabilities
over all patterns as closely as possible. In this alternative learning problem the desired
probabilities would be derived from training patterns containing both input (feature)
and output (category) information. The actual probability describes the states of a
network annealed with neither input nor output variables clamped.
We now make use of the distinction between configurations of “visible” units (the
input and output, denoted α), and the hidden states, denoted β, shown in Fig. 7.1.
For instance, whereas a and b (c.f., Eq. 4) refered to different configurations of the
full system, α and β sill specify visible and hidden configurations.
The probability of a visible configuration is the sum over all possible hidden con-
figurations:
P (α) =
∑
β
P (α, β)
=
∑
β
e−Eαβ/T
Z
(6)
14 CHAPTER 7. STOCHASTIC METHODS
input
output
h
id
d
en
wij
-1 +1 -1 -1
-1 +1 -1 +1+1 -1
+1
-1
-1
+1
-1
+1
-1i
j
β
visible
visible d
αi
αo
1 2
c1
Figure 7.7: When a network such as shown in Fig. 7.1 is used for learning, it is
important to distinguish between two types of visible units — the d input units and
c output units, which receive external feature and category information — as well as
the remaining, hidden units. The state of the full network is indexed by an integer
γ, and since here there are 17 binary nodes, γ is bounded 0 ≤ γ < 217. The state
of the visible nodes is described by α; moreover, αi describes the input and αo the
output (the superscripts are not indexes, but merely refer to the input and output,
respectively). The state of the hidden nodes is indexed by β.
where Eαβ is the system energy in the configuration defined by the visible and hidden
parts, and Z is again the full partition function. Equation 6 is based on Eq. 3 and
states simply that to find the probability of a given visible state α, we sum over all
possible hidden states. A natural measure of the difference between the actual and
the desired probability distributions is the relative entropy, Kullback-Leibler distance
or Kullback-Leibler divergence,
DKL(Q(α), P (α)) =
∑
α
Q(α)log
Q(α)
P (α)
. (7)
Naturally, DKL is non-negative and can be zero if and only if P (α) = Q(α) for all α
(Appendix ??). Note that Eq. 6 depends solely upon the visible units, not the hidden
units.
Learning is based on gradient descent in the relative entropy. A set of training
patterns defines Q(α), and we seek weights so that at some temperature T the actual
distribution P (α) matches Q(α) as closely as possible. Thus we take an untrained
network and update each weight according to:
∆wij = −η
∂DKL
∂wij
= η
∑
α
Q(α)
P (α)
∂P (α)
∂wij
, (8)
where η is a learning rate. While P depends on the weights, Q does not, and thus we
used ∂Q(α)/∂wij = 0. We take the derivative in Eq. 6 and find:
7.3. BOLTZMANN LEARNING 15
∂P (α)
∂wij
=
∑
β
e−Eαβsi(αβ)sj(αβ)/T
TZ
−
(∑
β
e−Eαβ/T
)∑
λµ
eEλµsi(λµ)sj(λµ)
TZ2
=
1
T
∑
β
si(αβ)sj(αβ)P (α, β)− P (α)E [sisj ]
. (9)
Here si(αβ) is the state of node i in the full configuration specified by α and β. Of
course, if node i is a visible one, then only the value of α is relevant; if the node is
a hidden one, then only the value of β is relevant. (Our notation unifies these two
cases.) The expectation value E [sisj ] is taken at temperature T . We gather terms
and find from Eqs. 8 & 9
∆wij =
η
T
∑
α
Q(α)
P (α)
∑
β
si(αβ)sj(αβ)P (α, β)−
∑
α
Q(α)E [sisj ]
=
η
T
∑
αβ
Q(α)P (β|α)si(αβ)sj(αβ)− E [sisj ]
=
η
T
EQ[sisj ]α clamped︸ ︷︷ ︸
learning
−E [sisj ]free︸ ︷︷ ︸
unlearning
(10)
where P (α, β) = P (β|α)P (α). We have defined
EQ[sisj ]α clamped =
∑
αβ
Q(α)P (β|α)si(αβ)sj(αβ) (11)
to be the correlation of the variables si and sj when the visible units are held fixed
— clamped — in visible configuration α, averaged according to the probabilities of
the training patterns, Q(α).
The first term on the right of Eq. 10 is informally referred to as the learning
component or teacher component (as the visible units are held to values given by learning
componentthe teacher), and the second term the unlearning or student component (where the
unlearning
component
variables are free to vary). If EQ[sisj ]α clamped = E [sisj ]free, then ∆wij = 0 and we
have achieved the desired weights. The unlearning component reduces spurious cor-
relations between units — spurious in that they are not due to the training patterns.
A learning algorithm based on the above derivation would present each pattern in
the full training set several times and adjust the weights by Eq. 10, just as we saw in
numerous other training methods such as backpropagation (Chap. ??).
Stochastic Learning of input-output associations
Now consider the problem of learning mappings from input to output — our real in-
terest in pattern recognition. Here we want the network to learn associations between
the (visible) states on the input units, denoted αi, and states on the output units,
denoted αo, as shown in Fig. 7.1. Formally, we want P (αo|αi) to match Q(αo|αi)
16 CHAPTER 7. STOCHASTIC METHODS
as closely as possible. The appropriate cost function here is the Kullback-Leibler
divergence weighted by the probability of each input pattern:
D̄KL(Q(α
o|αi), P (αo|αi)) =
∑
αi
P (αi)
∑
αo
Q(αo|αi)logQ(α
o|αi)
P (αo|αi)
. (12)
Just as in Eq. 8, learning involves changing weights to reduce this weighted distance,
i.e.,
∆wij = −η
∂D̄KL
∂wij
. (13)
The derivation of the full learning rule follows closely that leading to Eq. 11; the only
difference is that the input units are clamped in both the learning and unlearning
components (Problem 11). The result is that the weight update is
∆wij =
η
T
EQ[sisj ]αiαo clamped︸ ︷︷ ︸
learning
−E [sisj ]αi clamped︸ ︷︷ ︸
unlearning
. (14)
In Sect. 7.3.3 we shall present pseudocode for implementing the preferred, deter-
ministic version of Boltzmann learning, but first we can gain intuition into the general
method by considering the learning of a single pattern according to Eq. 14. Figure 7.8
shows a seven-unit network being trained with the input pattern s1 = +1, s2 = +1
and the output pattern s6 = −1, s7 = +1. In a typical 1-of-c representation, this
desired output signal would represent category ω2. Since during both training and
classification, the input units s1 and s2 are clamped at the value +1, we have shown
only the associated 25 = 32 configurations at the right. The energy before learn-
ing (Eq. 1), corresponding to randomly chosen weights, is shown in black. After the
weights are trained by Eq. 14 using the pattern shown, the energy is changed (shown
in red). Note particularly that all states having the desired output pattern have their
energies lowered through training, just as we need. Thus when these input states are
clamped and the remaining networked annealed, the desired output is more likely to
be found.
Equation 14 appears a bit different from those we have encountered in pattern
recognition, and it is worthwhile explaining it carefully. Figure 7.9 illustrates in
greater detail the learning of the single training pattern in Fig. 7.8. Because s1 and
s2 are clamped throughout, EQ[s1s2]αiαoclamped = 1 = E [s1s2]αiclamped, and thus the
weight w12 is not changed, as indeed given by Eq. 14. Consider a more general case,
involving s1 and s7. During the learning phase both units are clamped at +1 and
thus the correlation is EQ[s1s7] = +1. During the unlearning phase, the output s7 is
free to vary and the correlation is lower; in fact it happens to be negative. Thus, the
learning rule seeks to increase the magnitude of w17 so that the input s1 = +1 leads
to s7 = +1, as can be seen in the matrix on the right. Because hidden units are only
weakly correlated (or anticorrelated), the weights linking hidden units are changed
only slightly.
In learning a training set of many patterns, each pattern is presented in turn, and
the weights updated as just described. Learning ends when the actual output matches
the desired output for all patterns (cf. Sect. 7.3.4).
7.3. BOLTZMANN LEARNING 17
+
-
-
-
-
+
+
+
-
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-
+
+
+
+
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.
.
.
+
+
+
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+
+
+
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+
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-
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+
+
E
training pattern
after training
s
1
s
2
s
5
s
7
s
6
s
3 s
4
s7
s6
s5
s4
s3
s2
s1
before training
these
configurations
become more
probable after training
+ +
- +
input
output
γ
ω1 ω2
Figure 7.8: The fully connected seven-unit network at the left is being trained via
the Boltzmann learning algorithm with the input pattern s1 = +1, s2 = +1, and the
output values s6 = −1 and s7 = +1, representing categories ω1 and ω2, respectively.
All 25 = 32 configurations with s1 = +1, s2 = +1 are shown at the right, along
with their energy (Eq. 1). The black curve shows the energy before training; the
red curve shows the energy after training. Note particularly that after training all
configurations that represent the full training pattern have been lowered in energy,
i.e., have become more probable. Consequently, patterns that do not represent the
training pattern become less probable after training. Thus, after training, if the input
pattern s1 = +1, s2 = +1 is presented and the remaining network annealed, there is
an increased chance of yielding s6 = −1, s7 = +1, as desired.
7.3.2 Missing features and category constraints
A key benefit of Boltzmann training (including its preferred implementation, described
in Sect. 7.3.3, below) is its ability to deal with missing features, both during training
and during classification. If a deficient binary pattern is used for training, input units
corresponding to missing features are allowed to vary — they are temporarily treated
as (unclamped) hidden units rather than clamped input units. As a result, during
annealing such units assume values most consistent with the rest of the input pattern
and the current state of the network (Problem 14). Likewise, when a deficient pattern
is to be classified, any units corresponding to missing input features are not clamped,
and are allowed to assume any value.
Some subsidiary knowledge or constraints can be incorporated into a Boltzmann
network during classification. Suppose in a five-category problem it is somehow known
that a test pattern is neither in category ω1 nor ω4. (Such constraints could come
from context or stages subsequent to the classifier itself.) During classification, then,
the output units corresponding to ω1 and ω4 are clamped at si = −1 during the
anneal, and the final category read as usual. Of course in this example the possible
categories are then limited to the unclamped output units, for ω2, ω3 and ω5. Such
constraint imposition may lead to an improved classification rate (Problem 15).
18 CHAPTER 7. STOCHASTIC METHODS
s1 s3 s5s2 s4 s6 s7
s1
s3
s5
s2
s4
s6
s7
s1 s3 s5s2 s4 s6 s7
s1
s3
s5
s2
s4
s6
s7
s1 s3 s5s2 s4 s6 s7
s1
s3
s5
s2
s4
s6
s7
+1+0.5
EQ[s1 s2]αi αo clamped E[s1 s2]αi clamped
∆w =
EQ[s1 s2]αi αo clamped - E[s1 s2]αi clamped
learning unlearning weight update
0-0.5-1
Figure 7.9: Boltzmann learning of a single pattern is illustrated for the seven-node
network of Fig. 7.8. The (symmetric) matrix on the left shows the correlation of units
for the learning component, where the input units are clamped to s1 = +1, s2 = +1,
and the outputs to s6 = −1, s7 = +1. The middle matrix shows the unlearning
component, where the inputs are clamped but outputs are free to vary. The difference
between those matrices is shown on the right, and is proportional to the weight update
(Eq. 14). Notice, for instance, that because the correlation between s1 and s2 is
large in both the learning and unlearning components (because those variables are
clamped), there is no associated weight change, i.e., ∆w12 = 0. However, strong
correlations between s1 and s7 in the learning but not in the unlearning component
implies that the weight w17 should be increased, as can be seen in the weight update
matrix.
Pattern completion
The problem of pattern completion is to estimate the full pattern given just a part
of that pattern; as such, it is related to the problem of classification with missing
features. Pattern completion is naturally addressed in Boltzmann networks. A fully
interconnected network, with or without hidden units, is trained with a set of repre-
sentative patterns; as before, the visible units correspond to the feature components.
When a deficient pattern is presented, a subset of the visible units are clamped to
the components of a partial pattern, and the network annealed. The estimate of the
unknown features appears on the remaining visible units, as illustrated in Fig. 7.10
(Computer exercise 3). Such pattern completion in Boltzmann networks can be more
accurate when known category information is imposed at the output units.
Boltzmann networks without hidden or category units are related to so-called
Hopfield networks or Hopfield auto-association networks (Problem 12). Such networksHopfield
network store patterns but not their category labels. The learning rule for such networks does
not require the full Boltzmann learning of Eq. 14. Instead, weights are set to be
proportional to the correlation of the feature vectors, averaged over the training set,
7.3. BOLTZMANN LEARNING 19
deficient
pattern
presented
learned patterns
pattern
completed
by network
s
1
s
2
s
3s
4
s
5
s
6s
7
- +
s
8
s
9
s
10
s
11
s
12
s
1
s
2
s
3
s
4
s
5
s
6
s
7
visible
hidden
Figure 7.10: A Boltzmann network can be used for pattern completion, i.e., filling in
unknown features of a deficient pattern. Here, a twelve-unit network with five hidden
units has been trained with the 10 numeral patterns of a seven-segment digital display.
The diagram at the lower left shows the correspondence between the display segments
and nodes of the network; a black segment is represented by a +1 and a light gray
segment as a −1. Consider the deficient pattern consisting of s2 = −1, s5 = +1. If
these units are clamped and the full network annealed, the remaining five visible units
will assume values most probable given the clamped ones, as shown at the right.
wij ∝ EQ[sisj ], (15)
with wii = 0; further, there is no need to consider temperature. Such learning is of
course much faster than true Boltzmann learning using annealing. If a network fully
trained by Eq. 15 is nevertheless annealed, as in full Boltzmann learning, there is no
guarantee that the equilibrium correlations in the learning and unlearning phases are
equal, i.e., that ∆wij = 0 (Problem 13).
The successes of such Hopfield networks in true pattern recognition have been
modest, partly because the basic Hopfield network does not have as natural an output
representation for categorization problems. Occassionally, though they can be used in
simple low-dimensional pattern completion or auto-association problems. One of their
primary drawbacks is their limited capacity, analogous to the fact that a two-layer
network cannot implement arbitrary decision boundaries as can a three-layer net. In
particular, it has been shown that the number of d-dimensional random patterns that
can be stored is roughly 0.14d — very limited indeed. In a Boltzmann with hidden
units such as we have discussed, however, the number of hidden units can be increased
in order to allow more patterns to be stored.
Because Boltzmann networks include loops and feedback connections, the internal
representations learned at the hidden units are often difficult to interpret. Occasion-
ally, though, the pattern of weights from the input units suggests feature groupings
that are important for the classification task.
20 CHAPTER 7. STOCHASTIC METHODS
7.3.3 Deterministic Boltzmann learning
The computational complexity of stochastic Boltzmann learning in a network with
hidden units is very high. Each pattern must be presented several times, and every
anneal requires each unit to be polled several times. Just as mean-field annealing is
usually preferable to stochastic annealing, so too a mean-field version of Boltzmann
learning is preferable to the stochastic version. The basic approach in deterministic
Boltzmann learning is to use Eq. 14 with mean-field annealing and analog values for
the si. Recall, at the end of deterministic simulated annealing, the values of si are
±1, as required by the problem.
Specifically, if we let D be the set of training patterns x containing feature and
category information, the algorithm is:
Algorithm 3 (Deterministic Boltzmann learning)
1 begin initialize D, η, T (k), wij i, j = 1, . . . , N
2 do Randomly select training pattern x
3 Randomize states si
4 Anneal network with input and output clamped
5 At final, low T, calculate [sisj ]αiαoclamped
6 Randomize states si
7 Anneal network with input clamped but output free
8 At final, low T, calculate [sisj ]αiclamped
9 wij ← wij + η/T
[
[sisj ]αiαoclamped − [sisj ]αiclamped
]
10 until k = kmax or convergence criterion met
11 return wij
12 end
Using mean-field theory, it is possible to efficiently calculate approximations of the
mean of correlations entering the gradient. The analog state si of each unit replaces
its average value E [si] and could in theory be calculated by iteratively solving a set
of nonlinear equations. The mean of correlations is then calculated by making the
approximation E [sisj ] ≈ E [si]E [sj ] ≈ sisj , as shown in lines 5 & 8.
7.3.4 Initialization and setting parameters
As with virtually every classifier, there are several interrelated parameters that must
be set in a Boltzmann network. The first are the network topology and number of
hidden units. The number of visible units (input and output) is determined by the
dimensions of the binary feature vectors and number of categories. In the absence
of detailed information about the problem, we assume the network is fully intercon-
nected, and thus merely the number of hidden units must be set. A popular alternate
topology is obtained by eliminating interconnections among input units, as well as
among output units. (Such a network is faster to train but will be somewhat less
effective at pattern completion or classifying deficient patterns.) Of course, generally
speaking the harder the classification problem the more hidden units will be needed.
The question is then, how many hidden units should be used?
Suppose the training set D has n distinct patterns of input-output pairs. An
upper bound on the minimum number of hidden units is n — one for each pattern
— where for each pattern there is a corresponding unique hidden unit having value
si = +1 while all others are −1. This internal representation can be insured in the
7.3. BOLTZMANN LEARNING 21
following way: for the particular hidden unit i, set wij to be positive for each input
unit j corresponding to a +1 feature in its associated pattern; further set wij to be
negative for input units corresponding to a −1 feature. For the remaining hidden
units, the sign of the corresponding weights should be inverted. Next, the connection
from hidden unit i to the output unit corresponding to the known category should be
positive, and negative to all other output units. The resulting internal representation
is closely related to that in the probabilistic neural network implementation of Parzen
windows (Chap. ??). Naturally, this representation is undesirable as the number of
weights grows exponentially with the number of patterns. Training becomes slow;
furthermore generalization tends to be poor.
Since the states of the hidden units are binary valued, and since it takes
log2n�
bits to specify n different items, there must be at least
log2n� hidden units if there
is to be a distinct hidden configuration for each of the n patterns. Thus a lower
bound on the number of hidden units is
log2n�, which is necessary for a distinct
hidden configuration for each pattern. Nevertheless, this bound need not be tight, as
there may be no set of weights insuring a unique representation (Problem 16). Aside
from these bounds, it is hard to make firm statements about the number of hidden
units needed — this number depends upon the inherent difficulty of the classification
problem. It is traditional, then, to start with a somewhat large net and use weight
decay. Much as we saw in backpropagation (Chap. ??), a Boltzmann network with
“too many” hidden units and weights can be improved by means of weight decay.
During training, a small increment ! is added to wij when si and sj are both positive
or both negative during learning phase, but subtracted in the unlearning phase. It is
traditional to decrease ! throughout training. Such a version of weight decay tends
to reduce the effects on the weights due to spurious random correlations in units and
to eliminate unneeded weights, thereby improving generalization.
One of the benefits of Boltzmann networks over backpropagation networks is that
“too many” hidden units in a backpropagation network tend to degrade performance
more than “too many” in a Boltzmann network. This is because during learning, there
is stochastic averaging over states in a Boltzmann network which tends to smooth
decision boundaries; backpropagation networks have no such equivalent averaging.
Of course, this averaging comes at a higher computational burden for Boltzmann
networks.
The next matter to consider is weight initialization. Initializing all weights to
zero is acceptable, but leads to unnecessarily slow learning. In the absence of infor-
mation otherwise, we can expect that roughly half the weights will be positive and
half negative. In a network with fully interconnected hidden units there is nothing
to differentiate the individual hidden units; thus we can arbitrarily initialize roughly
half of the weights to have positive values and the rest negative. Learning speed is
increased if weights are initialized with random values within a proper range. Assume
a fully interconnected network having N units (and thus N − 1 ≈ N connections to
each unit). Assume further that at any instant each unit has an equal chance of being
in state si = +1 or si = −1. We seek initial weights that will make the net force
on each unit a random variable with variance 1.0, roughly the useful range shown in
Fig. 7.5. This implies weights should be initialized randomly throughout the range
−
√
3/N < wij < +1
√
3/N (Problem 17).
As mentioned, annealing schedules of the form T (k+1) = cT (k) for 0 < c < 1 are
generally used, with 0.8 < c < 0.99.
If a very large number of iterations — several thousand — are needed, even
c = 0.99 may be too small. In that case we can write c = e−1/k0 , and thus
22 CHAPTER 7. STOCHASTIC METHODS
T (k) = T (1)e−k/k0 , and k0 can be interpreted as a decay constant. The initial tem-
perature T (1) should be set high enough that virtually all candidate state transitions
are accepted. While this condition can be insured by choosing T (1) extremely high,
in order to reduce training time we seek the lowest adequate value of T (1). A lower
bound on the acceptable initial temperature depends upon the problem, but can be
set empirically by monitoring state transitions in short simulations at candidate tem-
peratures. Let m1 be the number of energy-decreasing transitions (these are always
accepted), and m2 the number of energy-increasing queries according to the anneal-
ing algorithm; let E+[∆E] denote the average increase in energy over such transitions.
Then, from Eq. 4 we find that the acceptance ratio is
R =
number of accepted transitions
number of proposed transitions
≈ m1 +m2 · exp[−E+[∆E]/T (1)]
m1 +m2
. (16)
Rearranging terms we see that the initial temperature obeys
T (1) =
E+[∆E]
ln[m2]− ln[m2R−m1(1−R)]
. (17)
For any initial temperature set by the designer, the acceptance ratio may or may
not be nearly the desired 1.0; nevertheless Eq. 17 will be obeyed. The appropriate
value for T (1) is found through a simple iterative procedure. First, set T (1) to zero
and perform a sequence of m0 trials (pollings of units); count empirically the number
of energetically favorable (m1) and energetically unfavorable (m2) transitions. In
general, m1 + m2 < m0 because many candidate energy increasing transitions are
rejected, according to Eq. 4. Next, use Eq. 17 to calculate a new, improved value of
T (1) from the observed m1 and m2. Perform another sequence of m0 trials, observe
new values for m1 and m2, recalculate T (1), and so on. Repeat this procedure until
m1 +m2 ≈ m0. The associated T (1) gives an acceptance ratio R ≈ 1, and is thus to
be used. In practice this method quickly yields a good starting temperature.
The next important parameter is the learning rate η in Eq. 14. Recall that the
learning is based on gradient descent in the weighted Kullback-Leibler divergence
between the actual and the desired distributions on the visible units. In Chap. ??
we derived bounds on the learning rate for multilayer neural networks by calculating
the curvature of the error, and finding the maximum value of the learning rate that
insured stability. This curvature was based on a Hessian matrix, the matrix of second-
order derivatives of the error with respect to the weights. In the case of an N -unit,
fully connected Boltzmann network, whose N(N − 1)/2 weights are described by a
vector w, this curvature is proportional to wtHw, where
H =
∂2D̄KL
∂w2
(18)
is the appropriate Hessian matrix and the Kullback-Liebler divergence is given by
Eq. 12. Given weak assumptions about the classification problem we can estimate this
Hessian matrix; the stability requirement is then simply η ≤ T 2/N2 (Problem 18).
Note that at large temperature T , a large learning rate is acceptable since the effective
error surface is smoothed by high randomness.
While not technically parameter setting, one heuristic that provides modest com-
putational speedup is to propose changing the states of several nodes simultaneously
early in an anneal. The change in energy and acceptance probability are calculated
7.4. *BOLTZMANN NETWORKS AND GRAPHICAL MODELS 23
as before. At the end of annealing, however, polling should be of single units in order
to accurately find the optimum configuration.
A method which occasionally improves the final solution is to update and store the
current best configuration during an anneal. If the basic annealing converges to a local
minimum that is worse than this stored configuration, this current optimal should be
used. This is a variant of the pocket algorithm which finds broad use in methods that pocket
algorithmdo not converge monotonically or can get caught in local minima (Chap ??).
There are two stopping criteria associated with Boltzmann learning. The first
determines when to stop a single anneal (associated with either the learning or the
unlearning components). Here, the final temperature should be so low that no ener-
getically unfavorable transitions are accepted. Such information is readily apparent
in the graph of the energy versus iteration number, such as shown at the right of
Fig. 7.3. All N variables should be polled individually at the end of the anneal, to
insure that the final configuration is indeed a local (though perhaps not global) energy
minimum.
The second stopping criterion controls the number of times each training pattern
is presented to the network. Of course the proper criterion depends upon the inherent
difficulty of the classification problem. In general, overtraining is less of a concern
in Boltzmann networks than it is in multilayer neural networks trained via gradient
descent. This is because the averaging over states in Boltzmann networks tends to
smooth decision boundaries while overtraining in multilayer neural networks tunes
the decision boundaries to the particular training set. A reasonable stopping criterion
for Boltzmann networks is to monitor the error on a validation set (Chap. ??), and
stop learning when this error no longer changes significantly.
7.4 *Boltzmann networks and graphical models
While we have considered fully interconnected Boltzmann networks, the learning al-
gorithm (Algorithm 3) applies equally well to networks with arbitrary connection
topologies. Furthermore, it is easy to modify Boltzmann learning in order to impose
constraints such as weight sharing. As a consequence, several popular recognition
architectures — so-called graphical models such as Bayesian belief networks and Hid-
den Markov Models — have counterparts in structured Boltzmann networks, and this
leads to new methods for training them.
Recall from Chap. ?? that Hidden Markov Models consist of several discrete hidden
and visible states; at each discrete time step t, the system is in a single hidden state
and emits a single visible state, denoted ω(t) and v(t), respectively. The transition
probabilities between hidden states at successive time steps are
aij = P (ωj(t+ 1)|ωi(t)) (19)
and between hidden and visible states at a given time are
bjk = P (vk(t)|ωj(t)). (20)
The Forward-Backward or Baum-Welch algorithm (Chap. ??, Algorithm ??) is tra-
ditionally used for learning these parameters from a pattern of Tf visible states∗
VTf = {v(1), v(2), . . . , v(Tf )}.
∗ Here we use Tf to count the number of discrete time steps in order to avoid confusion with the
temperature T in Boltzmann simulations.
24 CHAPTER 7. STOCHASTIC METHODS
..
...
..
...
..
...
..
...
..
...
..
...
h
id
d
en
u
n
it
s
Aij
Bjk
Aij
Bjk
Aij
Bjk Bjk
Aij
Bjk Bjk
....
t = 1 2 3 4 T
f
-1 T
f
visible units
Figure 7.11: A Hidden Markov Model can be “unfolded” in time to show a trellis,
which can be represented as a Boltzmann chain, as shown. The discrete hidden
states are grouped into vertical sets, fully interconnected by weights Aij (related to
the HMM transition probabilities aij). The discrete visible states are grouped into
horizontal sets, and are fully interconnected with the hidden states by weights Bjk
(related to transition probabilities bjk). Training the net with a single pattern, or list
of Tf visible states, consists of clamping the visible states and performing Boltzmann
learning throughout the full network, with the constraint that each of the time shifted
weights labeled by a particular Aij have the same numerical value.
Recall that a Hidden Markov model can be “unfolded” in time to yield a trellis
(Chap. ??, Fig. ??). A structured Boltzmann network with the same trellis topology
— a Boltzmann chain — can be used to implement the same classification as theBoltzmann
chain corresponding Hidden Markov Model (Fig. 7.11). Although it is often simpler to
work in a representation where discrete states have multiple values, we temporarily
work in a representation where the binary nodes take value si = 0 or +1, rather than
±1 as in previous discussions. In this representation, a special case of the general
energy (Eq. 1) includes terms for a particular sequence of visible, VTf , and hidden
states ωTf = {ω(1), ω(2), . . . , ω(Tf )} and can be written as
EωV = E[ω
Tf ,VTf ] = −
Tf−1∑
t=1
Aij −
Tf∑
t=1
Bjk (21)
where the particular values of Aij and Bjk terms depend implicitly upon the sequence.
The choice of binary state representation implies that only the weights linking nodes
that both have si = +1 appear in the energy. Each “legal” configuration — consisting
of a single visible unit and a single hidden unit at each time — implies a set of Aij
and Bjk (Problem 20). The partition function is the sum over all legal states,
Z =
∑
ωV
e−EωV/T , (22)
which insures normalization. The correspondence between the Boltzmann chain at
temperature T and the unfolded Hidden Markov model (trellis) implies
7.5. *EVOLUTIONARY METHODS 25
Aij = T ln aij and Bjk = T ln bjk. (23)
(As in our discussion of Hidden Markov Models, we assume the initial hidden state is
known and thus there is no need to consider the correspondence of prior probabilities in
the two approaches.) While the 0−1 binary representation of states in the structured
network clarifies the relationship to Hidden Markov Models through Eq. 21, the more
familiar representation si = ±1 works as well. Weights in the structured Boltzmann
network are trained according to the method of Sect. 7.3, though the relation to
transition probabilities in a Hidden Markov Model is no longer simple (Problem 21).
Other graphical models
In addition to Hidden Markov Models, a number of graphical models have analogs
in structured Boltzmann networks. One of the most general includes Bayesian belief
nets, directed acyclic graphs in which each node can be in one of a number of discrete
states, and nodes are interconnected with conditional probabilities (Chap. ??). As
in the case of Hidden Markov Models, the correspondence with Boltzmann networks
is clearest if the discrete states in the belief net are binary states; nevertheless in
practice multistate representations more naturally enforce the constraints and are
generally preferred (Computer exercise ??).
A particularly intriguing recognition problem arises when a temporal signal has
two inherent time scales, for instance the rapid daily behavior in a financial market
superimposed on slow seasonal variations. A standard Hidden Markov Model typically
has a single inherent time scale and hence is poorly suited to such problems. We might
seek to use two interconnected HMMs, possibly with different numbers of hidden
states. Alas, the Forward-Backward algorithm generally does not converge when
applied to a model having closed loops, as when two Hidden Markov Models have
cross connections.
Here the correspondence with Boltzmann networks is particularly helpful. We can
link two Boltzmann chains with cross connections, as shown in Fig. 7.12, to form
a Boltzmann zipper. The particular benefit of such an architecture is that it can Boltzmann
zipperlearn both short-time structure (through the “fast” component chain) as well as long-
time structure (through the “slow” chain). The cross connections, labeled by weight
matrix E in the figure, learn correlations between the “fast” and “slow” internal
representations. Unlike the case in Eq. 23, the E weights are not simply related to
transition probabilities, however (Problem ??).
Boltzmann zippers can address problems such as acoustic speech recognition,
where the fast chain learns the rapid transitions and structure of individual phonemes
while the slow component chain learns larger structure associated with prosody and
stress throughout a word or a full phrase. Related applications include speechreading
(lipreading), where the fast chain learns the acoustic transitions and the slow chain
the much slower transitions associated with the (visible) image of the talker’s lips,
jaw and tongue and body gestures, where fast hand motions are coupled to slower
large-scale motions of the arms and torso.
7.5 *Evolutionary methods
Inspired by the process of biological evolution, evolutionary methods of classifier de-
sign employ stochastic search for an optimal classifier. These admit a natural imple-
26 CHAPTER 7. STOCHASTIC METHODS
..
...
..
...
..
...
..
...
..
...
..
...
"f
a
st
"
h
id
d
en
u
n
it
s
A
B
A
B
A
B B
A
B B
"fast" visible units
..
.
..
..
.
....
..
.
..
....
"slow" visible units
"s
lo
w
"
h
id
d
en
u
n
it
s
C
D D D
E E E
....
Figure 7.12: A Boltzmann zipper consists of two Boltzmann chains (cf. Fig. 7.11),
whose hidden units are interconnected. The component chains differ in the rate
at which visible features are sampled, and thus they capture structure at different
temporal scales. Correlations are learned by the weights linking the hidden units,
here labeled E. It is somewhat more difficult to train linked Hidden Markov Models
to learn structure at different time scales.
mentation on massively parallel computers. In broad overview, such methods proceed
as follows. First, we create several classifiers — a population — each varying somewhatpopulation
from the other. Next, we judge or score each classifier on a representative version of
score the classification task, such as accuracy on a set of labeled examples. In keeping with
the analogy with biological evolution, the resulting (scalar) score is sometimes called
the fitness. Then we rank these classifiers according to their score and retain the bestfitness
classifiers, some portion of the total population. Again, in keeping with biological
terminology, this is called survival of the fittest.survival
of the
fittest
We now stochastically alter the classifiers to produce the next generation — the
children or offspring. Some offspring classifiers will have higher scores than their
offspring
parents in the previous generation, some will have lower scores. The overall process
parent
is then repeated for subsequent generation: the classifiers are scored, the best ones
retained, randomly altered to give yet another generation, and so on. In part because
of the ranking, each generation has, on average, a slightly higher score than the
previous one. The process is halted when the single best classifier in a generation has
a score that exceeds a desired criterion value.
The method employs stochastic variations, and these in turn depend upon the
fundamental representation of each classifier. There are two primary representations
we shall consider: a string of binary bits (in basic genetic algorithms), and snippets
of computer code (in genetic programming). In both cases, a key property is that
7.5. *EVOLUTIONARY METHODS 27
occasionally very large changes in classifier are introduced. The presence of such
large changes and random variations implies that evolutionary methods can find good
classifiers even in extremely complex discontinuous spaces or “fitness landscapes” that
are hard to address by techniques such as gradient descent.
7.5.1 Genetic Algorithms
In basic genetic algorithms, the fundamental representation of each classifier is a bi-
nary string, called a chromosome. The mapping from the chromosome to the features chromosome
and other aspects of the classifier depends upon the problem domain, and the designer
has great latitude in specifying this mapping. In pattern classification, the score is
usually chosen to be some monotonic function of the accuracy on a data set, possibly
with penalty term to avoid overfitting. We use a desired fitness, θ, as the stopping
criterion. Before we discuss these points in more depth, we first consider more specif-
ically the structure of the basic genetic algorithm, and then turn to the key notion of
genetic operators, used in the algorithm.
Algorithm 4 (Basic Genetic algorithm)
1 begin initialize θ, Pco, Pmut, L N -bit chromosomes
2 do Determine fitness of each chromosome, fi, i = 1, . . . , L
3 Rank the chromosomes
4 do Select two chromosomes with highest score
5 if Rand[0, 1) < Pco then crossover the pair at a randomly chosen bit
6 else change each bit with probability Pmut
7 Remove the parent chromosomes
8 until N offspring have been created
9 until Any chromosome’s score f exceeds θ
10 return Highest fitness chromosome (best classifier)
11 end
Figure 7.13 shows schematically the evolution of a population of classifiers given by
Algorithm 4.
Genetic operators
There are three primary genetic operators that govern reproduction, i.e., producing
offspring in the next generation described in lines 5 & 6 of Algorithm 4. The last two
of these introduce variation into the chromosomes (Fig. 7.14):
Replication: A chromosome is merely reproduced, unchanged.
Crossover: Crossover involves the mixing — “mating” — of two chromosomes. A mating
split point is chosen randomly along the length of either chromosome. The first
part of chromosome A is spliced to the last part of chromosome B, and vice
versa, thereby yielding two new chromosomes. The probability a given pair of
chromosomes will undergo crossover is given by Pco in Algorithm 4.
Mutation: Each bit in a single chromosome is given a small chance, Pmut, of being
changed from a 1 to a 0 or vice versa.
28 CHAPTER 7. STOCHASTIC METHODS
11010100101000
01001110111011
11100010110100
00001110010100
11001010101010
00101100100100
11110100101011
10001001010001
11010110101000
11110101101001
generation k
15
11
29
36
54
73
22
92
84
27
11010100101000 15
01001110111011 11
11100010110100 29
00001110010100 36
11001010101010 54
00101100100100 73
11110100101011 22
10001001010001 92
11010110101000 84
11110101101001 27
11010100101000
01001110111011
11100010110100
00001110010100
11001010101010
00101100100100
11110100101011
10001001010001
11010110101000
11110101101001
score rank
survival of
the fittest and
reproduction
generation k+1
N
i
f
f >
θ
t
h
e
n
e
n
dfi
parents offspring
Figure 7.13: A basic genetic algorithm is a stochastic iterative search method. Each
of the L classifiers in the population in generation k is represented by a string of
bits of length N , called a chromosome (on the left). Each classifier is judged or
scored according its performance on a classification task, giving L scalar values fi.
The chromosomes are then ranked according to these scores. The chromosomes are
considered in descending order of score, and operated upon by the genetic operators
of replication, crossover and mutation to form the next generation of chromosomes —
the offspring. The cycle repeats until a classifier exceeds the criterion score θ.
Other genetic operators may be employed, for instance inversion — where the chromo-
some is reversed front to back. This operator is used only rarely since inverting a
chromosome with a high score nearly always leads to one with very low score. Below
we shall briefly consider another operator, insertions.
Representation
When designing a classifier by means of genetic algorithms we must specify the map-
ping from a chromosome to properties of the classifier itself. Such mapping will depend
upon the form of classifier and problem domain, of course. One of the earliest and
simplest approaches is to let the bits specify features (such as pixels in a character
recognition problem) in a two-layer Perceptron with fixed weights (Chap. ??). The
primary benefit of this particular mapping is that different segments of the chromo-
some, which generally remain undisturbed under the crossover operator, may evolve
to recognize different portions of the input space such as the descender (lower) or the
ascender (upper) portions of typed characters. As a result, occasionally the crossover
operation will append a good segment for the ascender region in one chromosome
to a good segment for the descender region in another, thereby yielding an excellent
overall classifier.
Another mapping is to let different segments of the chromosome represent the
weights in a multilayer neural net with a fixed topology. Likewise, a chromosome
could represent a network topology itself, the presence of an individual bit implying
two particular neurons are interconnected. One of the most natural representations
is for the bits to specify properties of a decision tree classifier (Chap. ??), as shown
in Fig. 7.15.
7.5. *EVOLUTIONARY METHODS 29
11010100101001010101111010100011111010010
11010100101001010101111010100011111010010
11010100101001010101111010100011111010010
11011100100001110101111110110011101010010
11010100101001010101111010100011111010010
00101100001010001010100001010110101001110
11010100101001010101111011010110101001110
00101100001010001010100000100011111010010
replication mutationcrossover
parents
(generation k)
offspring
(generation k+1)
A
B
Figure 7.14: Three basic genetic operations are used to transform a population of
chromosomes at one generation to form a new generation. In replication, the chromo-
some is unchanged. Crossover involves the mixing or “mating” of two chromosomes
to yield two new chromosomes. A position along the chromosomes is chosen randomly
(red vertical line); then the first part of chromosome A is linked with the last part of
chromosome B, and vice versa. In mutation, each bit is given a small chance of being
changed from a 1 to a 0 or vice versa.
Scoring
For a c-category classification problem, it is generally most convenient to evolve c
dichotomizers, each to distinguish a different ωi from all other ωj for j �= i. During
classification, the test pattern is presented to each of the c dichotomizers and assigned
the label accordingly. The goal of classifier design is accuracy on future patterns, or if
decisions have associated costs, then low expected cost. Such goals should be reflected
in the method of scoring and selection in a genetic algorithm. Given sample patterns
representative version of the target classification task, it is natural to base the score on
the classification accuracy measured on the data set. As we have seen numerous times,
there is a danger that the classifier becomes “tuned” to the properties of the particular
data set, however. (We can informally broaden our usage of the term “overfitting”
from generic learning to apply to this search-based case as well.) One method for
avoiding such overfitting is penalizing classifier complexity, and thus the score should
have a term that penalizes overly large networks. Another method is to adjusting
the stopping criterion. Since the appropriate measure of classifier complexity and
the stopping criterion depend strongly on the problem, it is hard to make specific
guidelines in setting these parameters. Nevertheless, designers should be prepared to
explore these parameters in any practical application.
Selection
The process of selection specifies which chromosomes from one generation will be
sources for chromosomes in the next generation. Up to here, we have assumed that
the chromosomes would be ranked and selected in order of decreasing fitness until the
next generation is complete. This has the benefit of generally pushing the population
toward higher and higher scores. Nevertheless, the average improvement from one
generation to the next depends upon the variance in the scores at a given generation,
30 CHAPTER 7. STOCHASTIC METHODS
110101001
110101001001011010111001010110110001101100010001011001001010110
xi < θ xi < θ xi < θ xi < θ
xi < θ xi < θ
xi < θ
00 → x1
01 → x2
10 → x3
11 → x4
fea
tu
re
sig
n
threshold θ
ω1 ω2 ω1 ω2 ω1 ω2ω1 ω2
chromosome
Figure 7.15: One natural mapping is from a binary chromosome to a binary tree
classifier, illustrated here for a four-feature, monothetic tree dichotomizer. In this
example, each of the nodes computes a query of the form ±xi < θ? and is governed
by nine bits in the chromosome. The first bit specifies a sign, the next two bits
specify the feature queried. The remaining six bits are a binary representation of the
threshold θ. For instance, the left-most node encodes the rule +x3 < 41? (In practice,
larger trees would be used for problems with four features.)
and because this standard fitness-based selection need not give high variance, other
selection methods may prove superior.
The principle alternative selection scheme is fitness-proportional selection, or fitness-fitness-
proportional
selection
proportional reproduction, in which the probability that each chromosome is selected
is proportional to its fitness. While high-fitness chromosomes are preferentially se-
lected, occasionally low-fitness chromosomes are selected, and this may preserve di-
versity and increase variance of the population.
A minor modification of this method is to make the probability of selection propor-
tional to some monotonically increasing function of the fitness. If the function instead
has a positive second derivative, the probability that high-fitness chromosomes is en-
hanced. One version of this heuristic is inspired by the Boltzmann factor of Eq. 2;
the probability that chromosome i with fitness fi will be selected is
7.5. *EVOLUTIONARY METHODS 31
P (i) =
efi/T
E [efi/T ]
, (24)
where the expectation is over the current generation and T is a control parameter
loosely referred to as a temperature. Early in the evolution the temperature is set
high, giving all chromosomes roughly equal probability of being selected. Late in the
evolution the temperature is set lower so as to find the chromosomes in the region of
the optimal classifier. We can express such search by analogy to biology: early in the
search the population remains diverse and explores the fitness landscape in search of
promising areas; later the population exploits the specific fitness opportunities in a
small region of the space of possible classifiers.
7.5.2 Further heuristics
There are many additional heuristics that can occasionally be of use. One concerns
the adaptation of the crossover and mutation rates, Pco and Pmut. If these rates are
too low, the average improvement from one generation to the next will be small, and
the search unacceptably long. Conversely, if these rates are too high, the evolution
is undirected and similar to a highly inefficient random search. We can monitor the
average improvement in fitness of each generation and the mutation and crossover
rates as long as such improvement is rapid. In practice, this is done by encoding the
rates in the chromosomes themselves and allowing the genetic algorithm to select the
proper values.
Another heuristic is to use a ternary, or n-ary chromosomes rather than the tradi-
tional binary ones. These representations provide little or no benefit at the algorith-
mic level, but may make the mapping to the classifier itself more natural and easier
to compute. For instance, a ternary chromosome might be most appropriate if the
classifier is a decision tree with three-way splits.
Occasionally the mapping to the classifier will work for chromosomes of differ-
ent length. For example, if the bits in the chromosome specify weights in a neural
network, then longer chromosomes would describe networks with a larger number of
hidden units. In such a case we allow the insertion operator, which with a small insertion
probability inserts bits into the chromosome at a randomly chosen position. This
so-called “messy” genetic algorithm method has a more appropriate counterpart in
genetic programming, as we shall see in Sect. 7.6.
7.5.3 Why do they work?
Because there are many heuristics to choose as well as parameters to set, it is hard to
make firm theoretical statements about building classifiers by means of evolutionary
methods. The performance and search time depend upon the number of bits, the size
of a population, the mutation and crossover rates, choice of features and mapping
from chromosomes to the classifier itself, the inherent difficulty of the problem and
possibly parameters associated with other heuristics.
A genetic algorithm restricted to mere replication and mutation is, at base, a
version of stochastic random search. The incorporation of the crossover operator,
which mates two chromosomes, provides a qualitatively different search, one that
has no counterpart in stochastic grammars (Chap. ??). Crossover works by finding,
rewarding and recombining “good” segments of chromosomes, and the more faithfully
the segments of the chromosomes represent such functional building blocks, the better
32 CHAPTER 7. STOCHASTIC METHODS
we can expect genetic algorithms to perform. The only way to insure this is with prior
knowledge of the problem domain and the desired form of classifier.
7.6 *Genetic Programming
Genetic programming shares the same algorithmic structure of basic genetic algo-
rithms, but differs in the representation of each classifier. Instead of chromosomes
consisting of strings of bits, genetic programming uses snippets of computer programs
made up of mathematical operators and variables. As a result, the genetic operators
are somewhat different; moreover a new operator plays a significant role in genetic
programming.
The four principal operators in genetic programming are (Fig. 7.16):
Replication: A snippet is merely reproduced, unchanged.
Crossover: Crossover involves the mixing — “mating” — of two snippets. A splitmating
point is chosen from allowable locations in snippet A as well as from snippet B.
The first part of snippet A is spliced to the back part of chromosome B, and
vice versa, thereby yielding two new snippets.
Mutation: Each bit in a single snippet is given a small chance of being changed to
a different value. Such a change must be compatible with the syntax of the
total snippet. For instance, a number can be replaced by another number; a
mathematical operator that takes a single argument can be replaced by another
such operator, and so forth.
Insertion: Insertion consists in replacing a single element in the snippet with anotherinsertion
(short) snippet randomly chosen from a set.
In the c-category problem, it is simplest to form c dichotomizers just as in genetic
algorithms. If the output of the classifier is positive, the test pattern belongs to
category ωi, if negative, then it is NOT in ωi.
Representation
A program must be expressed in some language, and the choice affects the complexity
of the procedure. Syntactically rich languages such as C or C++ are complex and
somwhat difficult to work with. Here the syntactic simplicity of a language such asLisp
is advantageous. Many Lisp expressions can be written in the form (
parenthesized expression. For example, (+ X 2) and (* 3 (+ Y 5)) are valid Lisp
expressions for the arithmetic expressions x + 2 and 3(y + 5), respectively. These
expressions are easily represented by a binary tree, with the operator being specified
at the node and the operands appearing as the children (Fig. 7.17).
Whatever language is used, genetic programming operators used for mutation
should replace variables and constants with variables and constants, and operators
with functionally compatible operators. They should aslo be required to produce
syntactically valid results. Nevertheless, occassionally an ungrammatical code snippet
may be produced. For that reason, it is traditional to employ a wrapper — a routinewrapper
that decides whether the classifier is meaningful, and eliminates them if not.
7.6. SUMMARY 33
(OR (AND (NOT X0)(NOT X1))(AND X0 X1))
(OR (AND (NOT X0)(NOT X1))(AND X0 X1))
(OR (AND (NOT X0)(NOT X1))(AND X0 X1))
(OR (OR (NOT X1)(NOT X1))(AND X2 X1))
(OR (AND (NOT X0)(NOT X1))(AND X0 X1))
(OR (AND (X2)(NOT X0))(AND X2 X0))
(OR (AND (NOT X0)(NOT X1))(AND X2 X0))
(OR (AND (X2)(NOT X0))(AND X0 X1))
replication mutationcrossover
(OR (AND (NOT X0)(NOT X1))(AND X0 X1))
(OR (AND (NOT X0)(NOT X1))(AND (NOT X2) X1))
insertion
parents
(generation k)
offspring
(generation k+1)
A
B
(NOT X2)
Figure 7.16: Four basic genetic operations are used to transform a population of
snippets of code at one generation to form a new generation. In replication, the
snippet is unchanged. Crossover involves the mixing or “mating” of two snippets to
yield two new snippets. A position along the snippet A is randomly chosen from the
allowable locations (red vertical line); likewise one is chosen for snippet B. Then the
front portion of A is spliced to the back portion of B and vice versa. In mutation,
each element is given a small chance of being changed. There are several different
types of elements, and replacements must be of the same type. For instance, only a
number can replace another number; only a numerical operator that takes a single
argument can replace a similar operator, and so on. In insertion, a randomly selected
element is replaced by a compatible snippet, keeping the entire snippet grammatically
well formed and meaningful.
It is nearly impossible to make sound theoretical statements about genetic pro-
gramming and even the rules of thumb learned from simulations in one domain, such
as control or function optimization are of little value in another domain, such as clas-
sification problems. Of course, the method works best in problems that are matched
by the classifier representation, as simple operations such as multiplication, division,
square roots, logical NOT, and so on.
Nevertheless, we can state that as computation continues to decrease in cost, more
of the burden of solving classification problems will be assumed by computation rather
than careful analysis, and here techniques such as evolutionary ones will be of use in
classification research.
Summary
When a pattern recognition problem involves a model that is discrete or of such
high complexity that analytic or gradient descent methods are unlikely to work, we
may employ stochastic techniques — ones that at some level rely on randomness to
find model parameters. Simulated annealing, based on physical annealing of metals,
consists in randomly perturbing the system, and gradually decreasing the randomness
to a low final level, in order to find an optimal solution. Boltzmann learning trains the
weights in a network so that the probability of a desired final output is increased. Such
learning is based on gradient descent in the Kullback-Liebler divergence between two
distributions of visible states at the output units: one distribution describes these
units when clamped at the known category information, and the other when they
are free to assume values based on the activations throughout the network. Some
34 CHAPTER 7. STOCHASTIC METHODS
(- (+ (+ X0 X1) (* X2 (- X4 X0)))(+ (/ X3 X2)(* X2 X0)))
–
+
+
X0 X1 X2
X4 X0
X3 X2 X2 X0
* / *
+
–
/
*
X2 X4 X3
X4 X1
*
/
(/ (* X2 X4) (* X3 (/ X4 X1)))
(- (+ (+ X0 X1) (* X2 (- X4 X0)))(/ X4 X1))
–
+
+
X0 X1 X2
X4 X0
*
–
/
*
X2 X4 X3
*
(/ (* X2 X4) (* X3 (+ (/ X3 X2)(* X2 X0))))
X3 X2 X2 X0
/ *
+
/
X4 X1
parents
(generation k)
offspring
(generation k+1)
Figure 7.17: Unlike the decision trees of Fig. 7.15 and Chap. ??, the trees shown here
are merely a representation using the syntax of Lisp that implements a single function.
For instance, the upper-right (parent) tree implements x2x4
x3(x4/x1)
. Such functions are
used with an implied threshold or sign function when used for classification. Thus
the function will operate on the features of a test pattern and emit category ωi if the
function is positive, and NOT ωi otherwise.
graphical models, such as hidden Markov models and Bayes belief networks, have
counterparts in structured Boltzmann networks, and this leads to new applications of
Boltzmann learning.
Search methods based on evolution — genetic algorithms and genetic programming
— perform highly parallel stochastic searches in a space set by the designer. The fun-
damental representation used in genetic algorithms is a string of bits, or chromosome;
the representation in genetic programming is a snippet of computer code. Variation
is introduced by means of crossover, mutation and insertion. As with all classification
methods, the better the features, the better the solution. There are many heuristics
that can be employed and parameters that must be set. As the cost of computation
contiues to decline, computationally intensive methods, such as Boltzmann networks
and evolutionary methods, should become increasingly popular.
7.6. BIBLIOGRAPHICAL AND HISTORICAL REMARKS 35
Bibliographical and Historical Remarks
The general problem of search is of central interest in computer science and artificial
intelligence, and is far to expansive to treat here. Nevertheless, techniques such as
depth first, breadth first, branch-and-bound, A* [19], occassionally find use in fields
touching upon pattern recognition, and practitioners should have at least a passing
knowledge of them. Good overviews can be found in [33] and a number of textbooks
on artificial intelligence, such as [46, 67, 55]. For rigor and completeness, Knuth’s
book on the subject is without peer [32].
The infinite monkey theorem, attributed to Sir Arthur Eddington, states that if
there is a sufficiently large number of monkeys typing at typewriters, eventually one
will bang out the script to Hamlet. It reflects one extreme of the tradeoff between prior
knowledge about the location of a solution on the one hand and the effort of search
required to fit it on the other. Computers made available in the early 1950s permitted
the first automated attempts at highly stochastic search, most notably the pioneering
work of Metropolis and colleagues for simulating chemical processes [40]. One of the
earliest and most influential applications of stochastic methods for pattern recognition
was the Pandemonium learning method due to Selfridge, which used stochastic search
for input weights in a feed-forward network model [57]. Kirkpatrick, Gelatt and Vec-
chi [30], and independently Černý [64], introduced the Boltzmann factor to general
stochastic search methods, the first example of simulated annealing. The statistical
physics foundations of Boltzmann factors, at the present level of mathematical sophis-
tication, can be found in [31]. The physical model of stochastic binary components
was introduced by Wilhemlm Lenz in 1920, but became associated with his doctoral
student Ernst Ising several years thereafter, and first called the “Ising model” in a
paper by R. Peierls [50]. It has spawned a great deal of theoretical and simulation
research [20].
The use of simulated annealing for learning was proposed by Ackley, Hinton and
Sejnowski [2], a good book on the method is [1], which described the procedure for ini-
tializing the temperature in simulated annealing and was the inspiration for Fig. 7.10.
Peterson and Anderson introduced deterministic annealing and mean-field Boltzmann
learning and described some of the (rare) conditions when the mean-field approxima-
tion might lead to non-optimal solutions [51]. Hinton showed that the Boltzmann
learning rule performs steepest descent in weight space for deterministic algorithm
[21].
A number of papers explore structured Boltzmann networks, including Hopfield’s
influential paper on networks for pattern completion or auto-association [25]. The
linear storage capacity of Hopfield networks quoted in the text, and nlogn relationships
for partial storage, are derived in [66, 39, 65]. The learning rule described in that work
has roots in the Learning matrix of [59, 60]. Harmonium [58, 14], another two-layer
variant of a Boltzmann network is primarily of historical interest. The relation of
Boltzmann networks to graphical models such as Hidden Markov models has been
explored in [27, 37] and [56], which was the source for our discussion in Sect. 7.4.
Implementation of constraints for Boltzmann machines was introduced in [42] and a
second-order pruning algorithm was described in [49].
Boltzmann learning has been applied to a number of real-world pattern recognition
problems, most notably speech recognition [8, 52] and stochastic restoration of images
or pattern completion [16]. Because Boltzmann learning has high computational
burden yet a natural VLSI implementation, a number of special-purpose chips have
been fabricated [23, 43, 44]. The ordering of configurations in Fig. 7.3, in which
36 CHAPTER 7. STOCHASTIC METHODS
neighboring configurations differe in just one bit, is a version of a Gray code; an
elegant method for constructing such codes is described in [18, Sect. 5.16 – 5.17].
Some of the earliest work inspired by evolution was described in [12, 13], but the
computational power available was insufficient for anything but toy problems. Later,
Rechenberg’s “evolution strategies” were applied to optimization in aeronautical de-
sign problems [53]. His earliest work did not employ full populations of candidate
solutions, nor the key operation of crossover. Evolutionary programming saves good
parents while evolutionary strategies generally does not. Neither employ mating, i.e.,
crossover. Holland introduced genetic algorithms in 1975 [24], and like the algorithm
itself, researchers have explored a very wide range of problems in search, optimization
and pattern recognition. A review appears in [6], and there is an increasing number
of textbooks [17, 41], the latter with a more rigorous approach to the mathemat-
ics. Koza’s extensive books on Genetic Programming provide a good introduction,
and include several illustrative simulations [34, 35], though relatively little on pattern
recognition. There are several collections of papers on evolutionary techniques in pat-
tern recognition, including [48]. An intriguing effect due to the interaction of learning
and evolution is the Baldwin effect, where learning can influence the rate of evolution
[22]; it has been shown that too much learning (as well as too little learning) leads to
slower evolution [28]. Evolutionary methods can lead to “non-optimal” or inelegant
solutions, and there is computational evidence that this occurs in nature [61, 62].
Problems
⊕
Section 7.1
1. One version of the infinite monkey theorem states that a single (immortal) monkey
typing randomly will ultimately reproduce the script of Hamlet. Estimate the time
needed for this, assuming the monkey can type two characters per second, that the
play has 50 pages, each containing roughly 80 lines, and 40 characters per line. Assume
there are 30 possible characters (a through z), space, period, exclamation point and
carriage return. Compare this time to the estimated age of the universe, 1010 years.⊕
Section 7.2
2. Prove that for any optimization problem of the form of Eq. 1 having a non-
symmetric connection matrix, there is an equivalent optimization problem in which
the matrix is replaced by its symmetric part.
3. The complicated energy landscape in the left of Fig. 7.2 is misleading for a number
of reasons.
(a) Discuss the difference between the continuous space shown in that figure with
the discrete space for the true optimization problem.
(b) The figure shows a local minimum near the middle of the space. Given the
nature of the discrete space, are any states closer to any “middle”?
(c) Suppose the axes referred to continuous variables si (as in mean-field annealing).
If each si obeyed a sigmoid (Fig. 7.5), could the energy landscape be non-
monotonic, as is shown in Fig. 7.2?
4. Consider exhaustive search for the minimum of the energy given in Eq. 1 for
binary units and arbitrary connections wij . Suppose that on a uniprocessor it takes
7.6. PROBLEMS 37
10−8 seconds to calculate the energy for each configuration. How long will it take to
exhaustively search the space for N = 100 units? How long for N = 1000 units?
5. Suppose it takes a uniprocessor 10−10 seconds to perform a single multiply-
accumulate, wijsisj , in the calculation of the energy E = −1/2
∑
ij
wijsisj given in
Eq. 1.
(a) Make some simplifying assumptions and write a formula for the total time re-
quired to search exhaustively for the minimum energy in a fully connected net-
work of N nodes.
(b) Plot your function using a log-log scale for N = 1, . . . , 105.
(c) What size network, N , could be searched exhaustively in a day? A year? A
century?
6. Make and justify any necessary mathematical assumptions and show analytically
that at high temperature, every configuration in a network of N units interconnected
by weights is equally likely (cf. Fig. 7.1).
7. Derive the exponential form of the Boltzmann factor in the following way. Consider
an isolated set of M + N independent magnets, each of which can be in an si = +1
or si = −1 state. There is a uniform magnetic field applied and this means that the
energy of the si = +1 state has some positive energy, which we can arbitrarily set to
1; the si = −1 state has energy −1. The total energy of the system is therefore the
sum of the number pointing up, ku, minus the number pointing down, kd; that is,
ET = ku − kd. (Of course, ku + kd = M +N regardless of the total energy.)
The fundamental statistical assumptions describing this system are that the mag-
nets are independent, and that the probability a subsystem (viz., the N magnets),
has a particular energy is proportional to the number of configurations that have this
energy.
(a) Consider the subsystem of N magnets, which has energy EN . Write an expres-
sion for the number of configurations K(N,EN ) that have energy EN .
(b) As in part (a), write a general expression for the number of configurations in
the subsystem M magnets at energy EM , i.e., K(M,EM ).
(c) Since the two subsystems consist of independent magnets, total number of
ways the full system can have total energy ET = EN + EM is the product
K(N,EN )K(M,EM ). Write an analytic expression for this total number.
(d) In statistical physics, if M � N , the M -magnet subsystem is called the heat
reservoire or heat bath. Assume that M � N , and write a series expansion for
your answer to part (c).
(e) Use your answer in part (d) to show that the probability the N -unit system has
energgy EN has the form of a Boltzmann factor, e−EN .
8. Prove that the analog value of si given by Eq. 5 is the expected value of a binary
variable in temperature T in the following simple case. Consider a single binary
magnet whose s = +1 state has energy +E0 and s = −1 state has energy −E0, as
would occur if an external magnetic field has been applied.
38 CHAPTER 7. STOCHASTIC METHODS
(a) Construct the partition function Z by summing over the two possible states
γ′ = 0 and γ′ = 1 according to Eq. 3.
(b) Recall that the probability of finding the system in state s = +1 is given by a
Boltzmann factor divided by the partition function (Eq. 2). Define the (analog)
expected value of the state to be
s = E [s] = P (s = +1)(+1) + P (s = −1)(−1).
Show that this implies the analog state of a single magnet obeys Eq. 5.
(c) Argue that if the N − 1 other magnets in a large system can be assumed to give an
average field (this is the mean-field approximation), then the analog value of a single
magnet will obey a function of the form given in Eq. 5.
9. Consider Boltzmann networks applied to the exclusive-OR problem.
(a) A fully connected network consisting solely of two input units and a single output
unit, whose sign gives the class, cannot solve the exclusive-OR problem. Prove
this by writing a set of inequalities for the weights and show that they are
inconsistent.
(b) As in part (a), prove that a fully connected Boltzmann network consisting solely
of two input units and two output units representing the two categories cannot
solve the exclusive-OR problem.
(c) Prove that a Boltzmann network of part (b) with a single hidden unit can im-
plement the exclusive-OR problem.
10. Consider a fully-connected Boltzmann network with two input units, a single
hidden unit and a single (category) output unit. Construct by hand a set of weights
wij for i, j = 1, 2, 3, 4 which allows the net to solve the exclusive-OR problem for a
representation in which si = ±1.⊕
Section 7.3
11. Show all intermediate steps in the derivation of Eq. 14 from Eq. 12. Be sure
your notation distinguishes this case from that leading to Eq. 10.
12. Train a six-unit Hopfield network with the following three patterns using the
learning rule of Eq. 15.
x1 = {+1,+1,+1,−1,−1,−1}
x2 = {+1,−1,+1,−1,+1,−1}
x3 = {−1,+1,−1,−1,+1,+1}
(a) Verify that each of the patterns gives a local minium in energy by perturbing
each of the six units individually and monitoring the energy.
(b) Verify that the symmetric state si → −si for i = 1, . . . , 6 also gives a local energy
minimum of the same energy.
13. Repeat Problem 12 but with the eight-unit network and the following patterns:
x1 = {+1,+1,+1,−1,−1,−1,−1,+1}
x2 = {+1,−1,+1,+1,+1,−1,+1,−1}
x3 = {−1,+1,−1,−1,+1,+1,−1,+1}
7.6. PROBLEMS 39
14. show that a missing feature assumes the appropriate value when training a
deficient pattern in a Boltzmann network.
15. show how if constraints that a pattern is not in a set of categories improves the
recognition for the others.
16. The text states a lower bound on the number of hidden units needed in a
Boltzmann network trained with n patterns is
log2n�. This is, of course, the number
of hiddens needed to insure a distinct hidden representation for each pattern. Show
that this lower bound is not tight, as there may not be weights to insure such a
representation. Do this by considering a Boltzmann network with three input units,
three hiddens and a single output, addressing the 3-bit parity problem.
(a) Argue that the hidden representation must be equivalent to the input represen-
tation.
(b) Argue that there is no two-layer Boltzmann network (here, hidden to output)
that can solve the three-bit parity problem. Explain why this implies that the
log2n� bound is not tight.
17. Consider the problem of initializing theN weights in a fully connected Boltzmann
network. Let there be N − 1 ≈ N weights connected to each unit. Suppose too that
the chance that any particulat units will be in the si = +1 state is 0.5, and likewise
for the si = −1 state. We seek weights such that the variance of the net activation
of each unit is roughly 1.0, a reasonable measure of the end of the linear range of the
sigmoid nonlinearity. The variance of li is
VAR[li] =
N∑
j=1
V AR[wijsj ] = NV AR[wij ]V AR[sj ]..
Set V AR[li] = 1, and solve for V AR[wij ] and thereby show that weights should be
initialized randomly in the range −1
√
3/N < wij < +
√
3/N .
18. Show that under reasonable conditions, the learning rate η in Eq. 14 for a
Boltzmann network of N units should be bounded η ≤ T 2/N to insure stability as
follows:
(a) Take the derivative of Eq. 14 to prove that the Hessian is
H =
∂2D̄KL
∂w2
=
∂2D̄KL
∂wij∂wuv
=
1
T 2
[E [sisjsusv]− E [sisj ]E [susv]] .
(b) Use this to show that
wtHw ≤ 1
T 2
E
∑
ij
|wij |
2
.
(c) Suppose we normalize weights such that ‖w‖ = 1 and thus∑
ij
wij ≤
√
N.
Use this fact together with your answer to part (b) to show that the curvature
of the D̄KL obeys
wtHw ≤ 1
T 2
E
[(√
N
)2]
=
N
T 2
.
40 CHAPTER 7. STOCHASTIC METHODS
(d) Use the fact that stability demands the learning rate to be the inverse of the
curvature, along with your answer in (c), to show that the learning rate should
be bounded η ≤ T 2/N .
⊕
Section 7.4
19. For any HMM, there exists a Boltzmann chain that implements the equivalent
probability model. Show the converse is not true, that is, for every chain, there
exists an HMM. Use the fact that weights in a Boltzmann chain are bounded −∞ <
Aij , Bjk < +∞, but probabilities in an HMM are positive and sum to 1.
20. For a Boltzmann chain with Tf steps, c hidden units and xx visible units, how
many legal paths are there (cf. Fig. 7.11).
21. The discussion of the relation between Boltzmann chains and hidden Markov
models in the text assumed the initial hidden state was known. Show that if this
hidden state is not known, the energy of Eq. 21 has another term which describes the
prior probability the system is in a particular hidden state.
⊕
Section 7.5
22. Consider the populations of size L of N -bit chromosomes.
(a) Show the number of different populations is
(
L+2N−1
2N−1
)
.
(b) Assume some number 1 ≤ Ls ≤ L are selected for reproduction in a given
generation. Use your answer to part (a) to write an expression for the number
of possible sets of parents as a function of L and La. (It is just the set, not their
order that is relevant.)
(c) Show that your answer to part (b) reduces to that in part (a) for the case
La = L.
(d) Show that your answer to part (b) gives L in the case La = 1.
⊕
Section 7.6
23. For each of the below snippets, mark suitable positions for breaks for the crossover
operator.
(a) (* (X0 (+ x4 x8)) x5 (SQRT 5))
(b) (SQRT ( X0 (+ x4 x8)))
(c) (* (- (SIN X0) (* (TAN 3.4) (SQRT X4)))
(d) (* (X0 (+ x4 x8)) x5 (SQRT 5))
(e) Separate the following Lisp symbols into groups such that any member in a
group can be replaced by another through the mutation operator in genetic
programming:
{+, X3, NOR, *, X0, 5.5, SQRT, /, X5, SIN, -, -4.5, NOT, OR, 2.7, TAN}
7.6. COMPUTER EXERCISES 41
Computer exercises
Several of the exercises use the data in the following table.
ω1 ω2
xxxxx xxxxx
xxxxx xxxxx
xxxxx xxxxx
xxxxx xxxxx
xxxxx xxxxx
xxxxx xxxxx
xxxxx xxxxx
xxxxx xxxxx
xxxxx xxxxx
xxxxx xxxxx
⊕
Section 7.2
1. Consider the problem of searching for a global minimum of the energy given in
Eq. 1 for a system of N units, fully interconnected by weights randomly chosen in the
range −1/
√
N < wij < +1/
√
N . Let N = 10.
(a) Write a program to search through all 2N configurations to find global minima,
and apply it to your network. Verify that there are two “global” minima.
(b) Write a program to perform the following version of gradient descent. Let
the units be numbered and ordered i = 1, . . . , N for bookkeeping. For each
configuration, find the unit with the lowest index i which can be changed to
lower the total energy. Iteratively make this change until the system converges,
or it is clear that it will not converge.
(c) Perform a search as in part (b) but with random polling of units.
(d) Repeat parts (a – c) for N = 100 and N = 1000.
(e) Discuss your results, paying particular attention to convergence and the problem
of local minima.
2. Algorithm 1
⊕
Section 7.3
3. Train a Boltzmann network consisting of eight input units and ten category units
with the characters of a seven-segment display shown in Fig. 7.10.
(a) Use the network to classify each of the ten patterns, and thus verify that all
have been learned.
(b) Explore pattern completion in your network the following way. For each of the
28 possible patterns do pattern completion for several characters. Add hidden
units and show that better performance results for ambiguous characters
42 CHAPTER 7. STOCHASTIC METHODS
4. ;laskjdf⊕
Section 7.4
5. lskjdf⊕
Section 7.5
6. lksdfj⊕
Section 7.6
7. Consider a two-category problem with four features bounded region −1 ≤ xi ≤ +1
for i = 1, 2, 3, 4.
(a) Generate training points in each of two categories defined by
ω1 : x1 + 0.5x2 − 0.3x3 − 0.1x4 < 0.5
ω2 : −x1 + 0.2x2 + x3 − 0.6x4 < 0.2
by randomly selecting a point in the four-dimensional space. If it satisfies neither
of the two inequalities, delete the point. If it satisfies just one of the inequalities,
label its category accordingly. If it satisfies both inequalities, randomly choose
a label with probability 0.5. If it satisfies neither of the inequalities, discard the
point. Continue in this way until you have 50 points for each category.
(b) GP
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Index
annealing
deterministic
Algorithm, 11
mean-field, see simulated anneal-
ing, deterministic, see anneal-
ing, deterministic
schedule, 8, 21
ascender (character), 28
Baldwin effect, 36
Boltzmann
chain, 24
constant, 6
factor, 6
zipper, 25
Boltzmann learning, see learning, Boltz-
mann, 12–25
application, 35
deterministic
Algorithm, 20
stochastic, 13–19
Boltzmann network
weight initialization, 21
chromosome, 26, 27
clamp, 12, 15
classifier
representation, 28
configuration, 4
hidden, 13
constraint
imposition, 17
cooling schedule, see annealing, sched-
ule
correlation, 15
spurious, 15
unit, 20
crossover, 32
descender (character), 28
deterministic annealing, see simulated
annealing, deterministic
Eddington, Sir Arthur, 35
energy, 4
interaction, 4
landscape, 5, 9
energy (Lyapunov), 4
entropy
relative, see Kullback-Leibler dis-
tance, 14
evolution
strategies, 36
evolutionary method, 25–33
feature
missing, 17
fitness, 26, 27, 29
fittest
survival of, 26
force
magnet, 9
genetic
Algorithm, 27
algorithm, 3
operator, 27
genetic programming, 32–33
golf course landscape, 5
gradient descent, 3
Gray code, 36
greedy search, see search, greedy
Hamlet, 35
Harmonium, 35
hypercube, 9
infinite monkey theorem, 35
insertion, 28, 32
insertion operator, 31
inversion (genetic operator), 28
Kullback-Leibler distance, 14
weighted, 16
48
INDEX 49
Kullback-Leibler divergence, see Kullback-
Leibler distance
learning
Boltzmann, 3
application, 35
evolution interaction, 36
rate
Boltzmann, 14
learning component, 15
Lisp, 32
local minimum, 8
Lyapunov function, 4
magnet
analogy for optimization, 4
mating, see crossover, see crossover
maximum likelihood, 3
mutation, 32
neural network, 28
chromosome representation, 28
objective function, see Lyapunov func-
tion
offspring, 26, 27
one-of-c representation, 12
optimization problem, 4
overfitting, 27
and genetic algorithms, 29
Pandemonium, 35
parent
genetic algorithm, 26
partition function, 6, 14
Parzen window, 21
pattern
completion, 35
Boltzmann network, 18–19
deficient, 17
training, 14
Perceptron, 28
pocket algorithm, 23
poll, 7
population, 26
Probabilistic Neural Network, 21
pruning
Boltzmann network, 35
relative entropy, 14
replication, 32
response function, 9
score (evolutionary methods), see fit-
ness
search
exhaustive, 3
greedy, 4
stochastic, 4
selection
and genetic algorithm, 29
fitness-proportional, 30
sigmoid, 9
simulated annealing, 5–12
deterministic, 9–12
sequential, see simulated anneal-
ing, stochastic
stochastic
Algorithm, 8
statistical mechanics, 3
stopping criterion
and genetic algorithms, 29
student component, see unlearning com-
ponent
survival of the fittest, 26
teacher component, see learning com-
ponent
temperature
annealing, 5
in genetic algorithms, 31
temperature (randomness), 5
topology
Boltzmann net, 20
unlearning component, 15
visible unit, 13
weight decay
Boltzmann net, 21
wrapper, 32
Contents
8 Non-metric Methods 3
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
8.2 Decision trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
8.3 CART . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
8.3.1 Number of splits . . . . . . . . . . . . . . . . . . . . . . . . . . 6
8.3.2 Test selection and node impurity . . . . . . . . . . . . . . . . . 6
8.3.3 When to stop splitting . . . . . . . . . . . . . . . . . . . . . . . 11
8.3.4 Pruning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
8.3.5 Assignment of leaf node labels . . . . . . . . . . . . . . . . . . 13
Example 1: A simple tree . . . . . . . . . . . . . . . . . . . . . . . . . 14
8.3.6 Computational complexity . . . . . . . . . . . . . . . . . . . . . 15
8.3.7 Feature choice . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.3.8 Multivariate decision trees . . . . . . . . . . . . . . . . . . . . . 16
8.3.9 Priors and costs . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.3.10 Missing attributes . . . . . . . . . . . . . . . . . . . . . . . . . 18
Example 2: Surrogate splits and missing attributes . . . . . . . . . . . 19
8.4 Other tree methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
8.4.1 ID3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
8.4.2 C4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
8.4.3 Which tree classifier is best? . . . . . . . . . . . . . . . . . . . 22
8.5 *Recognition with strings . . . . . . . . . . . . . . . . . . . . . . . . . 22
8.5.1 String matching . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Algorithm 1: Naive string matching . . . . . . . . . . . . . . . . . . . . 25
Algorithm 2: Boyer-Moore string matching . . . . . . . . . . . . . . . 25
8.5.2 Edit distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Algorithm 3: Edit distance . . . . . . . . . . . . . . . . . . . . . . . . . 28
8.5.3 Computational complexity . . . . . . . . . . . . . . . . . . . . . 29
8.5.4 String matching with errors . . . . . . . . . . . . . . . . . . . . 29
8.5.5 String matching with the “don’t-care” symbol . . . . . . . . . . 30
8.6 Grammatical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
8.6.1 Grammars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
8.6.2 Types of string grammars . . . . . . . . . . . . . . . . . . . . . 33
Example 3: A grammar for pronouncing numbers . . . . . . . . . . . . 34
8.6.3 Recognition using grammars . . . . . . . . . . . . . . . . . . . 35
Algorithm 4: Bottom-up parsing . . . . . . . . . . . . . . . . . . . . . . 36
8.7 Grammatical inference . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Algorithm 5: Grammar inference (overview) . . . . . . . . . . . . . . . 39
Example 4: Grammar inference . . . . . . . . . . . . . . . . . . . . . . 40
1
2 CONTENTS
8.8 *Rule-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8.8.1 Learning rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Bibliographical and Historical Remarks . . . . . . . . . . . . . . . . . . . . 44
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Computer exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Chapter 8
Non-metric Methods
8.1 Introduction
We have considered pattern recognition based on feature vectors of real-valuedand discrete-valued numbers, and in all cases there has been a natural measure
of distance between such vectors. For instance in the nearest-neighbor classifier the
notion figures conspicuously — indeed it is the core of the technique — while for
neural networks the notion of similarity appears when two input vectors sufficiently
“close” lead to similar outputs. Most practical pattern recognition methods address
problems of this sort, where feature vectors are real-valued and there exists some
notion of metric.
But suppose a classification problem involves nominal data — for instance descrip- nominal
datations that are discrete and without any natural notion of similarity or even ordering.
Consider the use of information about teeth in the classification of fish and sea mam-
mals. Some teeth are small and fine (as in baleen whales) for straining tiny prey from
the sea. Others (as in sharks) coming in multiple rows. Some sea creatures, such as
walruses, have tusks. Yet others, such as squid, lack teeth altogether. There is no
clear notion of similarity (or metric) for this information about teeth: it is meaning-
less to consider the teeth of a baleen whale any more similar to or different from the
tusks of a walrus, than it is the distinctive rows of teeth in a shark from their absence
in a squid, for example.
Thus in this chapter our attention turns away from describing patterns by vec-
tors of real numbers and towardusing lists of attributes. A common approach is
to specify the values of a fixed number of properties by a property d-tuple For ex- property
d-tupleample, consider describing a piece of fruit by the four properties of color, texture,
taste and smell. Then a particular piece of fruit might be described by the 4-tuple
{red, shiny, sweet, small}, which is a shorthand for color = red, texture = shiny,
taste = sweet and size = small. Another common approach is to describe the pat-
tern by a variable length string of nominal attributes, such as a sequence of base pairs string
in a segment of DNA, e.g., “AGCTTCAGATTCCA.”∗ Such lists or strings might be them-
selves the output of other component classifiers of the type we have seen elsewhere.
For instance, we might train a neural network to recognize different component brush
∗ We often put strings between quotation marks, particularly if this will help to avoid ambiguities.
3
4 CHAPTER 8. NON-METRIC METHODS
strokes used in Chinese and Japanese characters (roughly a dozen basic forms); a
classifier would then accept as inputs a list of these nominal attributes and make the
final, full character classification.
How can we best use such nominal data for classification? Most importantly, how
can we efficiently learn categories using such non-metric data? If there is structure in
strings, how can it be represented? In considering such problems, we move beyond the
notion of continuous probability distributions and metrics toward discrete problems
that are addressed by rule-based or syntactic pattern recognition methods.
8.2 Decision trees
It is natural and intuitive to classify a pattern through a sequence of questions,
in which the next question asked depends on the answer to the current question.
This “20-questions” approach is particularly useful for non-metric data, since all
of the questions can be asked in a “yes/no” or “true/false”or “value(property) ∈
set of values” style that does not require any notion of metric.
Such a sequence of questions is displayed in a directed decision tree or simply tree,
where by convention the first or root node is displayed at the top, connected by succes-root node
sive (directional) links or branches to other nodes. These are similarly connected until
link
branch
we reach terminal or leaf nodes, which have no further links (Fig. 8.1). Sections 8.3 &
leaf
8.4 describe some generic methods for creating such trees, but let us first understand
how they are used for classification. The classification of a particular pattern begins
at the root node, which asks for the value of a particular property of the pattern. The
different links from the root node corresopnd to the different possible values. Based
on the answer we follow the appropriate link to a subsequent or descendent node. Indescendent
the trees we shall discuss, the links must be mutually distinct and exhaustive, i.e.,
one and only one link will be followed. The next step is to make the decision at the
appropriate subsequent node, which can be considered the root of a sub-tree. Wesub-tree
continue this way until we reach a leaf node, which has no further question. Each leaf
node bears a category label and the test pattern is assigned the category of the leaf
node reached.
The simple decision tree in Fig. 8.1 illustrates one benefit of trees over many other
classifiers such as neural networks: interpretability. It is a straightforward matter
to render the information in such a tree as logical expressions. Such interpretability
has two manifestations. First, we can easily interpret the decision for any particular
test pattern as the conjunction of decisions along the path to its corresponding leaf
node. Thus if the properties are {taste, color, shape, size}, the pattern x = {sweet,
yellow, thin, medium} is classified as Banana because it is (color = yellow) AND
(shape = thin).∗ Second, we can occasionally get clear interpretations of the cate-
gories themselves, by creating logical descriptions using conjunctions and disjunctions
(Problem 8). For instance the tree shows Apple = (green AND medium) OR (red
AND medium).
Rules derived from trees — especially large trees — are often quite complicated
and must be reduced to aid interpretation. For our example, one simple rule describes
Apple = (medium AND NOT yellow). Another benefit of trees is that they lead to
∗ We retain our convention of representing patterns in boldface even though they need not be true
vectors, i.e., they might contain nominal data that cannot be added or multiplied the way vector
components can. For this reason we use the terms “attribute” to represent both nominal data and
real-valued data, and reserve “feature” for real-valued data.
8.3. CART 5
Color?
Size? Size?Shape?
round
Size?
yellow
redgreen
thin mediumsmall smallbig
Grapefruit
big small
Watermelon Banana AppleApple
Lemon
Grape Taste?
sweet sour
Cherry Grape
m
edium
level 0
level 1
level 2
level 3
root
Figure 8.1: Classification in a basic decision tree proceeds from top to bottom. The
questions asked at each node concern a particular property of the pattern, and the
downward links correspond to the possible values. Successive nodes are visited until a
terminal or leaf node is reached, where the category label is read. Note that the same
question, Size?, appears in different places in the tree, and that different questions
can have different numbers of branches. Moreover, different leaf nodes, shown in pink,
can be labeled by the same category (e.g., Apple).
rapid classification, employing a sequence of typically simple queries. Finally, we note
that trees provide a natural way to incorporate prior knowledge from human experts.
In practice, though, such expert knowledge if of greatest use when the classification
problem is fairly simple and the training set is small.
8.3 CART
Now we turn to the matter of using training data to create or “grow” a decision tree.
We assume that we have a set D of labeled training data and we have decided on a
set of properties that can be used to discriminate patterns, but do not know how to
organize the tests into a tree. Clearly, any decision tree will progressively split the
set of training examples into smaller and smaller subsets. It would be ideal if all the
samples in each subset had the same category label. In that case, we would say that
each subset was pure, and could terminate that portion of the tree. Usually, however,
there is a mixture of labels in each subset, and thus for each branch we will have
to decide either to stop splitting and accept an imperfect decision, or instead select
another property and grow the tree further.
This suggests an obvious recursive tree-growing process: given the data repre-
sented at a node, either declare that node to be a leaf (and state what category to
assign to it), or find another property to use to split the data into subsets. How-
ever, this is only one example of a more generic tree-growing methodology know as
CART (Classification and Regression Trees). CART provides a general framework
that can be instatiated in various ways to produce different decision trees. In the
CART approach, six general kinds of questions arise:
1. Should the properties be restricted to binary-valued or allowed to be multi-
6 CHAPTER 8. NON-METRIC METHODS
valued? That is, how many decision outcomes or splits will there be at a node?split
2. Which property should be tested at a node?
3. When should a node be declared a leaf?
4. If the tree becomes “too large,” how can it be made smaller and simpler, i.e.,
pruned?
5. If a leaf node is impure, how should the category label be assigned?
6. How should missing data be handled?
We consider each of these questions in turn.
8.3.1 Number of splits
Each decision outcome at a node is called a split, since it corresponds to splitting a
subset of the training data. The root node splits the full training set; each successive
decision splits a proper subset of the data. The number of splits at a node is closely
related to question 2, specifying which particular split will be made at a node. In
general, the number of splits is set by the designer, and could vary throughout the tree,
as we saw in Fig. 8.1. The number of links descending from a node is sometimes called
the node’s branching factor or branching ratio, denoted B. However, every decisionbranching
factor (and hence every tree) can be represented using just binary decisions (Problem 2).
Thus the root node querying fruit color (B = 3) in our example could be replaced by
two nodes: the first would ask fruit = green?, and at the end of its “no” branch,
another node would ask fruit = yellow?. Because of the universal expressive power
of binary trees and the comparative simplicity in training, we shall concentrate on
such trees (Fig. 8.2).
8.3.2 Test selection and node impurity
Much of the work in designing trees focuses on deciding which property test or query
should be performed at each node.∗ With non-numeric data, there is no geometrical
interpretation of how the test at a node splits the data. However, for numerical
data, there is a simple way to visualize the decision boundaries that are produced
by decision trees. For example, suppose that the test at each node has the form “is
xi ≤ xis?” This leads to hyperplane decision boundaries that are perpendicular to the
coordinate axes, and to decision regions of the form illustrated in Fig. 8.3.
The fundamental principle underlying tree creation is that of simplicity: we prefer
decisions that lead to a simple, compact tree with few nodes. This is a version of
Occam’s razor, that the simplest model that explains data is the one to be preferred
(Chap. ??). To this end, we seek a property test T at each node N that makes the
data reaching the immediate descendent nodes as “pure” as possible. In formalizingpurity
this notion, it turns out to be more conveninet to define the impurity, rather than
∗ The problem is further complicated by the fact that there is no reason why the test at a node
has to involve only one property. One might well consider logical combinations of properties, such
as using (size = medium) AND (NOT (color = yellow))? as a test. Trees in which each test is
based on a single property are called monothetic; if the query at any of the nodes involves two or
more properties, the tree is called polythetic. For simplicity, we generally restrict our treatment to
monothetic trees. In all cases, the key requirement is that the decision at a node be well-defined
and unambiguous so that the response leads down one and only one branch.
8.3. CART 7
Grapefruit
Banana Apple
Lemon Cherry Grape
GrapeApple
Watermelon
yes no
no
no
no
no
no
no
no
yesyes
yes yes yes
yes yes
color = Green?
size = big?
size = medium?
color = yellow?
size = small?shape = round?
size = big? taste = sweet?
Figure 8.2: A tree with arbitrary branching factor at different nodes can always be
represented by a functionally equivalent binary tree, i.e., one having branching factor
B = 2 throughout. By convention the “yes” branch is on the left, the “no” branch on
the right. This binary tree contains the same information and implements the same
classification as that in Fig. 8.1.
the purity of a node. Several different mathematical measures of impurity have been
proposed, all of which have basically the same behavior. Let i(N) denote the impurity
of a nodeN . In all cases, we want i(N) to be 0 if all of the patterns that reach the node
bear the same category label, and to be large if the categories are equally represented.
The most popular measure is the entropy impurity (or occasionally information
impurity): entropy
impurity
i(N) = −
∑
j
P (ωj) log2 P (ωj), (1)
where P (ωj) is the fraction of patterns at node N that are in category ωj .∗ By
the well-known properties of entropy, if all the patterns are of the same category,
the impurity is 0; otherwise it is positive, with the greatest value occuring when the
different classes are equally likely.
Another definition of impurity is particularly useful in the two-category case.
Given the desire to have zero impurity when the node represents only patterns of
a single category, the simplest polynomial form is:
i(N) = P (ω1)P (ω2). (2)
This can be interpreted as a variance impurity since under reasonable assumptions it variance
impurity∗ Here we are a bit sloppy with notation, since we normally reserve P for probability and P̂ for
frequency ratios. We could be even more precise by writing P̂ (x ∈ ωj |N) — i.e., the fraction
of training patterns x at node N that are in category ωj , given that they have survived all the
previous decisions that led to the node N — but for the sake of simplicity we sill avoid such
notational overhead.
8 CHAPTER 8. NON-METRIC METHODS
R1
R2R2
R2
R2
R1 R1
R1
x1
x3
x2
x2
x1
R1
R2
R1
Figure 8.3: Monothetic decision trees create decision boundaries with portions per-
pendicular to the feature axes. The decision regions are marked R1 and R2 in these
two-dimensional and three-dimensional two-category examples. With a sufficiently
large tree, any decision boundary can be approximated arbitrarily well.
is related to the variance of a distribution associated with the two categories (Prob-
lem 10). A generalization of the variance impurity, applicable to two or more cate-
gories, is the Gini impurity:Gini
impurity
i(N) =
∑
i �=j
P (ωi)P (ωj) = 1 −
∑
j
P 2(ωj). (3)
This is just the expected error rate at nodeN if the category label is selected randomly
from the class distribution present at N . This criterion is more strongly peaked at
equal probabilities than is the entropy impurity (Fig. 8.4).
The misclassification impurity can be written asmisclassifi-
cation
impurity i(N) = 1 − max
j
P (ωj), (4)
and measures the minimum probability that a training pattern would be misclassified
atN . Of the impurity measures typically considered, this measure is the most strongly
peaked at equal probabilities. It has a discontinuous derivative, though, and this can
present problems when searching for an optimal decision over a continuous parameter
space. Figure 8.4 shows these impurity functions for a two-category case, as a function
of the probability of one of the categories.
We now come to the key question — given a partial tree down to node N , what
value s should we choose for the property test T? An obvious heuristic is to choose
the test that decreases the impurity as much as possible. The drop in impurity is
defined by
∆i(N) = i(N) − PLi(NL) − (1 − PL)i(NR), (5)
where NL and NR are the left and right descendent nodes, i(NL) and i(NR) their
impurities, and PL is the fraction of patterns at node N that will go to NL when
property test T is used. Then the “best” test value s is the choice for T that maximizes
∆i(T ). If the entropy impurity is used, then the impurity reduction corresponds to an
8.3. CART 9
0 1
P
i(P)
m
is
cl
as
si
fic
at
io
n
entropy
G
ini/variance
0.5
Figure 8.4: For the two-category case, the impurity functions peak at equal class fre-
quencies and the variance and the Gini impurity functions are identical. To facilitate
comparisons, the entropy, variance, Gini and misclassification impurities (given by
Eqs. 1 – 4, respectively) have been adjusted in scale and offset to facilitate compari-
son; such scale and offset does not directly affect learning or classification.
information gain provided by the query. Since each query in a binary tree is a single
“yes/no” one, the reduction in entropy impurity due to a split at a node cannot be
greater than one bit (Problem 5).
The way to find an optimal decision for a node depends upon the general form of
decision. Since the decision criteria are based on the extrema of the impurity func-
tions, we are free to change such a function by an additive constant or overall scale
factor and this will not affect which split is found. Designers typically choose functions
that are easy to compute, such as those based on a single feature or attribute, giving
a monothetic tree. If the form of the decisions is based on the nominal attributes,
we may have to perform extensive or exhaustive search over all possible subsets of
the training set to find the rule maximizing ∆i. If the attributes are real-valued,
one could use gradient descent algorithms to find a splitting hyperplane (Sect. 8.3.8),
giving a polythetic tree. An important reason for favoring binary trees is that the
decision at any node can generally be cast as a one-dimensional optimization problem.
If the branching factor B were instead greater than 2, a two- or higher-dimensional
optimization would be required; this is generally much more difficult (Computer ex-
ercise ??).
Sometimes there will be several decisions s that lead to the same reduction in
impurity and the question arises how to choose among them. For example, if the
features are real-valued and a split lying anywhere in a range xl < xs < xu for
the x variable leads to the same (maximum) impurity reduction, it is traditional to
choose either the midpoint or the weighted average — xs = (xl + xu)/2 or xs =
(1 − P )xl + xuP , respectively — where P is the probability a pattern goes to the
“left” under the decision. Computational simplicity may be the determining factor as
there are rarely deep theoretical reasons to favor one over another.
Note too that the optimization of Eq. 5 is local — done at a single node. As with
the vast majority of such greedy methods, there is no guarantee that successive locally greedy
methodoptimal decisions lead to the global optimum. In particular, there is no guarantee
that after training we have the smallest tree (Computer exercise ??). Nevertheless,
for every reasonable impurity measure and learning method, we can always continue
to split further to get the lowest possible impurity at the leafs (Problem ??). There
is no assurance that the impurity at a leaf node will be the zero, however: if two
10 CHAPTER 8. NON-METRIC METHODS
patterns have the same attribute description yet come from different categories, the
impurity will be greater than zero.
Occasionally during tree creation the misclassification impurity (Eq. 4) will not
decrease whereas the Gini impurity would (Problem ??); thus although classification
is our final goal, we may prefer the Gini impurity because it “anticipates” later splits
that will be useful. Consider a case where at node N there are 90 patterns in ω1 and
10 in ω2. Thus the misclassification impurity is 0.1. Suppose there are no splits that
guarantee a ω2 majority in either of the two descendent nodes. Then the misclassifi-
cation remains at 0.1 for all splits. Now consider a split which sends 70 ω1 patterns
to the right along with 0 ω2 patterns, and sends 20 ω1 and 10 ω2 to the left. This is
an attractive split but the misclassification impurity is still 0.1. On the other hand,
the Gini impurity for this split is less than the Gini for the parent node. In short,
the Gini impurity shows that this as a good split while the misclassification rate does
not.
In multiclass binary tree creation, the twoing criterion may be useful.∗ The overalltwoing
criterion goal is to find the split that best splits groups of the c categories, i.e., a candidate
“supercategory” C1 consisting of all patterns in some subset of the categories, and
candidate “supercategory” C2 as all remaining patterns. Let the class of categories
be C = {ω1, ω2, . . . , ωc}. At each node, the decision splits the categories into C1 =
{ωi1 , ωi2 , . . . , ωik} and C2 = C −C1. For every candidate split s, we compute a change
in impurity ∆i(s, C1) as though it corresponded to a standard two-class problem. That
is, we find the split s∗(C1) that maximizes the change in impurity. Finally, we find
the supercategory C∗1 which maximizes ∆i(s∗(C1), C1). The benefit of this impurity is
that it is strategic — it may learn the largest scale structure of the overall problem
(Problem 4).
It may be surprising, but the particular choice of an impurity function rarely seems
to affect the final classifier and its accuracy. An entropy impurity is frequently used
because of its computational simplicity and basis in information theory, though the
Gini impurity has received significant attention as well. In practice, the stopping
criterion and the pruning method — when to stop splitting nodes, and how to merge
leaf nodes — are more important than the impurity function itself in determining
final classifier accuracy, as we shall see.
Multi-way splits
Although we shall concentrate on binary trees, we briefly mention the matter of
allowing the branching ratio at each node to be set during training, a technique will
return to in a discussion of the ID3 algorithm (Sect. 8.4.1). In such a case, it is
tempting to use a multi-branch generalization of Eq. 5 of the form
∆i(s) = i(N) −
B∑
k=1
Pki(Nk), (6)
where Pk is the fraction of training patterns sent down the link to node Nk, and
B∑
k=1
Pk = 1. However, the drawback with Eq. 6 is that decisions with large B are
inherently favored over those with small B whether or not the large B splits in fact
represent meaningful structure in the data. For instance, even in random data, a
∗ The twoing criterion is not a true impurity measure.
8.3. CART 11
high-B split will reduce the impurity more than will a low-B split. To avoid this
drawback, the candidate change in impurity of Eq. 6 must be scaled, according to
∆iB(s) =
∆i(s)
−
B∑
k=1
Pklog2Pk
. (7)
a method based on the gain ratio impurity (Problem 17). Just as before, the optimal gain ratio
impuritysplit is the one maximizing ∆iB(s).
8.3.3 When to stop splitting
Consider now the problem of deciding when to stop splitting during the training of
a binary tree. If we continue to grow the tree fully until each leaf node corresponds
to the lowest impurity, then the data has typically been overfit (Chap. ??). In the
extreme but rare case, each leaf corresponds to a single training point and the full tree
is merely a convenient implementation of a lookup table; it thus cannot be expected
to generalize well in (noisy) problems having high Bayes error. Conversely, if splitting
is stopped too early, then the error on the training data is not sufficiently low and
hence performance may suffer.
How shall we decide when to stop splitting? One traditional approach is to use
techniques of Chap. ??, in particular cross-validation. That is, the tree is trained
using a subset of the data (for instance 90%), with the remaining (10%) kept as a
validation set. We continue splitting nodes in successive layers until the error on the
validation data is minimized.
Another method is to set a (small) threshold value in the reduction in impurity;
splitting is stopped if the best candidate split at a node reduces the impurity by
less than that pre-set amount, i.e., if maxs ∆i(s) ≤ β. This method has two main
benefits. First, unlike cross-validation, the tree is trained directly using all the training
data. Second, leaf nodes can lie in different levels of the tree, which is desirable
whenever the complexity of the data varies throughout the range of input. (Such an
unbalanced tree requires a different number of decisions for different test patterns.) balanced
treeA fundamental drawback of the method, however, is that it is often difficult to know
how to set the threshold because there is rarely a simple relationship between β and
the ultimate performance (Computer exercise 2). A very simple method is to stop
when a node represents fewer than some threshold number of points, say 10, or some
fixed percentage of the total training set, say 5%. This has a benefit analogous to
that in k-nearest-neighbor classifiers (Chap. ??); that is, the size of the partitions is
small in regions where data is dense, but large where the data is sparse.
Yet another method is to trade complexity for test accuracy by splitting until a
minimum in a new, global criterion function,
α · size+
∑
leaf nodes
i(N), (8)
is reached. Here size could represent the number of nodes or links and α is some
positive constant. (This is analogous to regularization methods in neural networks
that penalize connection weights or nodes.) If an impurity based on entropy is used
for i(N), then Eq. 8 finds support from minimum description length (MDL), which minimum
description
length
we shall consider again in Chap. ??. The sum of the impurities at the leaf nodes is a
measure of the uncertainty (in bits) in the training data given the model represented
12 CHAPTER 8. NON-METRIC METHODS
by the tree; the size of the tree is a measure of the complexity of the classifier itself
(which also could be measured in bits). A difficulty, however, is setting α, as it is not
always easy to find a simple relationship between α and the final classifier performance
(Computer exercise 3).
An alternative approach is to use a stopping criterion based on the statistical
significance of the reduction of impurity. During tree construction, we estimate the
distribution of all the ∆i for the current collection of nodes; we assume this is the
full distribution of ∆i. For any candidate node split, we then determine whether it
is statistically different from zero, for instance by a chi-squared test (cf. Sect. ??).
If a candidate split does not reduce the impurity significantly, splitting is stopped
(Problem 15).
A variation in this technique of hypothesis testing can be applied even withouthypothesis
testing strong assumptions on the distribution of ∆i. We seek to determine whether a can-
didate split is “meaningful,” that is, whether it differs significantly from a random
split. Suppose n patterns survive at node N (with n1 in ω1 and n2 in ω2); we wish to
decide whether a candidate split s differs significantly from a random one. Suppose
a particular candidate split s sends Pn patterns to the left branch, and (1 − P )n to
the right branch. A random split having this probability (i.e., the null hypothesis)
would place Pn1 of the ω1 patterns and Pn2 of the ω2 patterns to the left, and the
remaining to the right. We quantify the deviation of the results due to candidate split
s from the (weighted) random split by means of the chi-squared statistic, which inchi-squared
statistic this two-category case is
χ2 =
2∑
i=1
(niL − nie)2
nie
, (9)
where niL is the number of patterns in category ωi sent to the left under decision s,
and nie = Pni is the number expected by the random rule. The chi-squared statistic
vanishes if the candidate split s gives the same distribution as the random one, and
is larger the more s differs from the random one. When χ2 is greater than a critical
value, as given in a table (cf. Table ??), then we can reject the null hypothesis since
s differs “significantly” at some probability or confidence level, such as .01 or .05.confidence
level The critical values of the confidence depend upon the number of degrees of freedom,
which in the case just described is 1, since for a given probability P the single value
n1L specifies all other values (n1R, n2L and n2R). If the “most significant” split at a
node does not yield a χ2 exceeding the chosen confidence level threshold, splitting is
stopped.
8.3.4 Pruning
Occassionally, stopped splitting suffers from the lack of sufficient look ahead, a phe-
nomenon called the horizon effect. The determination of the optimal split at a nodehorizon
effect N is not influenced by decisions at N ’s descendent nodes, i.e., those at subsequent
levels. In stopped splitting, node N might be declared a leaf, cutting off the possi-
bility of beneficial splits in subsequent nodes; as such, a stopping condition may be
met “too early” for overall optimal recognition accuracy. Informally speaking, the
stopped splitting biases the learning algorithm toward trees in which the greatest
impurity reduction is near the root node.
The principal alternative approach to stopped splitting is pruning. In pruning, a
tree is grown fully, that is, until leaf nodes have minimum impurity — beyond any
8.3. CART 13
putative “horizon.” Then, all pairs of neighboring leaf nodes (i.e., ones linked to a
common antecedent node, one level above) are considered for elimination. Any pair
whose elimination yields a satisfactory (small) increase in impurity is eliminated, and
the common antecedent node declared a leaf. (This antecedent, in turn, could itself
be pruned.) Clearly, such merging or joining of the two leaf nodes is the inverse of merging
splitting. It is not unusual that after such pruning, the leaf nodes lie in a wide range
of levels and the tree is unbalanced.
Although it is most common to prune starting at the leaf nodes, this is not nec-
essary: cost-complexity pruning can replace a complex subtree with a leaf directly.
Further, C4.5 (Sect. 8.4.2) can eliminate an arbitrary test node, thereby replacing a
subtree by one of its branches.
The benefits of pruning are that it avoids the horizon effect; further, since there
is no training data held out for cross-validation, it directly uses all information in the
training set. Naturally, this comes at a greater computational expense than stopped
splitting, and for problems with large training sets, the expense can be prohibitive
(Computer exercise ??). For small problems, though, these computational costs are
low and pruning is generally to be preferred over stopped splitting. Incidentally, what
we have been calling stopped training and pruning are sometimes called pre-pruning
and post-pruning, respectively.
A conceptually different pruning method is based on rules. Each leaf has an
associated rule — the conjunction of the individual decisions from the root node,
through the tree, to the particular leaf. Thus the full tree can be described by a large
list of rules, one for each leaf. Occasionally, some of these rules can be simplified
if a series of decisions is redundant. Eliminating the irrelevant precondition rules
simplifies the description, but has no influence on the classifier function, including
its generalization ability. The predominant reason to prune, however, is to improve
generalization. In this case we therefore eliminate rules so as to improve accuracy on a
validation set (Computer exercise 6). This technique may even allow the elimination
of a rule corresponding to a node near the root.
One of the benefits of rule pruning is that it allows us to distinguish between the
contexts in which any particular node N is used. For instance, for some test pattern
x1 the decision rule at node N is necessary; for another test pattern x2 that rule is
irrelevant and thus N could be pruned. In traditional node pruning, we must either
keep N or prune it away. In rule pruning, however, we can eliminate it where it is
not necessary (i.e., for patterns such as x1) and retain it for others (such as x2).
A final benefit is that the reduced rule set may give improved interpretability.
Although rule pruning was not part of the original CART approach, such pruning
can be easily applied to CART trees. We shall consider an example of rule pruning
in Sect. 8.4.2.
8.3.5 Assignment of leaf node labels
Assigning category labels to the leaf nodes is the simplest step in tree construction. If
successive nodes are split as far as possible, and each leaf node corresponds to patterns
in a single category (zero impurity), then of course this category label is assigned to
the leaf. In the more typical case, where either stopped splitting or pruning is used
and the leaf nodes have positive impurity, each leaf should be labeled by the category
that has most points represented. An extremely small impurity is not necessarily
desirable, since it may be an indication that the tree is overfitting the training data.
Example 1 illustrates some of these steps.
14 CHAPTER 8. NON-METRIC METHODS
Example 1: A simple tree classifier
Consider the following n = 16 points in two dimensions for training a binary
CART tree (B = 2) using the entropy impurity (Eq. 1).
ω1 (black) ω2 (red)
x1 x2 x1 x2
.15 .83 .10 .29
.09 .55 .08 .15
.29 .35 .23 .16
.38 .70 .70 .19
.52 .48 .62 .47
.57 .73 .91 .27
.73 .75 .65 .90
.47 .06 .75 .36* (.32†)
x1 < 0.6
x2 < 0.32
x1 < 0.35
x2 < 0.61
x1 < 0.69
x2 < 0.33
x2 < 0.09 x1 < 0.6
x1 < 0.69
x1
x2
*
†
ω1
ω2
ω2
ω2ω1 ω1
ω1 ω1
ω1ω2
ω2
1.0
.88 .65
.81 1.0
1.0
.76.59
.92
0
.2
.4
.6
.8
1
.2 .4 .6 .8 1
x1
x2
0
.2
.4
.6
.8
1
.2 .4 .6 .8 1
R1
R2
R2
R2
R2R1 R1
R1
R1
R1
Training data and associated (unpruned) tree are shown at the top. The entropy
impurity at non-terminal nodes is shown in red and the impurity at each leaf is 0. If
the single training point marked * were instead slightly lower (marked †), the resulting
tree and decision regions would differ significantly, as shown at the bottom.
8.3. CART 15
The impurity of the root node is
i(Nroot) = −
2∑
i=1
P (ωi)log2 P (ωi) = −[.5log2.5 + .5log2.5] = 1.0.
For simplicity we consider candidate splits parallel to the feature axes, i.e., of the form
“is xi < xis?”. By exhaustive search of the n−1 positions for the x1 feature and n−1
positions for the x2 feature we find by Eq. 5 that the greatest reduction in the impurity
occurs near x1s = 0.6, and hence this becomes the decision criterion at the root node.
We continue for each sub-tree until each final node represents a single category (and
thus has the lowest impurity, 0), as shown in the figure. If pruning were invoked,
the pair of leaf nodes at the left would be the first to be deleted (gray shading) since
there the impurity is increased the least. In this example, stopped splitting with the
proper threshold would also give the same final network. In general, however, with
large trees and many pruning steps, pruning and stopped splitting need not lead to
the same final tree.
This particular training set shows how trees can be sensitive to details of the train-
ing points. If the ω2 point marked * in the top figure is moved slightly (marked †), the
tree and decision regions differ significantly, as shown at the bottom. Such instability
is due in large part to the discrete nature of decisions early in the tree learning.
Example 1 illustrates the informal notion of instability or sensitivity to training stability
points. Of course, if we train any common classifier with a slightly different training
set the final classification decisions will differ somewhat. If we train a CART classifier,
however, the alteration of even a single training point can lead to radically different
decisions overall. This is a consequence of the discrete and inherently greedy nature
of such tree creation. Instability often indicates that incremental and off-line versions
of the method will yield significantly different classifiers, even when trained on the
same data.
8.3.6 Computational complexity
Suppose we have n training patterns in d dimensions in a two-category problem, and
wish to construct a binary tree based on splits parallel to the feature axes using an
entropy impurity. What are the time and the space complexities?
At the root node (level 0) we must first sort the training data, O(nlogn) for each of
the d features or dimensions. The entropy calculation is O(n) + (n− 1)O(d) since we
examine n − 1 possible splitting points. Thus for the root node the time complexity
is O(dnlogn). Consider an average case, where roughly half the training points are
sent to each of the two branches. The above analysis implies that splitting each
node in level 1 has complexity O(d n/2 log(n/2)); since there are two such nodes
at that level, the total complexity is O(dnlog(n/2)). Similarly, for the level 2 we
have O(dnlog(n/4)), and so on. The total number of levels is O(log n). We sum the
terms for the levels and find that the total average time complexity is O(dn (log n)2).
The time complexity for recall is just the depth of the tree, i.e., the total number
of levels, is O(log n). The space complexity is simply the number of nodes, which,
given some simplifying assumptions (such as a single training point per leaf node), is
1 + 2 + 4 + ...+ n/2 ≈ n, that is, O(n) (Problem 9).
16 CHAPTER 8. NON-METRIC METHODS
We stress that these assumptions (for instance equal splits at each node) rarely
hold exactly; moreover, heuristics can be used to speed the search for splits dur-
ing training. Nevertheless, the result that for fixed dimension d the training is
O(dn2 log n) and classification O(log n) is a good rule of thumb; it illustrates how
training is far more computationally expensive than is classification, and that on
average this discrepancy grows as the problem gets larger.
There are several techniques for reducing the complexity during the training of
trees based on real-valued data. One of the simplest heuristics is to begin the search
for splits xis at the “middle” of the range of the training set, moving alternately
to progressively higher and lower values. Optimal splits always occur for decision
thresholds between adjacent points from different categories and thus one should test
only such ranges. These and related techniques generally provide only moderate
reductions in computation (Computer exercise ??). When the patterns consist of
nominal data, candidate splits could be over every subset of attributes, or just a
single entry, and the computational burden is best lowered using insight into features
(Problem 3).
8.3.7 Feature choice
As with most pattern recognition techniques, CART and other tree-based methods
work best if the “proper” features are used (Fig. 8.5). For real-valued vector data,
most standard preprocessing techniques can be used before creating a tree. Pre-
processing by principal components (Chap. ??) can be effective, since it finds the
“important” axes, and this generally leads to simple decisions at the nodes. If how-
ever the principal axes in one region differ significantly from those in another region,
then no single choice of axes overall will suffice. In that case we may need to employ
the techniques of Sect. 8.3.8, for instance allowing splits to be at arbitrary orientation,
often giving smaller and more compact trees.
8.3.8 Multivariate decision trees
If the “natural” splits of real-valued data do not fall parallel to the feature axes or the
full training data set differs significantly from simple or accommodating distributions,
then the above methods may be rather inefficient and lead to poor generalization
(Fig. 8.6); even pruning may be insufficient to give a good classifier. The simplest
solution is to allow splits that are not parallel to the feature axes, such as a general
linear classifier trained via gradient descent on a classification or sum-squared-error
criterion (Chap. ??). While such training may be slow for the nodes near the root if
the training set is large, training will be faster at nodes closer to the leafs since less
training data is used. Recall can remain quite fast since the linear functions at each
node can be computed rapidly.
8.3.9 Priors and costs
Up to now we have tacitly assumed that a category ωi is represented with the same
frequency in both the training and the test data. If this is not the case, we need
a method for controlling tree creation so as to have lower error on the actual final
classification task when the frequencies are different. The most direct method is to
“weight” samples to correct for the prior frequencies (Problem 16). Furthermore,
we may seek to minimize a general cost, rather than a strict misclassification or 0-1
8.3. CART 17
.2 .4 .6 .8 1
0
.2
.4
.6
.8
1
- 1.2 x1 + x2 < 0.1
x1 < 0.27
x2 < 0.32
x1 < 0.07
x2 < 0.6
x1 < 0.55
x2 < 0.86
x1 < 0.81
x1
x2
ω2 ω1
ω2
ω1
ω1
ω1
ω1
ω2
ω2
ω2R2
R1
R2
R1
.2 .4 .6 .8 1
0
.2
.4
.6
.8
1
x1
x2
Figure 8.5: If the class of node decisions does not match the form of the training data,
a very complicated decision tree will result, as shown at the top. Here decisions are
parallel to the axes while in fact the data is better split by boundaries along another
direction. If however “proper” decision forms are used (here, linear combinations of
the features), the tree can be quite simple, as shown at the bottom.
cost. As in Chap. ??, we represent such information in a cost matrix λij — the
cost of classifying a pattern as ωi when it is actually ωj . Cost information is easily
incorporated into a Gini impurity, giving the following weighted Gini impurity, weighted
Gini
impurity
i(N) =
∑
ij
λijP (ωi)P (ωj), (10)
which should be used during training. Costs can be incorporated into other impurity
measures as well (Problem 11).
18 CHAPTER 8. NON-METRIC METHODS
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.04 x1 + 0.16 x2 < 0.11
0.27 x1 - 0.44 x2 < -0.02
0.96 x1 - 1.77x2 < -0.45
5.43 x1 - 13.33 x2 < -6.03
x2 < 0.5
x2 < 0.56x1 < 0.95
x2 < 0.54
x1
ω1ω2
R2
R1
0
ω2
ω1
ω1
ω1
ω1
ω2
ω2
ω2
R1
R2
R2
R1
x2
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
x10
x2
Figure 8.6: One form of multivariate tree employs general linear decisions at each
node, giving splits along arbitrary directions in the feature space. In virtually all
interesting cases the training data is not linearly separable, and thus the LMS algo-
rithm is more useful than methods that require the data to be linearly separable, even
though the LMS need not yield a minimum in classification error (Chap. ??). The
tree at the bottom can be simplified by methods outlined in Sect. 8.4.2.
8.3.10 Missing attributes
Classification problems might have missing attributes during training, during classi-
fication, or both. Consider first training a tree classifier despite the fact that some
training patterns are missing attributes. A naive approach would be to delete from
consideration any such deficient patterns; however, this is quite wasteful and should bedeficient
pattern employed only if there are many complete patterns. A better technique is to proceed
as otherwise described above (Sec. 8.3.2), but instead calculate impurities at a node
N using only the attribute information present. Suppose there are n training points
at N and that each has three attributes, except one pattern that is missing attribute
x3. To find the best split at N , we calculate possible splits using all n points using
attribute x1, then all n points for attribute x2, then the n− 1 non-deficient points for
attribute x3. Each such split has an associated reduction in impurity, calculated as
before, though here with different numbers of patterns. As always, the desired split
is the one which gives the greatest decrease in impurity. The generalization of this
procedure to more features, to multiple patterns with missing attributes, and even to
8.3. CART 19
patterns with several missing attributes is straightforward, as is its use in classifying
non-deficient patterns (Problem 14).
Now consider how to create and use trees that can classify a deficient pattern. The
trees described above cannot directly handle test patterns lacking attributes (but see
Sect. 8.4.2), and thus if we suspect that such deficient test patterns will occur, we
must modify the training procedure discussed in Sect. 8.3.2. The basic approach
during classification is to use the traditional (“primary”) decision at a node whenever
possible (i.e., when the queries involves a feature that is present in the deficient test
pattern) but to use alternate queries whenever the test pattern is missing that feature.
During training then, in addition to the primary split, each non-terminal node
N is given an ordered set of surrogate splits, consisting of an attribute label and a surrogate
splitrule. The first such surrogate split maximizes the “predictive association” with the
predictive
association
primary split. A simple measure of the predictive association of two splits s1 and
s2 is merely the numerical count of patterns that are sent to the “left” by both s1
and s2 plus the count of the patterns sent to the “right” by both the splits. The
second surrogate split is defined similarly, being the one which uses another feature
and best approximates the primary split in this way. Of course, during classification
of a deficient test pattern, we use the first surrogate split that does not involve the test
pattern’s missing attributes. This missing value strategy corresponds to a linear model
replacing the pattern’s missing value by the value of the non-missing attribute most
strongly correlated with it (Problem ??). This strategy uses to maximum advantage
the (local) associations among the attributes to decide the split when attribute values
are missing. A method closely related to surrogate splits is that of virtual values, in virtual
valuewhich the missing attribute is assigned its most likely value.
Example 2: Surrogate splits and missing attributes
Consider the creation of a monothetic tree using an entropy impurity and the
following ten training points. Since the tree will be used to classify test patterns with
missing features, we will give each node surrogate splits.
ω1 :
x1
07
8
,
x2
18
9
,
x3
29
0
,
x4
41
1
,
x5
52
2
ω2 :
y1
33
3
,
y2
60
4
,
y3
74
5
,
y4
85
6
,
y5
96
7
.
Through exhaustive search along all three features, we find the primary split at the
root node should be “x1 < 5.5?”, which sends {x1,x2,x3,x4,x5,y1} to the left and
{y2,y3,y4,y5} to the right, as shown in the figure.
We now seek the first surrogate split at the root node; such a split must be based
on either the x2 or the x3 feature. Through exhaustive search we find that the split
“x3 < 3.5?” has the highest predictive association with the primary split — a value
of 8, since 8 patterns are sent to matching directions by each rule, as shown in the
figure. The second surrogate split must be along the only remaining feature, x2. We
find that for this feature the rule “x2 < 3.5?” has the highest predictive association
20 CHAPTER 8. NON-METRIC METHODS
with the primary split, a value of 6. (This, incidentally, is not the optimal x2 split for
impurity reduction — we use it because it best approximates the preferred, primary
split.) While the above describes the training of the root node, training of other nodes
is conceptually the same, though computationally less complex because fewer points
need be considered.
x1<5.5?
x1, x2, x3, x4, x5, y1 y2, y3, y4, y5
x3<3.5? x2<3.5?
primary split first surrogate split second surrogate split
predictive association
with primary split = 8
predictive association
with primary split = 6
x3, x4, x5, y1 y2, y3, y4, y5,
x1, x2
x4, x5, y1,
y2
y3, y4, y5,
x1, x2, x3
Of all possible splits based on a single feature, the primary split, “x1 < 5.5?”, mini-
mizes the entropy impurity of the full training set. The first surrogate split at the root
node must use a feature other than x1; its threshold is set in order to best approxi-
mate the action of the primary split. In this case “x3 < 3.5?” is the first surrogate
split. Likewise, here the second surrogate split must use the x2 feature; its threshold
is chosen to best approximate the action of the primary split. In this case “x2 < 3.5?”
is the second surrogate split. The pink shaded band marks those patterns sent to the
matching direction as the primary split. The number of patterns in the shading is
thus the predictive association with the primary split.
During classification, any test pattern containing feature x1 would be queried using
the primary split, “x1 ≤ 5.5?” Consider though the deficient test pattern (∗, 2, 4)t,
where * is the missing x1 feature. Since the primary split cannot be used, we turn
instead to the first surrogate split, “x3 ≤ 3.5?”, which sends this point to the right.
Likewise, the test pattern (∗, 2, ∗)t would be queried by the second surrogate split,
“x2 ≤ 3.5?”, and sent to the left.
Sometimes the fact that an attribute is missing can be informative. For instance,
in medical diagnosis, the fact that an attribute (such as blood sugar level) is missing
might imply that the physician had some reason not to measure it. As such, a missing
attribute could be represented as a new feature, and used in classification.
8.4 Other tree methods
Virtually all tree-based classification techniques can incorporate the fundamental tech-
niques described above. In fact that discussion expanded beyond the core ideas in
the earliest presentations of CART. While most tree-growing algorithms use an en-
tropy impurity, there are many choices for stopping rules, for pruning methods and
for the treatment of missing attributes. Here we discuss just two other popular tree
algorithms.
8.4.1 ID3
ID3 received its name because it was the third in a series of identification or “ID”
procedures. It is intended for use with nominal (unordered) inputs only. If the problem
8.4. OTHER TREE METHODS 21
involves real-valued variables, they are first binned into intervals, each interval being
treated as an unordered nominal attribute. Every split has a branching factor Bj ,
where Bj is the number of discrete attribute bins of the variable j chosen for splitting.
In practice these are seldom binary and thus a gain ratio impurity should be used
(Sect. 8.3.2). Such trees have their number of levels equal to the number of input
variables. The algorithm continues until all nodes are pure or there are no more
variables to split on. While there is thus no pruning in standard presentations of the
ID3 algorithm, it is straightforward to incorporate pruning along the ideas presented
above (Computer exercise 4).
8.4.2 C4.5
The C4.5 algorithm, the successor and refinement of ID3, is the most popular in a
series of “classification” tree methods. In it, real-valued variables are treated the same
as in CART. Multi-way (B > 2) splits are used with nominal data, as in ID3 with a
gain ratio impurity based on Eq. 7. The algorithm uses heuristics for pruning derived
based on the statistical significance of splits.
A clear difference between C4.5 and CART involves classifying patterns with miss-
ing features. During training there are no special accommodations for subsequent
classification of deficient patterns in C4.5; in particular, there are no surrogate splits
precomputed. Instead, if node N with branching factor B queries the missing feature
in a deficient test pattern, C4.5 follows all B possible answers to the descendent nodes
and ultimately B leaf nodes. The final classification is based on the labels of the B
leaf nodes, weighted by the decision probabilities at N . (These probabilities are sim-
ply those of decisions at N on the training data.) Each of N ’s immediate descendent
nodes can be considered the root of a sub-tree implementing part of the full classifica-
tion model. This missing-attribute scheme corresponds to weighting these sub-models
by the probability any training pattern at N would go to the corresponding outcome
of the decision. This method does not exploit statistical correlations between different
features of the training points, whereas the method of surrogate splits in CART does.
Since C4.5 does not compute surrogate splits and hence does not need to store them,
this algorithm may be preferred over CART if space complexity (storage) is a major
concern.
The C4.5 algorithm has the provision for pruning based on the rules derived from
the learned tree. Each leaf node has an associated rule — the conjunction of the
decisions leading from the root node, through the tree, to that leaf. A technique
called C4.5Rules deletes redundant antecedents in such rules. To understand this, C4.5Rules
consider the left-most leaf in the tree at the bottom of Fig. 8.6, which corresponds to
the rule
IF
[
(0.40×1 + 0.16×2 < 0.11)
AND (0.27x1 − 0.44x2 < −0.02)
AND (0.96x1 − 1.77x2 < −0.45)
AND (5.43x1 − 13.33x2 < −6.03)
]
THEN x ∈ ω1.
This rule can be simplified to give
IF
[
( 0.40x1 + 0.16x2 < 0.11)
22 CHAPTER 8. NON-METRIC METHODS
AND (5.43x1 − 13.33x2 < −6.03)
]
THEN x ∈ ω1,
as should be evident in that figure. Note especially that information corresponding to
nodes near the root can be pruned by C4.5Rules. This is more general than impurity
based pruning methods, which instead merge leaf nodes.
8.4.3 Which tree classifier is best?
In Chap. ?? we shall consider the problem of comparing different classifiers, including
trees. Here, rather than directly comparing typical implementations of CART, ID3,
C4.5 and other numerous tree methods, it is more instructive to consider variations
within the different component steps. After all, with care one can generate a tree using
any reasonable feature processing, impurity measure, stopping criterion or pruning
method. Many of the basic principles applicable throughout pattern classification
guide us here. Of course, if the designer has insight into feature preprocessing, this
should be exploited. The binning of real-valued features used in early versions of ID3
does not take full advantage of order information, and thus ID3 should be applied
to such data only if computational costs are otherwise too high. It has been found
that an entropy impurity works acceptably in most cases, and is a natural default. In
general, pruning is to be preferred over stopped training and cross-validation, since it
takes advantage of more of the information in the training set; however, pruning large
training sets can be computationally expensive. The pruning of rules is less useful
for problems that have high noise and are at base statistical in nature, but such
pruning can often simplify classifiers for problems where the data were generated
by rules themselves. Likewise, decision trees are poor at inferring simple concepts,
for instance whether more than half of the binary (discrete) attributes have value
+1. As with most classification methods, one gains expertise and insight through
experimentation on a wide range of problems. No single tree algorithm dominates or
is dominated by others.
It has been found that trees yield classifiers with accuracy comparable to other
methods we have discussed, such as neural networks and nearest-neighbor classifiers,
especially when specific prior information about the appropriate form of classifier is
lacking. Tree-based classifiers are particularly useful with non-metric data and as
such they are an important tool in pattern recognition research.
8.5 *Recognition with strings
Suppose the patterns are represented as ordered sequences or strings of discrete items,
as in a sequence of letters in an English word or in DNA bases in a gene sequence,
such as “AGCTTCGAATC.” (The letters A, G, C and T stand for the nucleic acids adenine,
guanine, cytosine and thymine.) Pattern classification based on such strings of discrete
symbols differs in a number of ways from the more commonly used techniques we
have addressed up to here. Because the string elements — called characters, letterscharacter
or symbols — are nominal, there is no obvious notion of distance between strings.
There is a further difficulty arising from the fact that strings need not be of the
same length. While such strings are surely not vectors, we nevertheless broaden our
familiar boldface notation to now apply to strings as well, e.g., x = “AGCTTC,” though
we will often refer to them as patterns, strings, templates or general words. (Of course,word
8.5. *RECOGNITION WITH STRINGS 23
there is no requirement that these be meaningful words in a natural language such as
English or French.) A particularly long string is denoted text. Any contiguous string text
that is part of x is called a substring, segment, or more frequently a factor of x. For
factorexample, “GCT” is a factor of “AGCTTC.”
There is a large number of problems in computations on strings. The ones that
are of greatest importance in pattern recognition are:
String matching: Given x and text, test whether x is a factor of text, and if so,
where it appears.
Edit distance: Given two strings x and y, compute the minimum number of ba-
sic operations — character insertions, deletions and exchanges — needed to
transform x into y.
String matching with errors: Given x and text, find the locations in text where
the “cost” or “distance” of x to any factor of text is minimal.
String matching with the “don’t care” symbol: This is the same as basic string
matching, but with a special symbol, ©/ , the don’t care symbol, which can match don’t care
symbolany other symbol.
We should begin by understanding the several ways in which these string opera-
tions are used in pattern classification. Basic string matching can be viewed as an
extreme case of template matching, as in finding a particular English word within a
large electronic corpus such as a novel or digital repository. Alternatively, suppose
we have a large text such as Herman Melville’s Moby Dick, and we want to classify
it as either most relevant to the topic of fish or to the topic of hunting. Test strings
or keywords for the fish topic might include “salmon,” “whale,” “fishing,” “ocean,” keyword
while those for hunting might include “gun,” “bullet,” “shoot,” and so on. String
matching would determine the number of occurrences of such keywords in the text.
A simple count of the keyword occurrences could then be used to classify the text
according to topic. (Other, more sophisticated methods for this latter stage would
generally be preferable.)
The problem of string matching with the don’t care symbol is closely related
to standard string matching, even though the best algorithms for the two types of
problems differ, as we shall see. Suppose, for instance, that in DNA sequence analysis
we have a segment of DNA, such as x = “AGCCG©/©/©/©/©/GACTG,” where the first and last
sections (called motifs) are important for coding a protein while the middle section,
which consists of five characters, is nevertheless known to be inert and to have no
function. If we are given an extremely long DNA sequence (the text), string matching
with the don’t care symbol using the pattern x containing ©/ symbols would determine
if text is in the class of sequences that could yield the particular protein.
The string operation that finds greatest use in pattern classification is based on
edit distance, and is best understood in terms of the nearest-neighbor algorithm
(Chap. ??). Recall that in that algorithm each training pattern or prototype is stored
along with its category label; an unknown test pattern is then classified by its near-
est prototype. Suppose now that the prototypes are strings and we seek to classify
a novel test string by its “nearest” stored string. For instance an acoustic speech
recognizer might label every 10-ms interval with the most likely phoneme present in
an utterance, giving a string of discrete phoneme labels such as “tttoooonn.” Edit
distance would then be used to find the “nearest” stored training pattern, so that its
category label can be read.
24 CHAPTER 8. NON-METRIC METHODS
The difficulty in this approach is that there is no obvious notion of metric or
distance between strings. In order to proceed, then, we must introduce some measure
of distance between the strings. The resulting edit distance is the minimum number
of fundamental operations needed to transform the test string into a prototype string,
as we shall see.
The string-matching-with-errors problem contains aspects of both the basic string
matching and the edit distance problems. The goal is to find all locations in text
where x is “close” to the substring or factor of text. This measure of closeness is
chosen to be an edit distance. Thus the string-matching-with-errors problem finds
use in the same types of problems as basic string matching, the only difference being
that there is a certain “tolerance” for a match. It finds use, for example, in searching
digital texts for possibly misspelled versions of a given target word.
Naturally, deciding which strings to consider is highly problem-dependent. Nev-
ertheless, given target strings and the relevance of tolerances, and so on, the string
matching problems just outlined are conceptually very simple; the challenge arises
when the problems are large, such as searching for a segment within the roughly
3 × 109 base pairs in the human genome, the 3 × 107 characters in an electronic ver-
sion of War and Peace or the more than 1013 characters in a very large digital
repository. For such cases, the effort is in finding tricks and heuristics that make the
problem computationally tractable.
We now consider these four string operations in greater detail.
8.5.1 String matching
The most fundamental and useful operation in string matching is testing whether a
candidate string x is a factor of text. Naturally we assume the number of characters
in text, denoted length[text] or |text|, is greater than that in x, and for most com-
putationally interesting cases |text| � |x|. Each discrete character is taken from an
alphabet A, for example binary or decimal numerals, the English letters, or four DNAalphabet
bases, i.e., A = {0, 1} or {0,1,2,...,9} or {a,b,c,...,z} or {A,G,C,T}, respec-
tively. A shift, s, is an offset needed to align the first character of x with charactershift
number s + 1 in text. The basic string matching problem is to find whether there
exists a valid shift, i.e., one where there is a perfect match between each character invalid shift
x and the corresponding one in text. The general string-matching problem is to list
all valid shifts (Fig. 8.7).
a b a c d b d a c b b a c d a c
b d a c
text
x
s = 5
Figure 8.7: The general string-matching problem is to find all shifts s for which the
pattern x appears in text. Any such shift is called valid. In this case x = “bdac” is
indeed a factor of text, and s = 5 is the only valid shift.
The most straightforward approach in string matching is to test each possible shift
s in turn, as given in the naive string-matching algorithm.
8.5. *RECOGNITION WITH STRINGS 25
Algorithm 1 (Naive string matching)
1 begin initialize A,x, text, n← length[text],m← length[x]
2 s← 0
3 while s ≤ n−m
4 if x[1...m] = text[s+ 1...s+m]
5 then print “pattern occurs at shift” s
6 s← s+ 1
7 return
8 end
Algorithm 1 is hardly optimal — it takes time Θ((n−m+1)m) in the worst case; if x
and text are random, however, the algorithm is efficient (Problem 18). The weakness
in the naive string-matching algorithm is that information from one candidate shift
s is not exploited when seeking a subsequent candidate shift. A more sophisticated
method, the Boyer-Moore algorithm, uses such information in a clever way.
Algorithm 2 (Boyer-Moore string matching)
1 begin initialize A,x, text, n← length[text],m← length[x]
2 F(x) ← last-occurrence function
3 G(x) ← good-suffix function
4 s← 0
5 while s ≤ n−m
6 do j ← m
7 while j > 0 and x[j] = text[s+ j]
8 do j ← j − 1
9 if j = 0
10 then print “pattern occurs at shift” s
11 s← s+ G(0)
12 else s← s+ max
[
G(j), j − F(text[s+ j])
]
13 return
14 end
Postponing for the moment considerations of the functions F and G, we can see that
the Boyer-Moore algorithm resembles the naive string-matching algorithm, but with
two exceptions. First, at each candidate shift s, the character comparisons are done
in reverse order, i.e., from right to left (line 8). Second, according to lines 11 & 12,
the increment to a new shift apparently need not be 1.
The power of Algorithm 2 lies in two heuristics that allow it to skip the examination
of a large number shifts and hence character comparisons: the good-suffix heuristic and
the bad-character heuristic operate independently and in parallel. After a mismatch
is detected, each heuristic proposes an amount by which s can be safely increased
without missing a valid shift; the larger of these proposed shifts is selected and s is
increased accordingly.
The bad-character heuristic utilizes the rightmost character in text that does not bad-
character
heuristic
match the aligned character in x. Because character comparisons proceed right-to-
left, this “bad character” is found as efficiently as possible. Since the current shift s is
invalid, no more character comparisons are needed and a shift increment can be made.
The bad-character heuristic proposes incrementing the shift by an amount to align
the rightmost occurrence of the bad character in x with the bad character identified
in text. This guarantees that no valid shifts have been skipped (Fig. 8.8).
26 CHAPTER 8. NON-METRIC METHODS
s
r o b a b i l i t i e s _ f o r _ e s t i m a tp
bad character good suffix
e s
e s t i m a t e s
s + 3
r o b a b i l i t i e s _ f o r _ e s t i m a tp e s
e s t i m a t e s
s + 7
r o b a b i l i t i e s _ f o r _ e s t i m a tp e s
e s t i m a t e s
proposed by
bad-character heuristic
proposed by
good-suffix heuristic
Figure 8.8: String matching by the Boyer-Moore algorithm takes advantage of infor-
mation obtained at one shift s to propose the next shift; the algorithm is generally
much less computationally expensive than naive string matching, which always incre-
ments shifts by a single character. The top figure shows the alignment of text and
pattern x for an invalid shift s. Character comparisons proceed right to left, and
the first two such comparisons are a match — the good suffix is “es.” The first
(right-most) mismatched character in text, here “i,” is called the bad character. The
bad-character heuristic proposes incrementing the shift to align the right-most “i”
in x with the bad character “i” in text — a shift increment of 3, as shown in the
middle figure. The bottom figure shows the effect of the good-suffix heuristic, which
proposes incrementing the shift the least amount that will align the good suffix, “es”
in x, with that in text — here an increment of 7. Lines 11 & 12 of the Boyer-Moore
algorithm select the larger of the two proposed shift increments, i.e., 7 in this case.
Although not shown in this figure, after the mismatch is detected at shift s+ 7, both
the bad-character and the good-suffix heuristics propose an increment of yet another
7 characters, thereby finding a valid shift.
Now consider the good-suffix heuristic, which operates in parallel with the bad- good-
suffix
heuristic
character heuristic, and also proposes a safe shift increment. A general suffix of x issuffix
a factor or substring of x that contains the final character in x. (Likewise, a prefix
prefix contains the initial character in x.) At shift s the rightmost contiguous characters in
text that match those in x are called the good suffix, or “matching suffix.” As before,good
suffix because character comparisons are made right-to-left, the good suffix is found with
the minimum number of comparisons. Once a character mismatch has been found, the
good-suffix heuristic proposes to increment the shift so as to align the next occurrence
of the good suffix in x with that identified in text. This insures that no valid shift has
been skipped. Given the two shift increments proposed by the two heuristics, line 12
8.5. *RECOGNITION WITH STRINGS 27
of the Boyer-Moore algorithm chooses the larger.
These heuristics rely on the functions F and G. The last-occurrence function, last-
occurrence
function
F(x), is merely a table containing every letter in the alphabet and the position of its
rightmost occurrence in x. For the pattern in Fig. 8.8, the table would contain: a, 6;
e, 8; i, 4; m, 5; s, 9; and t, 8. All 20 other letters in the English alphabet are assigned
a value 0, signifying that they do not appear in x. The construction of this table is
simple (Problem 22) and need be done just once; it does not significanly affect the
computational cost of the Boyer-Moore algorithm.
The good-suffix function, G(x), creates a table which for each suffix gives the good-
suffix
function
location of its other occurrences in x. In the example in Fig. 8.8, the suffix s (the
last character in “estimates”) also occurs at position 2 in x. Further, the suffix “es”
occurs at position 1 in x. The suffix “tes” does not appear elsewhere in x and hence
it, and all other suffixes, are assigned the value 0. In sum, then, the table of G(x)
would have just two non-zero entries: s, 2 and es, 1.
In practice, these heuristics make the Boyer-Moore one of the most attractive
string-matching algorithms on serial computers. Other powerful methods quickly be-
come conceptually more involved and are generally based on precomputing functions
of x that enable efficient shift increments, or dividing the problem for efficient parallel
computation.
Many applications require a text to be searched for several strings, as in the case
of keyword search through a digital text. Occasionally, some of these search strings
are themselves factors of other search strings. Presumably we would not want to
acknowledge a match of a short string if it were also part of a match for a longer string.
Thus if our keywords included “beat,” “eat,” and “be,” we would want our search to
return only the string match of “beat” from text = “when chris beats the drum,”
not the shorter strings “eat” and “be,” which are nevertheless “there” in text. This is
an example of the subset-superset problem. Although there may be much bookkeeping subset-
superset
problem
associated with imposing such a strict bias for longer sequences over shorter ones, the
approach is conceptually straightforward (Computer exercise 9).
8.5.2 Edit distance
The fundamental idea underlying pattern recognition using edit distance is based on
the nearest-neighbor algorithm (Chap. ??). We store a full training set of strings
and their associated category labels. During classification, a test string is compared
to each stored string and a “distance” or score is computed; the test string is then
assigned the category label of the “nearest” string in the training set.
Unlike the case using real-valued vectors discussed in Chap. ??, there is no single
obvious measure of the similarity or difference between two strings. For instance, it
is not clear whether “abbccc” is closer to “aabbcc” or to “abbcccb.” To proceed,
then, we introduce a measure of the difference between two strings. Such an edit
distance between x and y describes how many fundamental operations are required
to transform x into y. These fundamental operations are:
substitutions: A character in x is replaced by the corresponding character in y.
insertions: A character in y is inserted into x, thereby increasing the length of x by
one character.
deletions: A character in x is deleted, thereby decreasing the length of x by one
character.
28 CHAPTER 8. NON-METRIC METHODS
Occasionally we also consider a fourth operation, interchange, or “twiddle,” or transpo-interchange
sition, which interchanges two neighboring characters in x. Thus, one could transform
x = “asp” into y = “sap” with a single interchange. Because such an interchange
can always be expressed as two substitutions, for simplicity we shall not consider
interchanges.
Let C be an m×n matrix of integers associated with a cost or “distance” and let
δ(·, ·) denote a generalization of the Kronecker delta function, having value 1 if the
two arguments (characters) match and 0 otherwise. The basic edit-distance algorithm
is then:
Algorithm 3 (Edit distance)
1 begin initialize A,x,y,m← length[x], n← length[y]
2 C[0, 0] ← 0
3 i← 0
4 do i← i+ 1
5 C[i, 0] ← i
6 until i = m
7 j ← 0
8 do j ← j + 1
9 C[0, j] ← j
10 until j = n
11 i← 0; j ← 0
12 do i← i+ 1
13 do j ← j + 1
14 C[i, j] = min
[
C[i− 1, j] + 1︸ ︷︷ ︸
insertion
,C[i, j − 1] + 1︸ ︷︷ ︸
deletion
,C[i− 1, j − 1] + 1 − δ(x[i],y[j])︸ ︷︷ ︸
no change/exchange
]
15 until j = n
16 until i = m
17 return C[m,n]
18 end
Lines 4 – 10 initialize the left column and top row of C with the integer number
of “steps” away from i = 0, j = 0. The core of this algorithm, line 14, finds the
minimum cost in each entry of C, column by column (Fig. 8.9). Algorithm 3 is
thus greedy in that each column of the distance or cost matrix is filled using merely
the costs in the previous column. Linear programming techniques can also be used
to find a global minimum, though this nearly always requires greater computational
effort (Problem 27).
If insertions and deletions are equally costly, then the symmetry property of a
metric holds. However, we can broaden the applicability of the algorithm by allowing
in line 14 different costs for the fundamental operations; for example insertions might
cost twice as much as substitutions. In such a broader case, properties of symmetry
and the triangle inequality no longer hold and edit distance is not a true metric
(Problem 28).
As shown in Fig. 8.9, x = “excused” can be transformed to y = “exhausted”
through one substitution and two insertions. The table shows the steps of this trans-
formation, along with the computed entries of the cost matrix C. For the case shown,
where each fundamental operation has a cost of 1, the edit distance is given by the
value of the cost matrix at the sink, i.e., C[7,9] = 3.
8.5. *RECOGNITION WITH STRINGS 29
deletion:
remove letter of x
insertion:
insert letter of y into x
exchange:
replace letter of x by letter of y
no change
e x h a u s t e d
e
x
c
u
s
e
d
x
y
source
sink
j
i
0
0 n
m
0 1 2 3 4 5 6 7 8 9
1 0 1 2 3 4 5 6 7 8
2 1 0 1 2 3 4 5 6 7
3 2 1 1 2 3 4 5 6 7
4 3 2 2 2 2 3 4 5 6
5 4 3 3 3 3 2 3 4 5
6 5 4 4 4 4 3 3 3 4
7 6 5 5 5 5 4 4 4 3
Figure 8.9: The edit distance calculation for strings x and y can be illustrated in
a table. Algorithm 3 begins at source, i = 0, j = 0, and fills in the cost matrix C,
column by column (shown in red), until the full edit distance is placed at the sink,
C[i = m, j = n]. The edit distance between “excused” and “exhausted” is thus 3.
x excused source string C[0,0] = 0
exhused substitute h for c C[3,3] = 1
exhaused insert a C[3,4] = 2
exhausted insert t C[5,7] = 3
y exhausted target string C[7,9] = 3
8.5.3 Computational complexity
Algorithm 3 is O(mn) in time, of course; it is O(m) in space (memory) since only the
entries in the previous column need be stored when computing C[i, j] for i = 0 to m.
Because of the importance of string matching and edit distance throughout computer
science, a number of algorithms have been proposed. We need not delve into the
details here (but see the Bibliography) except to say that there are sophisticated
string-matching algorithms with time complexity O(m+ n).
8.5.4 String matching with errors
There are several versions of the string-matching-with-errors problem; the one that
concerns us is this: given a pattern x and text, find the shift for which the edit
distance between x and a factor of text is minimum. The algorithm for the string-
matching-with-errors problem is very similar to that for edit distance. Let E be a
matrix of costs, analogous to C in Algorithm 3. We seek a shift for which the edit
distance to a factor of text is minimum, or formally min[C(x,y)] where y is any factor
of text. To this end, the algorithm must compute its new cost E whose entries are
E[i, j] = min[C(x[1…i],y[1…j])].
The principal difference between the algorithms for the two problems (i.e., with
or without errors) is that we initialize E[0,j] to 0 in the string matching with errors
problem, instead of to j in lines 4 – 10 of the basic string matching algorithm. This
initialization of E expresses the fact that the “empty” prefix of x matches an empty
factor of text, and contributes no cost.
30 CHAPTER 8. NON-METRIC METHODS
s = 11
h e _ p a c e _ t r i c t u r e s _ i n _ t
s t r u t r
sl d
c u e
character mismatch
best pattern match:
one character mismatch
edit distance = 1
Figure 8.10: The string matching with errors problem is to find the shift s for which the
edit distance between x and an aligned factor of text is minimum. In this illustration,
the minimum edit distance is 1, corresponding to the character exchange u → i and
the shift s = 11 is the location.
Two minor heuristics for reducing computational effort are relevant to the string-
matching-with-errors problem. The first is that except in highly unusual cases, the
length of the candidate factors of text that need be considered are roughly equal
to length[x]. Second, for each candidate shift, the edit-distance calculation can be
terminated if it already exceeds the current minimum. In practice, this latter heuris-
tic can reduce the computational burden significantly. Otherwise, the algorithm for
string matching with errors is virtually the same as that for edit distance (Computer
exercise 10).
8.5.5 String matching with the “don’t-care” symbol
String matching with the “don’t-care” symbol, ©/ , is formally the same as basic string
matching, but the ©/ in either x or text is said to match any character (Fig. 8.11).
s
c h _ p a t e r s i n _ l o n g s t rr n g
p a t t r s
/
/
/
/
_
pattern match
text
x
/ /
Figure 8.11: String matching with don’t care symbol is the same as basic string
matching except the ©/ symbol — in either text or x — matches any character. The
figure shows the only valid shift.
An obvious approach to string matching with the don’t care symbol is to modify
the naive string-matching algorithm to include a condition for matching the don’t
care symbol. Such an approach, however, retains the computational inefficiencies of
naive string matching (Problem 29). Further, extending the Boyer-Moore algorithm
to include ©/ is somewhat difficult and inefficient. The most effective methods are
based on fundamental methods in computer arithmetic and, while fascinating, would
take us away from our central concerns of pattern recognition (cf. Bibliography). The
use of this technique in pattern recognition is the same as string matching, with a
particular type of “tolerance.”
While learning is a general and fundamental technique throughout pattern recog-
8.6. GRAMMATICAL METHODS 31
nition, it has found limited use in recognition with basic string matching. This is
because the designer typically knows precisely which strings are being sought — they
do not need to be learned. Learning can, of course, be based on the outputs of a
string-matching algorithm, as part of a larger pattern recognition system.
8.6 Grammatical methods
Up to here, we have not been concerned with any detailed models that might underly
the generation of the sequence of characters in a string. We now turn to the case
where rules of a particular sort were used to generate the strings and thus where their
structure is fundamental. Often this structure is hierarchical, where at the highest
or most abstract level a sequence is very simple, but at subsequent levels there is
greater and greater complexity. For instance, at its most abstract level, the string
“The history book clearly describes several wars” is merely a sentence. At
a somewhat more detailed level it can be described as a noun phrase followed by a
verb phrase. The noun phrase can be expanded at yet a subsequent level, as can the
verb phrase. The expansion ends when we reach the words “The,” “history,” and
so forth — items that are considered the “characters,” atomic and without further
structure. Consider too strings representing valid telephone numbers — local, national
and international. Such numbers conform to a strict structure: either a country code
is present or it is not; if not, then the domestic national code may or may not be
present; if a country code is present, then there is a set of permissible city codes and
for each city there is a set of permissible area codes and individual local numbers, and
so on.
As we shall see, such structure is easily specified in a grammar, and when such
structure is present the use of a grammar for recognition can improve accuracy. For in-
stance, grammatical methods can be used to provide constraints for a full system that
uses a statistical recognizer as a component. Consider an optical character recogni-
tion system that recognizes and interprets mathematical equations based on a scanned
pixel image. The mathematical symbols often have specific “slots” that can be filled
with certain other symbols; this can be specified by a grammar. Thus an integral sign
has two slots, for upper and lower limits, and these can be filled by only a limited set
of symbols. (Indeed, a grammar is used in many mathematical typesetting programs
in order to prevent authors from creating meaningless “equations.”) A full system
that recognizes the integral sign could use a grammar to limit the number of candi-
date categories for a particular slot, and this increases the accuracy of the full system.
Similarly, consider the problem of recognizing phone numbers within acoustic speech
in an automatic dialing application. A statistical or Hidden-Markov-Model acoustic
recognizer might perform word spotting and pick out number words such as “eight”
and “hundred.” A subsequent stage based on a formal grammar would then exploit
the fact that telephone numbers are highly constrained, as mentioned.
We shall study the case when crisp rules specify how the representation at one
level leads to a more expanded and complicated representation at the next level. We
sometimes call a string generated by a set of rules a sentence; the rules are specified sentence
by a grammar, denoted G. (Naturally, there is no requirement that these be related
in any way to sentences in natural language such as English.) In pattern recognition,
we are given a sentence and a grammar, and seek to determine whether the sentence
was generated by G.
32 CHAPTER 8. NON-METRIC METHODS
8.6.1 Grammars
The notion of a grammar is very general and powerful. Formally, a grammar G
consists of four components:
symbols: Every sentence consists of a string of characters (which are also called
primitive symbols, terminal symbols or letters), taken from an alphabet A. For
bookkeeping, it is also convenient to include the null or empty string denoted ),null
string which has length zero; if ) is appended to any string x, the result is again x.
variables: These are also called non-terminal symbols, intermediate symbols or oc-
casionally internal symbols, and are taken from a set I.
root symbol: The root symbol or starting symbol is a special internal symbol, theroot
symbol source from which all sequences are derived. The root symbol is taken from a
set S.
productions: The set of production rules, rewrite rules, or simply rules, denoted P,production
rule specify how to transform a set of variables and symbols into other variables and
symbols. These rules determine the core structures that can be produced by the
grammar. For instance if A is an internal symbol and c a terminal symbol, the
rewrite rule cA→ cc means that any time the segment cA appears in a string,
it can be replaced by cc.
Thus we denote a general grammar by its alphabet, its variables, its particular root
symbol, and the rewrite rules: G = (A, I,S,P). The language generated by gram-language
mar, denoted L(G), is the set of all strings (possibly infinite in number) that can be
generated by G.
Consider two examples; the first is quite simple and abstract. Let A = {a, b, c},
S = S, I = {A,B,C}, and P =
p1: S → aSBA OR aBA p2: AB → BA
p3: bB → bb p4: bA → bc
p5: cA → cc p6: aB → ab
.
(In order to make the list of rewrite rules more compact, we shall condense rules
having the same left hand side by means of the OR on the right hand side. Thus rule
p1 is a condensation of the two rules S → aSBA and S → aBA.) If we start with S
and apply the rewrite rules in the following orders, we have the following two cases:
root S root S
p1 aBA p1 aSBA
p6 abA p1 aaBABA
p4 abc p6 aabABA
p2 aabBAA
p3 aabbAA
p4 aabbcA
p5 aabbcc
After the rewrite rules have been applied in these sequences, no more symbols match
the left-hand side of any rewrite rule, and the process is complete. Such a trans-
formation from the root symbol to a final string is called a production. These twoproduction
productions show that abc and aabbcc are in the language generated by G. In fact,
it can be shown (Problem 38) that this grammar generates the language L(G) =
{anbncn|n ≥ 1}.
8.6. GRAMMATICAL METHODS 33
A much more complicated grammar underlies the English language, of course. The
alphabet consists of all English words, A = {the, history, book, sold, over, 1000,
copies, . . . }, and the intermediate symbols are the parts of speech: I = {〈noun〉,
〈verb〉, 〈noun phrase〉, 〈verb phrase〉, 〈adjective〉, 〈adverb〉, 〈adverbial phrase〉}.
The root symbol here is S = 〈sentence〉. A restricted set of the production rules in
English includes:
P =
〈sentence〉 → 〈noun phrase〉〈verb phrase〉
〈noun phrase〉 → 〈adjective〉〈noun phrase〉
〈verb phrase〉 → 〈verb phrase〉〈adverbial phrase〉
〈noun〉 → book OR theorem OR . . .
〈verb〉 → describes OR buys OR holds OR . . .
〈adverb〉 → over OR . . .
This subset of the rules of English grammar does not prevent the generation of mean-
ingless sentences, of course. For instance, the nonsense sentence “Squishy green
dreams hop heuristically” can be derived in this subset of English grammar. Fig-
ure 8.12 shows the steps of a production in a derivation tree, where the root symbol derivation
treeis displayed at the top and the terminal symbols at the bottom.
The
history
sold
over
copies
1000
book
Figure 8.12: This derivation tree illustrates how a portion of English grammar can
transform the root symbol, here 〈sentence〉, into a particular sentence or string of
elements, here English words, which are read from left to right.
8.6.2 Types of string grammars
There are four main types of grammar, arising from different types of structure in the
productions. As we have seen, a rewrite rule is of the form α→ β, where α and β are
strings made up of intermediate and terminal symbols.
Type 0: Free or unrestricted Free grammars have no restrictions on the rewrite
rules and thus they provide no constraints or structure on the strings they can
34 CHAPTER 8. NON-METRIC METHODS
produce. While in principle they can express an arbitrary set of rules, this
generality comes at the tremendous expense of possibly unbounded learning
time. Knowing that a string is derived from a type 0 grammar provides no
information and as such, type 0 grammars in general have but little use in
pattern recognition.
Type 1: Context-sensitive A grammar is called context-sensitive if every rewrite
rule is of the form
αIβ → αxβ
where α and β are any strings made up of intermediate and terminal symbols,
I is an intermediate symbol and x is an intermediate or terminal symbol (other
than )). We say that “I can be rewritten as x in the context of α on the left
and β on the right.”
Type 2: Context-free A grammar is called context free if every production is of
the form
I → x
where I is an intermediate symbol and x an intermediate or terminal symbol
(other than )). Clearly, unlike a type 1 grammar, here there is no need for a
“context” for the rewriting of I by x.
Type 3: Finite State or Regular A grammar is called regular if every rewrite rule
is of the form
α→ zβ OR α→ z
where α and β are made up of intermediate symbols and z is a terminal symbol
(other than )). Such grammars are also called finite state because they can be
generated by a finite state machine, which we shall see in Fig. 8.16.
A language generated by a grammar of type i is called a type i language. It can be
shown that the class of grammars of type i includes all grammars of type i+ 1; thus
there is a strict hierarchy in grammars.
Any context-free grammar can be converted into one in Chomsky normal form
(CNF). Such a grammar has all rules of the formChomsky
normal
form A→ BC and A→ z
where A, B and C are intermediate symbols (that is, they are in I) and z is a terminal
symbol. For every context-free grammar G, there is another G′ in Chomsky normal
form such that L(G) = L(G′) (Problem 36).
Example 3: A grammar for pronouncing numbers
In order to understand these issues better, consider a grammar that yields pro-
nunciation of any number between 1 and 999,999. The alphabet has 29 basic terminal
symbols, i.e., the spoken words
A = {one, two, . . . , ten, eleven, . . . , twenty, thirty, . . . , ninety, hundred, thousand}.
8.6. GRAMMATICAL METHODS 35
There are six non-terminal symbols, corresponding to general six-digit, three-digit,
and two-digit numbers, the numbers between ten and nineteen, and so forth, as will
be clear below:
I = {digits6, digits3, digits2, digit1, teens, tys}.
The root node corresponds to a general number up to six digits in length:
S = digits6.
The set of rewrite rules is based on a knowledge of English:
P =
digits6 → digits3 thousand digits3
digits6 → digits3 thousand OR digits3
digits3 → digit1 hundred digits2
digits3 → digit1 hundred OR digits2
digits2 → teens OR tys OR tys digit1 OR digit1
digit1 → one OR two OR . . . nine
teens→ ten OR eleven OR . . . nineteen
tys→ twenty OR thirty OR . . . OR ninety
The grammar takes digit6 and applies the productions until the elements in the
final alphabet are produced, as shown in the figure. Because it contains rewrite rules
such as digits6 → digits3 thousand, this grammar cannot be type 3. It is easy to
confirm that it is type 2.
digits3 thousand digits3
digit1 hundred digits2
tys digit1
thirty nine
six
digits2
teens
fourteen
digit6
digits3 thousand digits3
digit1
two
639,014 2,953
digit1 hundred digits2
tys digit1
fifty three
nine
digit6
These two derivation trees show how the grammar G yields the pronunciation of
639,014 and 2,953. The final string of terminal symbols is read from left to right.
8.6.3 Recognition using grammars
Recognition using grammars is formally very similar to the general approaches used
throughout pattern recognition. Suppose we suspect that a test sentence was gen-
erated by one of c different grammars, G1, G2, . . . , Gc, which can be considered as
different models or classes. A test sentence x is classified according to which gram-
mar could have produced it, or equivalently, the language L(Gi) of which x is a
member.
Up to now we have worked forward — forming a derivation from a root node to
a final sentence. For recognition, though, we must employ the inverse process: that
is, given a particular x, find a derivation in G that leads to x. This process, called
parsing, is virtually always much more difficult than forming a derivation. We now parsing
36 CHAPTER 8. NON-METRIC METHODS
discuss one general approach to parsing, and briefly mention two others.
Bottom-up parsing
Bottom-up parsing starts with the test sentence x, and seeks to simplify it, so as to
represent it as the root symbol. The basic approach is to use candidate productions
from P “backwards,” i.e., find rewrite rules whose right hand side matches part of
the current string, and replace that part with a segment that could have produced it.
This is the general method in the Cocke-Younger-Kasami algorithm, which fills a parse
table from the “bottom up.” The grammar must be expressed in Chomsky normalparse
table form and thus the productions P must all be of the form A → BC, a broad but
not all inclusive category of grammars. Entries in the table are candidate strings in a
portion of a valid derivation. If the table contains the source symbol S, then indeed
we can work forward from S and derive the test sentence, and hence x ∈ L(G). In
the following, xi (for i = 1, . . . n) represents the individual terminal characters in the
string to be parsed.
Algorithm 4 (Bottom-up parsing)
1 begin initialize G = (A, I, S,P),x = x1x2 . . . xn
2 i← 0
3 do i← i+ 1
4 Vi1 ← {A | A→ xi}
5 until i = n
6 j ← 1
7 do j ← j + 1
8 i← 0
9 do i← i+ 1
10 Vij ← ∅
11 k ← 0
12 do k ← k + 1
13 Vij ← Vij ∪ {A | A→ BC ∈ P, B ∈ Vik and C ∈ Vi+k,j−k}
14 until k = j − 1
15 until i = n− j + 1
16 until j = n
17 if S ∈ V1n then print “parse of” x “successful in G”
18 return
19 end
Consider the operation of Algorithm 4 in the following simple abstract example.
Let the grammar G have two terminal and three intermediate symbols: A = {a, b},
and I = {A,B,C}. The root symbol is S, and there are just four production rules:
P =
p1 : S → AB OR BC
p2 : A → BA OR a
p3 : B → CC OR b
p4 : C → AB OR a
.
Figure 8.13 shows the parse table generated by Algorithm 4 for the input string x
= “baaba.” Along the bottom are the characters xi of this string. Lines 2 through
5 of the algorithm fill in the first (j = 1) row with any internal symbols that derive
the corresponding character in x. The i = 1 and i = 4 entries of that bottom row are
filled with B, since rewrite rule p3: B → b. Likewise the remaining entries are filled
with both A and C, as a result of rewrite rules p2 and p4.
8.6. GRAMMATICAL METHODS 37
The core computation in the algorithm is performed in line 13, which fills entries
throughout the table with symbols that could produce segments in lower rows, and
hence might be part of a valid derivation (if indeed one is found). For instance,
the i = 1, j = 2 entries contain any symbols that could produce segments in the
row beneath it. Thus this entry contains S because by rule p1 : S → BC, and also
contains A because by rule p2 : A → BA. According to the innermost loop over k
(lines 12 – 14), we seek the left hand side for rules that span a range. For instance,
the i = 3, j = 3 entry contains B because for k = 2 and rule p3: B → CC (as shown
in Fig. 8.14).
B
i
j
5
4
3
2
1
1 2 3 4 5
A,C A,C B A,C
S,A B S,C S,A
B B
S,A,C
S,A,C
0
0
b a a b a
strings of length 1
strings of length 2
strings of length 3
strings of length 4
strings of length 5
target string x
Figure 8.13: The bottom-up parsing algorithm fills the parse table with symbols that
might be part of a valid derivation. The pink lines are not provided by the algorithm,
but when read downward from the root symbol confirm that a valid derivation exists.
Figure 8.14 shows the cells that are searched when filling a particular cell in the
parse table. The sequence sweeps vertically up to the cell in question, while diagonally
down from the cell in question; this guarantees that the all paths from the top cell
in a valid derivation can be found. If the top cell contains the root symbol S (and
possibly other symbols), then indeed the string is successfully parsed. That is, there
exists a valid production leading from S to the target string x.
To understand how this table is filled, consider first the j = 1 row. The j = 4, i = 1
cell contains B, because according to rewrite rule p3, B is the only intermediate
symbol that could yield b in the query sentence, directly below. The same logic holds
for the i = 1, j = 1 cell. The remaining three cells for j = 1 contain A and C,
since these are the only intermediate variables that can derive a. Incidentally, the
derivation in Fig. 8.15 confirms that the parse is valid.
The computational complexity of bottom-up parsing performed by Algorithm 4 is
high. The innermost loop of line 13 is executed n or fewer times, while lines 7 & 9
are O(n2), which is also the space complexity. The time complexity is O(n3).
Top-down and other methods of parsing
As its name suggests, top-down parsing starts with the root node and successively
applies productions from P, with the goal of finding a derivation of the test sentence
x. Since it is rare that the sentence is derived in a single production, it is necessary to
specify some criteria to guide the choice of which rewrite rule to apply. Such criteria
could include beginning the parse at the first (left) character in the sentence (that
is, finding a small set of rewrite rules that yield the first character), then iteratively
expanding the production to derive subsequent characters.
38 CHAPTER 8. NON-METRIC METHODS
V
i,k
V
i+1,j-1
V
ij
k=1
V
i,k
V
i+2,j-2
V
ij
k=2
V
i,k
V
i+3,j-3
V
ij
k=3
V
i,k
V
i+4,j-4
V
ij
k=4
i
j
i
j
i
j
i
j
Figure 8.14: The innermost loop of Algorithm 4 seeks to fill a cell Vij (outlined in
red) by the left-hand side of any rewrite rule whose right-hand side corresponds to
symbols in the two shaded cells. As k is incremented, the cells queried move vertically
upward to the cell in question, and diagonally down from that cell. The shaded cells
show the possible right-hand sides in a derivation, as illustrated by the pink lines in
Fig. 8.13.
b
S
A B
AB
a
CC
a
a
BA
b
Figure 8.15: This valid derivation of “babaa” in G can be read from the pink lines in
the parse table of Fig. 8.13 generated by the bottom-up parse algorithm.
The bottom-up and top-down parsers just described are quite general and there
are a number of parsing algorithms which differ in space and time complexities. Many
parsing methods depend upon the model underlying the grammar. One popular such
model is finite state machines. Such a machine consists of nodes and transition links;finite
state
machine
each node can emit a symbol, as shown in Fig. 8.16.
8.7 Grammatical inference
In many applications, the grammar is designed by hand. Nevertheless, learning plays
an extremely important role in pattern recognition research and it is natural that
we attempt to learn a grammar from example sentences it generates. When seeking
to follow that general approach we are immediately struck by differences between
the areas addressed by grammatical methods and those that can be described as
statistical. First, for most languages there are many — often an infinite number
of — grammars that can produce it. If two grammars G1 and G2 generate the
same language (and no other sentences), then this ambiguity is of no consequence;
recognition will be the same. However, since training is always based on a finite set
of samples, the problem is underspecified. There are an infinite number of grammars
consistent with the training data, and thus we cannot recover the source grammar
uniquely.
8.7. GRAMMATICAL INFERENCE 39
S A
the
mouse
cow
B
was
C
found
seen
D
by
under
E
the
F
barn
farmer
G
Figure 8.16: One type of finite state machine consists of nodes that can emit terminal
symbols (“the,” “mouse,” etc.) and transition to another node. Such operation can be
described by a grammar. For instance, the rewrite rules for this finite state machine
include S → theA, A→ mouseB OR cowB, and so on. Clearly these rules imply this
finite state machine implements a type 3 grammar. The final internal node (shaded)
would lead to the null symbol ).
There are two main techniques used to make the problem of inferring a grammar
from instances tractable. The first is to use both positive and negative instances. That
is, we use a set D+ of sentences known to be derivable in the grammar; we also use
a set D− that are known to be not derivable in the grammar. In a multicategory
case, it is common to take the positive instances in Gi and use them for negative
examples in Gj for j �= i. Even with both positive and negative instances, a finite
training set rarely specifies the grammar uniquely. Thus our second technique is to
impose conditions and constraints. A trivial illustration is that we demand that the
alphabet of the candidate grammar contain only those symbols that appear in the
training sentences. Moreover, we demand that every production rule in the grammar
be used. We seek the “simplest” grammar that explains the training instances where
“simple” generally refers to the total number of rewrite rules, or the sum of their
lengths, or other natural criterion. These are versions of Occam’s razor, that the
simplest explanation of the data is to be preferred (Chap ??).
In broad overview, learning proceeds as follows. An initial grammar G0 is guessed.
Often it is useful to specify the type of grammar (1, 2 or 3), and thus place constraints
on the forms of the candidate rewrite rules. In the absence of other prior information,
it is traditional to make G0 as simple as possible and gradually expand the set of
productions as needed. Positive training sentences x+i are selected from D+ one by
one. If x+i cannot be parsed by the grammar, then new rewrite rules are proposed
for P. A new rule is accepted if and only if it is used for a successful parse of x+i and
does not allow any negative samples to be parsed.
In greater detail, then, an algorithm for inferring the grammar is:
Algorithm 5 (Grammatical inference (overview))
1 begin initialize D+,D−, G0
2 n+ ← |D+ | (number of instances in D+)
3 S ← S
4 A ← set of characters in D+
5 i← 0
6 do i← i+ 1
7 read x+i from D+
8 if x+i cannot be parsed by G
9 then do propose additional productions to P and variables to I
10 accept updates if G parses x+i but no string in D−
11 until i = n+
12 eliminate redundant productions
40 CHAPTER 8. NON-METRIC METHODS
13 return G← {A, I,S,P}
14 end
Informally, Algorithm 5 continually adds new rewrite rules as required by the
successive sentences selected from D+ so long as the candidate rewrite rule does not
allow a sentence in D− to be parsed. Line 9 does not state how to choose the specific
candidate rewrite rule, but in practice the rule may be chosen from a predefined set
(with simpler rules selected first), or based on specific knowledge of the underlying
models generating the sentences.
Example 4: Grammar inference
Consider inferring a grammarG from the following positive and negative examples:
D+ = {a, aaa, aaab, aab}, and D− = {ab, abc, abb, aabb}. Clearly the alphabet of G
is A = {a, b}. We posit a single internal symbol for G0, and the simplest rewrite rule
P = {S → A}.
i x+i P P produces D− ?
1 a
S → A
A → a No
2 aaa
S → A
A → a
A → aA
No
3 aaab
S → A
A → a
A → aA
A → ab
Yes: ab ∈ D−
3 aaab
S → A
A → a
A → aA
A → aab
No
4 aab
S → A
A → a
A → aA
A → aab
No
The table shows the progress of the algorithm. The first positive instance, a,
demands a rewrite rule A → a. This rule does not allow any sentences in D− to be
derived, and thus is accepted for P. When i = 3, the proposed rule A → ab indeed
allows x+3 to be derived, but the rule is rejected because it also derives a sentence
in D−. Instead, the next proposed rule, A → aab is accepted. The final grammar
inferred has the four rewrite rules shown at the bottom of the table.
The method of grammatical inference just described is quite general. It is made
more specialized by placing restrictions on the types of candidate rewrite rules, cor-
responding to the designer’s assumptions about the type of grammar (1, 2 or 3). For
a type 3 grammar, we can consider learning in terms of the finite state machine. In
that case, learning consists of adding nodes and links (cf. Bibliography).
8.8. *RULE-BASED METHODS 41
8.8 *Rule-based methods
In problems where classes can be characterized by general relationships among entities,
rather than by instances per se, it becomes attractive to build classifiers based on rules.
Rule-based methods are integral to expert systems in artificial intelligence, but since
they have found only modest use in pattern recognition, we shall give merely a short
overview. We shall focus on a broad class of if-then rules for representing and learning if-then
rulesuch relationships.
A very simple if-then rule is
IF Swims(x) AND HasScales(x) THEN Fish(x),
which means, of course, that if an object x has the property that it swims, and the
property that it has scales, then it is a fish. Rules have the great benefits that they
are easily interpreted and can be used in database applications where information is
encoded in relations. A drawback is that there is no natural notion of probability and
it is somewhat difficult, therefore, to use rules when there is high noise and a large
Bayes error.
A predicate, such as Man(·), HasTeeth(·) and AreMarried(·,·), is a test that predicate
returns a value of logical True or False.∗ Such predicates can apply to problems where
the data are numerical non-numerical, linguistic, strings, or any of a broad class of
types. The choice of predicates and their evaluation depend strongly on the problem,
of course, and in practice these are generally more difficult tasks than learning the
rules. For instance, Fig. 8.17 below illustrates the use of rules in categorizing a
structure as an arch. Such a rule might involve predicates such as Touch(·, ·) or
Supports(·, ·, ·) which address whether two blocks touch, or whether two blocks
support a third. It is a very difficult problem in computer vision to evaluate such
predicates based on a pixel image taken of the scene.
There are two main types of if-then rules: propositional (variable-free) and first- propositional
logicorder. A propositional rule describes a particular instance, as in
first-order
logicIF Male(Bill) AND IsMarried(Bill) THEN IsHusband(Bill),
where Bill is a particular atomic item. Because its properties are fixed, Bill is an
example of a (logical) constant. The deficiency of propositional logic is that it provides constant
no general way to represent general relations among a large number of instances. For
example, even if we knew Male(Edward) and IsMarried(Edward) are both True, the
above rule would not allow us to infer that Edward is is a husband, since that rule
applies only to the particular constant Bill.
This deficiency is overcome in first-order logic, which permit rules with variables, variable
such as
IF Eats(x, y) AND HasFlesh(x) THEN Carnivore(y),
where here x and y are the variables. This rule states that for any items x and y, if
y eats x and x has flesh, then y is a carnivore. Clearly this is a very powerful sum-
mary of an enormous wealth of examples — first-order rules are far more expressive
∗ We shall ignore cases where a predicate is Undefined.
42 CHAPTER 8. NON-METRIC METHODS
than classical propositional logic. The power of first-order logic is illustrated in the
following rules:
IF Male(x) AND IsMarried(y, z) THEN IsHusband(x),
IF Parent(x, y) AND Parent(y, z) THEN GrandParent(x, z)
and
IF Spouse(x, y) THEN Spouse(y, x).
A rule from first-order logic can also apply to constants, for instance:
IF Eats(Mouse, Cat) AND Mammal(Mouse) THEN Carnivore(Cat),
where Cat and Mouse are two particular constants.
If-then rules can also incorporate functions, which return numerical values, asfunction
illustrated in the following:
IF Male(x) AND (Age(x) < 16) THEN Boy(x),
where (Age(x) is a function that returns a numerical age in years while the expression
or term (Age(x) < 16) returns either logical True or False. In sum, the above ruleterm
states that a male younger than 16 years old is a boy. If we were to use decision trees
or statistical techniques, we would not be able to learn this rule perfectly, even given
a tremendously large number of examples.
It is clear given a set of first-order rules how to use them in pattern classification:
we merely present the unknown item and evaluate the propositions and rules. Thus
consider the long rule
IF IsBlock(x) AND IsBlock(y) AND IsBlock(z)
AND Touch(x, y) AND Touch(x, z) AND NotTouch(y, z) (11)
AND Supports(x, y, z) THEN Arch(x, y, z),
where Supports(x,y,z) means that x is supported by both y and z. We stress that
designing algorithms to implement IsBlock(·), Supports(·,·,·) and so on can be
extremely difficult; there is little we can say about them here other than that nearly
always building these component algorithms represents the greatest effort in designing
the overall classifier. Nevertheless, given reliable such algorithms, the rule could be
used to classify simple structures as an arch or non-arch (Fig. 8.17).
8.8.1 Learning rules
Now we turn briefly to the learning of such if-then rules. We have already seen several
ways to learn rules. For instance, we can train a decision tree via CART, ID3, C4.5
or other algorithm, and then simplify the tree to extract rules (Sect. 8.4). For cases
where the underlying data arises from a grammar, we can infer the particular rules via
the methods in Sect. 8.7. A key distinction in the approach we now discuss is that they
can learn sets of first-order rules containing variables. As in grammatical inference,
our approach to learning rules from a set of positive and negative examples, D+ and
8.8. *RULE-BASED METHODS 43
y z
x
y
x
y z
x
z
Figure 8.17: The rule in Eq. 11 identifies the figure on the left as an example of Arch,
but not the other two figures. In practice, it is very difficult to develop subsystems that
evaluate the propositions themselves, for instance Touch(x,y) and Supports(x,y,z).
D−, is to learn a single rule, delete the examples that it explains, and iterate. Such
sequential covering learning algorithms lead to a disjunctive set of rules that “cover”sequential
covering the training data. After such training it is traditional to simplify the resulting logical
rule by means of standard logical methods.
The designer must specify the predicates and functions, based on a prior knowledge
of the problem domain. The algorithm begins by considering the most general rules
using these predicates and functions, and finds the “best” simple rule. Here, “best”
means that the rule describes the largest number of training examples. Then, the
algorithm searches among all refinements of the best rule, choosing the refinement
that too is “best.” This process is iterated until no more refinements can be added,
or when the number of items described is maximum. In this way a single, though
possibly complex, if-then rule has been learned (Fig. 8.18). The sequential covering
algorithm iterates this process and returns a set of rules. Because of its greedy nature,
the algorithm need not learn the smallest set of rules.
IF Swims(x)
THEN Fish(x)=T
IF (Width(x)>2m)
THEN Fish(x)=F
IF HasHair(x)
THEN Fish(x)=F
IF Runs(x)
THEN Fish(x)=F
IF HasEyes(x)
THEN Fish(x)=T
IF Swims(x)
HasHair(x)
THEN Fish(x)=F
IF Swims(x)
Runs(x)
THEN Fish(x)=F
IF Swims(x)
LaysEggs(x)
THEN Fish(x)=T
IF Swims(x)
HasScales(x)
THEN Fish(x)=T
IF Swims(x)
(Weight(x)>9kg)
THEN Fish(x)=F
IF Swims(x)
HasScales(x)
HasEyes(x)
THEN Fish(x)=T
IF Swims(x)
HasScales(x)
HasGills(x)
THEN Fish(x)=T
IF Swims(x)
HasScales(x)
(Length(x)>5m)
THEN Fish(x)=F
IF
THEN Fish(x)=T
Figure 8.18: In sequential covering, candidate rules are searched through successive
refinements. First, the “best” rule having a single conditional predicate is found, i.e.,
the one explaining most training data. Next, other candidate predicates are added,
the best compound rule selected, and so forth.
A general approach is to search first through all rules having a single attribute.
44 CHAPTER 8. NON-METRIC METHODS
Next, consider the rule having a single conjunction of two predicates, then these
conjunctions, and so on. Note that this greedy algorithm need not be optimal — that
is, it need not yield the most compact rule.
Summary
Non-metric data consists of lists of nominal attributes; such lists might be unordered
or ordered (strings). Tree-based methods such as CART, ID3 and C4.5 rely on answers
to a series of questions (typically binary) for classification. The designer selects the
form of question and the tree is grown, starting at the root node, by finding splits
of data that make the representation more “pure.” There are several acceptable
impurity measures, such as misclassification, variance and Gini; the entropy impurity,
however, has found greatest use. To avoid overfitting and to improve generalization,
one must either employ stopped splitting (declaring a node with non-zero impurity to
be a leaf), or instead prune a tree trained to minimum impurity leafs. Tree classifiers
are very flexible, and can be used in a wide range of applications, including those with
data that is metric, non-metric or in combination.
When comparing patterns that consist of strings of non-numeric symbols, we use
edit distance — a measure of the number of fundamental operations (insertions, dele-
tions, exchanges) that must be performed to transform one string into another. While
the general edit distance is not a true metric, edit distance can nevertheless be used
for nearest-neighbor classification. String matching is finding whether a test string
appears in a longer text. The requirement of a perfect match in basic string matching
can be relaxed, as in string matching with errors, or with the don’t care symbol. These
basic string and pattern recognition ideas are simple and straightforward, addressing
them in large problems requires algorithms that are computationally efficient.
Grammatical methods assume the strings are generated from certain classes of
rules, which can be described by an underlying grammar. A grammar G consists of
an alphabet, intermediate symbols, a starting or root symbol and most importantly
a set of rewrite rules, or productions. The four different types of grammars — free,
context-sensitive, context-free, and regular — make different assumptions about the
nature of the transformations. Parsing is the task of accepting a string x and deter-
mining whether it is a member of the language generated by G, and if so, finding a
derivation. Grammatical methods find greatest use in highly structured environments,
particularly where structure lies at many levels. Grammatical inference generally uses
positive and negative example strings (i.e., ones in the language generated by G and
ones not in that language), to infer a set of productions.
Rule-based systems use either propositional logic (variable-free) or first-order logic
to describe a category. In broad overview, rules can be learned by sequentially “cov-
ering” elements in the training set by successively more complex compound rules.
Bibliographical and Historical Remarks
Most work on decision trees addresses problems in continuous features, though a
key property of the method is that they apply to nominal data too. Some of the
foundations of tree-based classifiers stem from the Concept Learning System described
in [42], but the important book on CART [10] provided a strong statistics foundation
and revived interest in the approach. Quinlan has been a leading exponent of tree
classifiers, introducing ID3 [66], C4.5 [69], as well as the application of minimum
8.8. BIBLIOGRAPHICAL AND HISTORICAL REMARKS 45
description length for pruning [71, 56]. A good overview is [61], and a comparison
of multivariate decision tree methods is given in [11]. Splitting and pruning criteria
based on probabilities are explored in [53], and the use of an interesting information
metric for this end is described in [52]. The Gini index was first used in analysis
of variance in categorical data [47]. Incremental or on-line learning in decision trees
is explored in [85]. The missing variable problem in trees is addressed in [10, 67],
which describe methods more general than those presented here. An unusual parallel
“neural” search through trees was presented in [78].
The use of edit distance began in the 1970s [64]; a key paper by Wagner and Fischer
proposed the fundamental Algorithm 3 and showed that it was optimal [88]. The
explosion of digital information, especially natural language text, has motivated work
on string matching and related operations. An excellent survey is [5] and two thorough
books are [23, 82]. The computational complexity of string algorithms is presented
in [21, Chapter 34]. The fast string matching method of Algorithm 2 was introduced
in [9]; its complexity and speedups and improvements were discussed in [18, 35, 24,
4, 40, 83]. String edit distance that permits block-level transpositions is discussed
in [48]. Some sophisticated string operations — two-dimensional string matching,
longest common subsequence and graph matching — have found only occasionally
use in pattern recognition. Statistical methods applied to strings are discussed in
[26]. Finite-state automata have been applied to several problems in string matching
[23, Chapter 7], as well as time series prediction and switching, for instance converting
from an alphanumeric representation to a binary representation [43]. String matching
has been applied to the recognition DNA sequences and text, and is essential in most
pattern recognition and template matching involving large databases of text [14].
There is a growing literature on special purpose hardware for string operations, of
which the Splash-2 system [12] is a leading example.
The foundations of a formal study of grammar, including the classification of
grammars, began with the landmark book by Chomsky [16]. An early exposition
of grammatical inference [39, Chapter 6] was the source for much of the discussion
here. Recognition based on parsing (Latin pars orationis or “part of speech”) has
been fundamental in automatic language recognition. Some of the earliest work on
three-dimensional object recognition relied on complex grammars which described
the relationships of corners and edges, in block structures such arches and towers.
It was found that such systems were very brittle; they failed whenever there were
errors in feature extraction, due to occlusion and even minor misspecifications of the
model. For the most part, then, grammatical methods have been abandoned for object
recognition and scene analysis [60, 25]. Grammatical methods have been applied to
the recognition of some simple, highly structured diagrams, such as electrical circuits,
simple maps and even Chinese/Japanese characters. For useful surveys of the basic
ideas in syntactic pattern recognition see [33, 34, 32, 13, 62, 14], for parsing see [28, 3],
for grammatical inference see [59]. The complexity of parsing type 3 is linear in the
length of the string, type 2 is low-order polynomial, type 1 is exponential; pointers to
the relevant literature appear in [76]. There has been a great deal of work on parsing
natural language and speech, and a good textbook on artificial intelligence addressing
this topic and much more is [75]. There is much work on inferring grammars from
instances, such as Crespi-Reghizzi algorithm (context free) [22]. If queries can be
presented interactively, the learning of a grammar can be speeded [81].
The methods described in this chapter have been expanded to allow for stochastic
grammars, where there are probabilities associated with rules [20]. A grammar can
be considered a specification of a prior probability for a class; for instance, a uniform
46 CHAPTER 8. NON-METRIC METHODS
prior over all (legal) strings in the language L. Error-correcting parsers have been
used when random variations arise in an underlying stochastic grammar [50, 84]. One
can also apply probability measures to languages [8].
Rule-based methods have formed the foundation of expert systems, and have been
applied extensively through many branches of artificial intelligence such as planning,
navigation and prediction; their use in pattern recognition has been modest, however.
Early influential systems include DENDRAL, for inferring chemical structure from
mass spectra [29], PROSPECTOR, for finding mineral deposits [38], and MYCIN,
for medical diagnosis [79]. Early use of rule induction for pattern recognition include
that of Michalski [57, 58]. Figure 8.17 was inspired by Winston’s influential work
on learning simple geometrical structures and relationships [91]. Learning rules can
be called inductive logic programming; Clark and Niblett have made a number of
contributions to the field, particularly their CN2 induction algorithm [17]. Quinlan,
who has contributed much to the theory and application of tree-based classifiers,
describes his FOIL algorithm, which uses a minimum description length criterion to
stop the learning of first-order rules [68]. Texts on inductive logic include [46, 63] and
general machine learning, including inferencing [44, 61].
Problems
⊕
Section 8.2
1. When a test pattern is classified by a decision tree, that pattern is subjected
to a sequence of queries, corresponding to the nodes along a path from root to leaf.
Prove that for any decision tree, there is a functionally equivalent tree in which every
such path consists of distinct queries. That is, given an arbitrary tree prove that
it is always possible to construct an equivalent tree in which no test pattern is ever
subjected to the same query twice.⊕
Section 8.3
2. Consider classification trees that are non-binary.
(a) Prove that for any arbitrary tree, with possibly unequal branching ratios through-
out, there exists a binary tree that implements the same classification function.
(b) Consider a tree with just two levels — a root node connected to B leaf nodes
(B ≥ 2). What are the upper and the lower limits on the number of levels in a
functionally equivalent binary tree, as a function of B?
(c) As in part (b), what are the upper and lower limits on the number of nodes in
a functionally equivalent binary tree?
3. Compare the computational complexities of a monothetic and a polythetic tree
classifier trained on the same data as follows. Suppose there are n/2 training patterns
in each of two categories. Every pattern has d attributes, each of which can take on
k discrete values. Assume that the best split evenly divides the set of patterns.
(a) How many levels will there be in the monothetic tree? The polythetic tree?
(b) In terms of the variables given, what is the complexity of finding the optimal
split at the root of a monothetic tree? A polythetic tree?
8.8. PROBLEMS 47
(c) Compare the total complexities for training the two trees fully.
4. The task here is to find the computational complexity of training a tree classifier
using the twoing impurity where candidate splits are based on a single feature. Sup-
pose there are c classes, ω1, ω2, …, ωc, each with n/c patterns that are d-dimensional.
Proceed as follows:
(a) How many possible non-trivial divisions into two supercategories are there at
the root node?
(b) For any one of these candidate supercategory divisions, what is the computa-
tional complexity of finding the split that minimizes the entropy impurity?
(c) Use your results from parts (a) & (b) to find the computational complexity of
finding the split at the root node.
(d) Suppose for simplicity that each split divides the patterns into equal subsets
and furthermore that each leaf node corresponds to a single pattern. In terms
of the variables given, what will be the expected number of levels of the tree?
(e) Naturally, the number of classes represented at any particular node will depend
upon the level in the tree; at the root all c categories must be considered, while at
the level just above the leaves, only 2 categories must be considered. (The pairs
of particular classes represented will depend upon the particular node.) State
some natural simplifying assumptions, and determine the number of candidate
classes at any node as a function of level. (You may need to use the floor or
ceiling notation, �x or !x”, in your answer, as described in the Appendix.)
(f) Use your results from part (e) and the number of patterns to find the compu-
tational complexity at an arbitrary level L.
(g) Use all your results to find the computational complexity of training the full
tree classifier.
(h) Suppose there n = 210 patterns, each of which is d = 6 dimensional, evenly
divided among c = 16 categories. Suppose that on a uniprocessor a fundamental
computation requires roughly 10−10 seconds. Roughly how long will it take to
train your classifier using the twoing criterion? How long will it take to classify
a single test pattern?
5. Consider training a binary tree using the entropy impurity, and refer to Eqs. 1 &
5.
(a) Prove that the decrease in entropy impurity provided by a single yes/no query
can never be greater than one bit.
(b) For the two trees in Example 1, verify that each split reduces the impurity
and that this reduction is never greater than 1 bit. Explain nevertheless why
the impurity at a node can be lower than at its descendent, as occurs in that
Example.
(c) Generalize your result from part (a) to the case with arbitrary branching ratio
B ≥ 2.
48 CHAPTER 8. NON-METRIC METHODS
6. Let P (ω1), . . . , P (ωc) denote the probabilities of c classes at node N of a binary
classification tree, and
c∑
j=1
P (ωj) = 1. Suppose the impurity i(P (ω1), . . . , P (ωc)) at
N is a strictly concave function of the probabilities. That is, for any probabilities
ia = i(P
a(ω1), . . . , P
a(ωc))
ib = i(P
b(ω1), . . . , P
b(ωc))
and
i∗(α) = i(α1P
a(ω1) + (1 − α1)P b(ω1), . . . , αcP a(ωc) + (1 − αc)P b(ωc)),
then for 0 ≤ αj ≤ 1 and
c∑
j=1
αj = 1, we have
ia ≤ i∗ ≤ ib.
(a) Prove that for any split, we have ∆i(s, t) ≥ 0, with equality if and only if
P (ωj |TL) = P (ωj |TR) = P (ωj |T ), for j = 1, . . . , c. In other words, for a concave
impurity function, splitting never increases the impurity.
(b) Prove that entropy impurity (Eq. 1) is a concave function.
(c) Prove that Gini impurity (Eq. 3) is a concave function.
7. Show that the surrogate split method described in the text corresponds to the
assumption that the missing feature (attribute) is the one most informative.
8. Consider a two-category problem and the following training patterns, each having
four binary attributes:
ω1 ω2
0110 1011
1010 0000
0011 0100
1111 1110
(a) Use the entropy impurity (Eq. 1) to create by hand an unpruned classifier for
this data.
(b) Apply simple logical reduction methods to your tree in order to express each
category by the simplest logical expression, i.e., with the fewest ANDs and ORs.
9. Show that the time complexity of recall in an unpruned, fully trained tree classifier
with uniform branching ratio is O(log n) where n is the number of training patterns.
For uniform branching factor, B, state the exact functional form of the number of
queries applied to a test pattern as a function of B.
10. Consider impurity functions for a two-category classification problem as a func-
tion of P (ω1) (and implicitly P (ω2) = 1−P (ω1)). Show that the simplest reasonable
polynomial form for the impurity is related to the sample variance as follows:
(a) Consider restricting impurity functions to the family of polynomials in P (ω1).
Explain why i must be at least quadratic in P (ω1).
8.8. PROBLEMS 49
(b) Write the simplest quadratic form for P (ω1) given the boundary conditions
i(P (ω1) = 0) = i(P (ω1) = 1) = 0; show that your impurity function can be
written i ∝ P (ω1)P (ω2).
(c) Suppose all patterns in category ω1 are assigned the value 1.0, while all those in
ω2 the value 0.0, thereby giving a bimodal distribution. Show that your impurity
measure is proportional to the sample variance of this full distribution. Interpret
your answer in words.
11. Show how general costs, represented in a cost matrix λij , can be incorporated
into the misclassification impurity (Eq. 4) during the training of a multicategory tree.
12. In this problem you are asked to create tree classifiers for a one-dimensional two-
category problem in the limit of large number of points, where P (ω1) = P (ω2) = 1/2,
p(x|ω1) ∼ N(0, 1) and p(x|ω2) ∼ N(1, 2), and all nodes have decisions of the form “is
x ≤ xs” for some threshold xs. Each binary tree will be small — just a root node plus
two other (non-terminal) nodes and four leaf nodes. For each of the four impurity
measures below, state the splitting criteria (i.e., the value xs at each of the three
non-terminal nodes), as well as the final test error. Whenever possible, express your
answers functionally, possibly using the error function erf(·), as well as numerically.
(a) Entropy impurity (Eq. 1).
(b) Gini impurity (Eq. 3).
(c) Misclassification impurity (Eq. 4).
(d) Another splitting rule is based on the so-called Kolmogorov-Smirnov test. Let Kolmogorov-
Smirnov
test
the cumulative distributions for a single variable x for each categories be Fi(x)
for i = 1, 2. The splitting criterion is the maximum difference in the cumulative
distributions, i.e.,
max
xs
|F1(xs) − F2(xs)|.
(e) Using the methods of Chap. ??, calculate the Bayes decision rule, and the Bayes
error.
13. Repeat Problem 12 but for two one-dimensional Cauchy distributions,
p(x|ωi) =
1
πbi
· 1
1 +
(
x−ai
bi
)2 , i = 1, 2,
with P (ω1) = P (ω2) = 1/2, a1 = 0, b1 = 1, a2 = 1 and b2 = 2. (Here error functions
are not needed.)
14. Generalize the missing attribute problem to the case of several missing features,
and to several deficient patterns. Specifically, write pseudocode for creating a binary
decision tree where each d-dimensional pattern can have multiple missing features.
15. During the growing of a decision tree, a node represents the following six-
dimensional binary patterns:
50 CHAPTER 8. NON-METRIC METHODS
ω1 ω2
110101 011100
101001 010100
100001 011010
101101 010000
010101 001000
111001 010100
100101 111000
011000 110101
Candidate decision are based on single feature values.
(a) Which feature should be used for splitting?
(b) Recall the use of statistical significance for stopped splitting. What is the null
hypothesis in this example?
(c) Calculate chi-squared for your decision in part (a). Does it differ significantly
from the null hypothesis at the 0.01 confidence level? Should splitting be
stopped?
(d) Repeat part (c) for the 0.05 level.
16. Consider the following patterns, each having four binary-valued attributes:
ω1 ω2
1100 1100
0000 1111
1010 1110
0011 0111
Note especially that the first patterns in the two categories are the same.
(a) Create by hand a binary classification tree for this data. Train your tree so that
the leaf nodes have the lowest impurity possible.
(b) Suppose it is known that during testing the prior probabilities of the two cat-
egories will not be equal, but instead P (ω1) = 2P (ω2). Modify your training
method and use the above data to form a tree for this case.
⊕
Section 8.4
17. Consider training a binary decision tree to classify two-component patterns from
two categories. The first component is binary, 0 or 1, while the second component
has six possible values, A through F:
ω1 ω2
1A 0A
0E 0C
0B 1C
1B 0F
1F 0B
0D 1D
8.8. PROBLEMS 51
Compare splitting the root node based on the first feature with splitting it on the
second feature in the following way.
(a) Use an entropy impurity with a two-way split (i.e., B = 2) on the first feature
and a six-way split on the second feature.
(b) Repeat (a) but using a gain ratio impurity.
(c) In light of your above answers discuss the value of gain ratio impurity in cases
where splits have different branching ratios.
⊕
Section 8.5
18. Consider strings x and text, of length m and n, respectively, from an alphabet A
consisting of d characters. Assume that the naive string-matching algorithm (Algo-
rithm 1) exits the implied loop in line 4 as soon as a mismatch occurs. Prove that the
number of character-to-character comparisons made on average for random strings is
(n−m+ 1)1 − d
−m
1 − d−1
≤ 2(n−m+ 1).
19. Consider string matching using the Boyer-Moore algorithm (Algorithm 2) based
on the trinary alphabet A = {a, b, c}. Apply the good-suffix function G and the
last-occurrence function F to each of the following strings:
(a) “acaccacbac”
(b) “abababcbcbaaabcbaa”
(c) “cccaaababaccc”
(d) “abbabbabbcbbabbcbba”
20. Consider the string-matching problem illustrated in the top of Fig. 8.8. Assume
text began at the first character of “probabilities.”
(a) How many basic character comparisons are required by the naive string-matching
algorithm (Algorithm 1) to find a valid shift?
(b) How many basic character comparisons are required by the Boyer-Moore string
matching algorithm (Algorithm 2)?
21. For each of the texts below, determine the number of fundamental character
comparisons needed to find all valid shifts for the test string x = “abcca” using
the naive string-matching algorithm (Algorithm 1) and the Boyer-Moore algorithm
(Algorithm 2).
(a) “abcccdabacabbca”
(b) “dadadadadadadad”
(c) “abcbcabcabcabc”
(d) “accabcababacca”
52 CHAPTER 8. NON-METRIC METHODS
(e) “bbccacbccabbcca”
22. Write pseudocode for an efficient construction of the last-occurrence function F
used in the Boyer-Moore algorithm (Algorithm 2). Let d be the number of elements
in the alphabet A, and m the length of string x.
(a) What is the time complexity of your algorithm in the worst case?
(b) What is the space complexity of your algorithm in the worst case?
(c) How many fundamental operations are required to compute F for the 26-
letter English alphabet for x = “bonbon”? For x = “marmalade”? For x =
“abcdabdabcaabcda”?
23. Consider the training data from the trinary alphabet A = {a, b, c} in the table
ω1 ω2 ω3
aabbc bccba caaaa
ababcc bbbca cbcaab
babbcc cbbaaaa baaca
Use the simple edit distance to classify each of the below strings. If there are ambi-
guities in the classification, state which two (or all three) categories are candidates.
(a) “abacc”
(b) “abca”
(c) “ccbba”
(d) “bbaaac”
24. Repeat Problem 23 using its training data but the following test data:
(a) “ccab”
(b) “abdca”
(c) “abc”
(d) “bacaca”
25. Repeat Problem 23 but assume that the cost of different string transformations
are not equal. In particular, assume that an interchange is twice as costly as an
insertion or a deletion.
26. Consider edit distance with positive but otherwise arbitrary costs associated
with each of the fundamental operations of insertion, deletion and substitution.
(a) Which of the criteria for a metric are always obeyed and which not necessarily
obeyed?
(b) For any criteria that are not always obeyed, construct a counterexample.
27. Algorithm 3 employs a greedy heuristic for computing the edit distance between
two strings x and y; it need not give a global optimum. In the following, let |x| = n1
and |y| = n2.
8.8. PROBLEMS 53
(a) State the computational complexity of an exhaustive examination of all trans-
formations of x into y. (Assume that no transformation need be considered if
it leads to a string shorter than Min[n1, n2] or longer than Max[n1, n2].)
(b) Recall from Chap. ?? the basic approach of linear programming. Write pseu-
docode that would apply linear programming to the calculation of edit distances.
28. Consider the general edit distance with positive costs and whether it has the
four properties of a true metric: non-negativity, reflexivity, symmetry and the triangle
inequality.
29. Consider strings x and text, of length m and n, respectively, from an alphabet
A consisting of d characters.
(a) Modify the pseudocode of the naive string-matching algorithm to include the
don’t care symbol.
(b) Employ the assumptions of Problem 18 but also that x has exactly k don’t
care symbols while text has none. Find the number of character-to-character
comparisons made on average for otherwise random strings.
(c) Show that in the limit of k = 0 your answer is closely related to that of Prob-
lem 18.
(d) What is your answer in part (b) in the limit k = m?
⊕
Section 8.6
30. Mathematical expressions in the computer language Lisp are of the form
(operation operand1 operand2) where spaces delineate potentially ambiguous sym-
bols and expressions can be nested, for example (quotient (plus 4 9) 6).
(a) Write a simple grammar for the four basic arithmetic operations plus, difference,
times and quotient, applied to positive single-digit integers. Be sure to include
parentheses in your alphabet.
(b) Determine by hand whether each of the following candidate Lisp expressions can
be derived in your grammar, and if so, show a corresponding derivation tree.
• (times (plus (difference 5 9)(times 3 8))(quotient 2 6))
• (7 difference 2)
• (quotient (7 plus 2) (plus 6 3))
• ((plus) (6 2))
• (difference (plus 5 9) (difference 6 8)) .
31. Consider the language L(G) = {anb|n ≥ 1}.
(a) Construct by hand a grammar that generates this language.
(b) Use G to form derivation trees for the strings “ab” and “aaaaab.”
32. Consider the grammar G: A = {a, b, c}, S = S, I = {A,B} and
P = {S → cAb, A→ aBa, B → aBa, B → cb}.
54 CHAPTER 8. NON-METRIC METHODS
(a) What type of grammar is G?
(b) Prove that this grammar generates the language L(G) = {cancbanb|n ≥ 1}.
(c) Draw the derivation tree the following two strings: “caacbaab” and “cacbab.”
33. A palindrome is a sequence of characters that reads the same forward andpalindrome
backward, such as “i,” “tat,” “boob,” and “sitonapotatopanotis.”
(a) Write a grammar that generates all palindromes using 26 English letters (no
spaces). Use your grammar to show a derivation tree for “noon” and “bib.”
(b) What type is your grammar (0, 1, 2 or 3)?
(c) Write a grammar that generates all words that consist of a single letter followed
by a palindrome. Use your grammar to show a derivation tree for “pi,” “too,”
and “stat.”
34. Consider the grammar G in Example 3.
(a) How many possible derivations are there in G for numbers 1 through 999?
(b) How many possible derivations are there for numbers 1 through 999,999?
(c) Does the grammar allow any of the numbers (up to six digits) to be pronounced
in more than one way?
35. Recall that ) is the empty string, defined to have zero length, and no man-
ifestation in a final string. Consider the following grammar G: A = {a}, S = S,
I = {A,B,C,D,E} and eight rewrite rules:
P =
S → ACaB Ca → aaC
CB → DB CB → E
aD → Da aD → AC
aE → Ea AE → )
.
(a) Note how A and B mark the beginning and end of the sentence, respectively,
and that C is a marker that doubles the number of as (while moving from left
to right through the word). Prove that the language generated by this grammar
is L(G) = {a2n |n > 0}.
(b) Show a derivation tree for “aaaa” and for “aaaaaaaa” (cf. Computer exer-
cise ??).
36. Explore the notion of Chomsky normal form in the following way.
(a) Show that the grammar G with A = {a, b}, S = S, I = {A,B} and rewrite
rules:
P =
S → bA OR aB
A → bAA OR aS OR a
B → aBB OR bS OR b
,
is not in Chomsky normal form.
8.8. PROBLEMS 55
(b) Show that grammar G′ with A = {a, b}, S = S, I = {A,B,Ca, Cb, D1, D2},
and
P =
S → CbA OR CaB D1 → AA
A → CaS OR CbD1 OR a D2 → BB
B → CbS OR CaD2 OR b Ca → a
Cb → b
.
is in Chomsky normal form.
(c) Show that G and G′ are equivalent by converting the rewrite rules of G into
those of G′ in the following way. Note that the rules A → a and B → b of G
are already acceptable. Now convert other rules of G appropriate for Chomsky
normal form. First replace S → bA in G by S → CbA and Cb → b. Likewise,
replace A → aS by A → CaS and Ca → a. Continue in this way, keeping in
mind the final form of the rewrite rules of G′.
(d) Give a derivation of “aabab” in G and in G′.
37. Prove that each of the following languages are not context-free.
(a) L(G) = {aibjck|i < j < k}.
(b) L(G) = {ai|i a prime}.
38. Consider a grammar with A = {a, b, c}, S = S, I = {A,B}, and
P =
S → aSBA OR aBA AB → BA
bB → bb bA → bc
cA → cc aB → ab
.
Prove that this grammar generates the language L(G) = {anbncn|n ≥ 1}.
39. Try to parse by hand the following utterances. For each successful parse, show
the corresponding derivation tree.
• three hundred forty two thousand six hundred nineteen
• thirteen
• nine hundred thousand
• two thousand six hundred thousand five
• one hundred sixty eleven
⊕
Section 8.7
40. Let D1 = {ab, abb, abbb} and D2 = {ba, aba, babb} be positive training examples
from two grammars, G1 and G2, respectively.
(a) Suppose both grammars are of type 3. Generate some candidate rewrite rules.
(b) Infer grammar G1 using D2 as negative examples.
(c) Infer grammar G2 using D1 as negative examples.
56 CHAPTER 8. NON-METRIC METHODS
(d) Use your trained grammars to classify the following sentences; label any sentence
that cannot be parsed in either grammar as “ambiguous”: “bba,”“abab,”“bbb”and
“abbbb.”
⊕
Section 8.8
41. For each of the below, write an rule giving an equivalent relation using any of
the following predicates: Male(·), Female(·), Parent(·,·), Married(·,·),
(a) Sister(·,·), where Sister(x,y) = True means that x is the sister of y.
(b) Father(·,·), where Father(x,y) = True means that x is the father of y.
(c) Grandmother(·,·), where Grandmother(x,y) = True means that x is the grand-
mother of y.
(d) Husband(·,·), where Husband(x,y) = True means that x is the husband of y.
(e) IsWife(·), where IsWife(x) = True means that simply that x is a wife.
(f) Siblings(·,·)
(g) FirstCousins(·,·)
Computer exercises
Several exercises will make use of the following data sampled from three categories.
Each of the five features takes on a discrete feature, indicated by the range listed at
the along the top. Note particularly that there are different number of samples in
each category, and that the number of possible values for the features is not the same.
For instance, the first feature can take on four values (A − D, inclusive), while the
last feature can take on just two (M −N).
sample category A−D E −G H − J K − L M −N
1 ω1 A E H K M
2 ω1 B E I L M
3 ω1 A G I L N
4 ω1 B G H K M
5 ω1 A G I L M
6 ω2 B F I L M
7 ω2 B F J L N
8 ω2 B E I L N
9 ω2 C G J K N
10 ω2 C G J L M
11 ω2 D G J K M
12 ω2 B D I L M
13 ω3 D E H K N
14 ω3 A E H K N
15 ω3 D E H L N
16 ω3 D F J L N
17 ω3 A F H K N
18 ω3 D E J L M
8.8. COMPUTER EXERCISES 57
⊕
Section 8.3
1. Write a general program for growing a binary tree and use it to train a tree fully
using the data from the three categories in the table, using an entropy impurity.
(a) Use the (unpruned) tree to classify the following patterns: {A,E, I, L,N},
{D,E, J,K,N}, {B,F, J,K,M}, and {C,D, J, L,N}.
(b) Prune one pair of leafs, increasing the entropy impurity as little as possible.
(c) Modify your program to allow for non-binary splits, where the branching ratio
B as is determined at each node during training. Train a new tree fully using a
gain ratio impurity and then classify the points in (a).
2. Recall that one criterion for stopping the growing of a decision tree is to halt
splitting when the best split reduces the impurity by less than some threshold value,
that is, when maxs ∆i(s) ≤ β where s indicates the split and β is the threshold.
Explore the relationship between classifier generalization and β through the following
simulations.
(a) Generate 200 training points, 100 each for two two-dimensional Gaussian dis-
tributions: p(x|ω1) ∼ N(
(−0.25
0
)
, I) and p(x|ω2) ∼ N(
(
+0.25
0
)
, I). Also use your
program to generate an independent test set of 100 points, 50 each of the cate-
gories.
(b) Write a program to grow a tree classifier, where a node is not split if maxs ∆i(s) ≤
β.
(c) Plot the generalization error of your tree classifier versus β for β = 0.01 → 1 in
steps of 0.01, as estimated on the test data generated in part (a).
(d) In light of your plot, discuss the relationship between β and generalization error.
3. Repeat all parts of Computer exercise 2, but instead of considering β, focus
instead on the role of α as used in Eq. 8.⊕
Section 8.4
4. Write a program for training an ID3 decision tree in which the branching ratio
B at each node is equal to the number of discrete “binned” possible values for each
attribute. Use a gain ratio impurity.
(a) Use your program to train a tree fully with the ω1 and ω2 patterns in the table
above.
(b) Use your tree to classify {B,G, I,K,N} and {C,D, J, L,M}.
(c) Write a logical expression which describes the decisions in part (b). Simplify
these expressions.
(d) Convert the information in your tree into a single logical expression which de-
scribes the ω1 category. Repeat for the ω2 category.
5. Consider the issue of tree-based classifiers and deficient patterns.
58 CHAPTER 8. NON-METRIC METHODS
(a) Write a program to generate a binary decision tree for categories ω1 and ω2
using samples points 1 – 10 from the table above and an entropy impurity. For
each decision node store the primary split and four surrogate splits.
(b) Use your tree to classify the following patterns, where as usual * denotes a
missing feature.
• {A,F,H,K,M}
• {∗, G,H,K,M}
• {C,F, I, L,N}
• {B, ∗, ∗,K,N}
(c) Now write a program to train a tree using deficient points. Train with sample
points 1 – 10 from the table, used in part (a), as well as the following four points:
• ω1: {∗, F, I,K,N}
• ω1: {B,G,H,K, ∗}
• ω2: {C,G, ∗, L,N}
• ω2: {∗, F, I,K,N}
(d) Use your tree from part (c) to classify the test points in part (b).
6. Train a tree classifier to distinguish all three categories ωi, i = 1, 2, 3, using all
20 sample points in the table above. Use an entropy criterion without pruning or
stopping.
(a) Express your tree as a set of rules.
(b) Through exhaustive search, find the rule or rules, which when deleted, lead to
the smallest increase in classification error as estimated by the training data.
⊕
Section 8.5
7. Write a program to implement the naive string-matching algorithm (Algorithm 1).
Insert a conditional branch so as to exit the innermost loop whenever a mismatch
occurs (i.e., the shift is found to be invalid). Add a line to count the total number of
character-to-character comparisons in the complete string search.
(a) Write a small program to generate a text of n characters, taken from an alphabet
having d characters. Let d = 5 and use your program to generate a text of length
n = 1000 and a test string x of length m = 10.
(b) Compare the number of character-to-character comparisons performed by your
program with the theoretical result quoted in Problem 18 for all pairs of the
following parameters: m = {10, 15, 20} and n = {100, 1000, 10000}.
8. Write a program to implement the Boyer-Moore algorithm (Algorithm 2) in the
following steps. Throughout let the alphabet have d characters.
(a) Write a routine for constructing the good-suffix function G. Let d = 3 and apply
your routine to the strings x1 = “abcbab”and x2 = “babab.”
8.8. COMPUTER EXERCISES 59
(b) Write a routine for constructing the last-occurrence function F . Apply it to the
strings x1 and x2 of part (a).
(c) Write an implementation of the Boyer-Moore algorithm incorporating your rou-
tines from parts (a) and (b). Generate a text of n = 10000 characters chosen
from the alphabet A = {a, b, c}. Use your program to search for x1 in text, and
again x2 in text.
(d) Make some statistical assumptions to estimate the number of occurrences of x1
and x2 in text, and compare that number to your answers in part (c).
9. Write an algorithm for addressing the subset-superset problem in string matching.
That is, search a text with several strings, some of which are factors of others.
(a) Let x1 = “beats,” x2 = “beat,” x3 = “be,” x4 = “at,” x5 = “eat,” x6 = “sat.”
Search for these possible factors in text = “beats beats beats . . . beats︸ ︷︷ ︸
100 ×
,” but
do not return any strings that are factors of other test strings found in text.
(b) Repeat with text consisting of 100 appended copies of “repeatable ,” and the
test items “repeatable,” “pea,” “table,” “tab,” “able,” “peat,” and “a.”
10. String matching with errors. Test on segments xxxx⊕
Section 8.6
11. Write a parser for the grammar described in the text: A = {a, b}, I = {A,B},
S = S and P =
p1 : S → AB OR BC
p2 : A → BA OR a
p3 : B → CC OR b
p4 : C → AB OR a
.
Use your program to attempt to parse each of the following strings. In all cases,
show the parse tree; for each successful parse show moreover the corresponding deriva-
tion tree.
• “aaaabbab”
• “ba”
• “baabab”
• “babab”
• “aaa”
• “baaa”
⊕
Section 8.7
12. Write a program to infer a grammar G from the following positive and negative
examples:
• D+ = {abc, aabbcc, aaabbbccc}
• D− = {abbc, abcc, aabcc}
60 CHAPTER 8. NON-METRIC METHODS
Take the following as candidate rewrite rules:
S → aSBA AB → BA cB → aC
S → bSBA BA → AB bA → bc
S → aBA bB → bb bC → bc
S → aSB bC → ba aB → ab
S → aSA cA → cc aB → ca
Proceed as follows:
(a) Implement the general bottom-up parser of Algorithm 4.
(b) Implement the general grammatical inferencing method of Algorithm 5.
(c) Use your programs in conjunction to infer G from the data.
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Index
d-tuple, see tuple
A, see alphabet
©/ , see “don’t care” symbol
20-questions, 4
adenine, 22
alphabet, 24
artificial intelligence, 41
bad-character heuristic, 25
bit, 9, 11
branch, 4
branching factor, 6
ID3, 21
branching ratio, see branching factor
C4.5, see classifier, tree, C4.5
character, 22
chi-squared statistic, see statistic, chi-
squared
Chomsky normal form, 34
classification
tree
CART, 21
Classification and regression tree, see
classifier, tree, CART
classifier
stability, see stability, classifier
tree
binary, 6
C4.5, 21
CART, 5–20
computational complexity, 15
ID3, 20
Concept Learning System, 44
confidence level, 12
constant (logical), 41
cost matrix, 17
cytosine, 22
data
nominal, 22
decision
primary, 19
decision tree, 4–22
interpretability, 4
deficient pattern, see pattern, deficient
degree of freedom, 12
derivation tree, 33
descendent node, see node, descendent
distance
edit, 27–29
distribution
cumulative, 49
DNA
base pair, 22
“don’t care” symbol, 23
edit distance, see distance, edit
Algorithm, 28
empty string, see string, null
expert systems, 41
factor, see string, factor
finite state machine, 38
first-order logic, see logic, first-order
fish
tooth, 3
function, 42
genome
human, 24
good-suffix function, 27
good-suffix heuristic, 26
grammar
free, 33
learning
Algorithm, 39
type 0, see grammar, free
unrestricted, see grammar, free
greedy method, 9
guanine, 22
horizon effect, 12
67
68 INDEX
hypothesis testing, 12
ID3, see classifier, tree, ID3
if-then rule, 41
impurity
entropy, 7
gain ratio, 11
Gini, 8, 10
information, see impurity, entropy
Kolmogorov-Smirnov, 49
misclassification, 8
scaling, 9
statistical significance, 12
variance, 7
weighted, 17
inductive logic programming, 46
interchange operation, 28
joining
node, see merging, node
keyword, 23
Kolmogorov-Smirnov test, 49
Kronecker delta, 28
language, 32
last-occurrence function, 27
leaf node, 4
letter, see string
link
tree, see branch
logic
first-order, 41
lookup table, 11
MDL, see minimum description length
Melville, Herman, 23
merging
node, 13
minimum description length, 11
misclassification
impurity, see impurity, misclassi-
fication
missing attribute
tree, 18
Moby Dick, 23
motif, 23
nearest-neighbor classifier, 11
node
descendent, 4, 12
leaf
label, 13
neighboring, 13
root, 4
terminal, see node, leaf
nominal data, 3
null string, see string, null
Occam’s razor, 6
palindrome, 54
parse
table, 36
parsing
bottom-up, 37
Algorithm, 36
pattern
deficient, 18
pattern recognition
syntactic, 4
post-pruning, see pruning
pre-pruning, see stopped training
predicate, 41
prefix, 26
empty, 29
primary decision, 19
production, 32
property d-tuple, see tuple
propositional logic, see logic, proposi-
tional
protein, 23
purity, 6
root node, see node, root
segment, see string, factor
sentence, 31
sequential covering, 43
shift
valid, 24
similarity, 3
split, 6
surrogate, 19
stability
classifier, 15
statistic
chi-squared, 12
statistical significance, 12
stopped splitting, 11–12
string, 3, 22
factor, 23
INDEX 69
matching, 24–27
Algorithm, 25
Boyer-Moore Algorithm, 25
with don’t-care symbol, 30
with errors, 29
null, 32
string operations, 22
sub-tree, 4
subset-superset problem, 27
substring, see string, factor
suffix, 26
matching, see good suffix
surrogate split, see split, surrogate, 21
symbol, see character
starting, 32
syntactic pattern recognition, see pat-
tern recognition, syntactic
template
string matching, see string match-
ing, template
template matching
string, 23
term (logical), 42
text
string matching, 23
thymine, 22
Tolstoy, Leo, 24
transposition, see interchange
tree, see decision tree
balanced, 11
binary, 6
derivation, 33
interpretation of, 4
missing attribute, 18–19
multi-variate, 16
polythetic, 9
priors, 16
pruning, 12–13
simplicity, 6
unbalanced, 13
tuple, 3
twiddle, see interchange
twoing criterion, 10
valid shift, 24
variable (logical), 41
variable-free logic, see logic, proposi-
tional
variance impurity, see impurity, vari-
ance
virtual value, 19
War and Peace, 24
word, 22
Contents
9 Algorithm-independent machine learning 3
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
9.2 Lack of inherent superiority of any classifier . . . . . . . . . . . . . . . 4
9.2.1 No Free Lunch Theorem . . . . . . . . . . . . . . . . . . . . . . 4
Example 1: No Free Lunch for binary data . . . . . . . . . . . . . . . . 6
9.2.2 *Ugly Duckling Theorem . . . . . . . . . . . . . . . . . . . . . 8
9.2.3 Minimum description length (MDL) . . . . . . . . . . . . . . . 11
9.2.4 Minimum description length principle . . . . . . . . . . . . . . 13
9.2.5 Overfitting avoidance and Occam’s razor . . . . . . . . . . . . . 14
9.3 Bias and variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
9.3.1 Bias and variance for regression . . . . . . . . . . . . . . . . . . 15
9.3.2 Bias and variance for classification . . . . . . . . . . . . . . . . 18
9.4 *Resampling for estimating statistics . . . . . . . . . . . . . . . . . . . 20
9.4.1 Jackknife . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Example 2: Jackknife estimate of bias and variance of the mode . . . . 23
9.4.2 Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
9.5 Resampling for classifier design . . . . . . . . . . . . . . . . . . . . . . 25
9.5.1 Bagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
9.5.2 Boosting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Algorithm 1: AdaBoost . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
9.5.3 Learning with queries . . . . . . . . . . . . . . . . . . . . . . . 29
9.5.4 Arcing, learning with queries, bias and variance . . . . . . . . . 31
9.6 Estimating and comparing classifiers . . . . . . . . . . . . . . . . . . . 32
9.6.1 Parametric models . . . . . . . . . . . . . . . . . . . . . . . . . 33
9.6.2 Cross validation . . . . . . . . . . . . . . . . . . . . . . . . . . 33
9.6.3 Jackknife and bootstrap estimation of classification accuracy . 35
9.6.4 Maximum-likelihood model comparison . . . . . . . . . . . . . 36
9.6.5 Bayesian model comparison . . . . . . . . . . . . . . . . . . . . 37
9.6.6 The problem-average error rate . . . . . . . . . . . . . . . . . . 39
9.6.7 Predicting final performance from learning curves . . . . . . . . 42
9.6.8 The capacity of a separating plane . . . . . . . . . . . . . . . . 44
9.7 Combining classifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9.7.1 Component classifiers with discriminant functions . . . . . . . 46
9.7.2 Component classifiers without discriminant functions . . . . . . 48
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Bibliographical and Historical Remarks . . . . . . . . . . . . . . . . . . . . 50
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Computer exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1
2 CONTENTS
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Chapter 9
Algorithm-independent
machine learning
9.1 Introduction
I n the previous chapters we have seen many learning algorithms and techniques forpattern recognition. When confronting such a range of algorithms, every reader
has wondered at one time or another which one is “best.” Of course, some algorithms
may be preferred because of their lower computational complexity; others may be
preferred because they take into account some prior knowledge of the form of the
data (e.g., discrete, continuous, unordered list, string, ...). Nevertheless there are
classification problems for which such issues are of little or no concern, or we wish
to compare algorithms that are equivalent in regard to them. In these cases we are
left with the question: Are there any reasons to favor one algorithm over another?
For instance, given two classifiers that perform equally well on the training set, it
is frequently asserted that the simpler classifier can be expected to perform better
on a test set. But is this version of Occam’s razor really so evident? Likewise, Occam’s
razorwe frequently prefer or impose smoothness on a classifier’s decision functions. Do
simpler or “smoother” classifiers generalize better, and if so, why? In this chapter
we address these and related questions concerning the foundations and philosophical
underpinnings of statistical pattern classification. Now that the reader has intuition
and experience with individual algorithms, these issues in the theory of learning may
be better understood.
In some fields there are strict conservation laws and constraint laws — such as the
conservation of energy, charge and momentum in physics, or the second law of ther-
modynamics, which states that the entropy of an isolated system can never decrease.
These hold regardless of the number and configuration of the forces at play. Given
the usefulness of such laws, we naturally ask: are there analogous results in pattern
recognition, ones that do not depend upon the particular choice of classifier or learn-
ing method? Are there any fundamental results that hold regardless of the cleverness
of the designer, the number and distribution of the patterns, and the nature of the
classification task?
Of course it is very valuable to know that there exists a constraint on classifier
3
4 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
accuracy, the Bayes limit, and it is sometimes useful to compare performance to this
theoretical limit. Alas in practice we rarely if ever know the Bayes error rate. Even if
we did know this error rate, it would not help us much in designing a classifier; thus
the Bayes error is generally of theoretical interest. What other fundamental principles
and properties might be of greater use in designing classifiers?
Before we address such problems, we should clarify the meaning of the title of this
chapter. “Algorithm-independent” here refers, first, to those mathematical founda-
tions that do not depend upon the particular classifier or learning algorithm used. Our
upcoming discussion of bias and variance is just as valid for methods based on neural
networks as for the nearest-neighbor or for model-dependent maximum likelihood.
Second, we mean techniques that can be used in conjunction with different learning
algorithms, or provide guidance in their use. For example, cross validation and re-
sampling techniques can be used with any of a large number of training methods. Of
course by the very general notion of an algorithm these too are algorithms, technically
speaking, but we discuss them in this chapter because of their breadth of applicability
and independence from the details of the learning techniques encountered up to here.
In this chapter we shall see, first, that no pattern classification method is inher-
ently superior to any other, or even to random guessing; it is the type of problem,
prior distribution and other information that determine which form of classifier should
provide the best performance. We shall then explore several ways to quantify and ad-
just the “match” between a learning algorithm and the problem it addresses. In any
particular problem there are differences between classifiers, of course, and thus we
show that with certain assumptions we can estimate their accuracy (even for instance
before the candidate classifier is fully trained) and compare different classifiers. Fi-
nally, we shall see methods for integrating component or “expert” classifiers, which
themselves might implement any of a number of algorithms.
We shall present the results that are most important for pattern recognition prac-
titioners, occasionally skipping over mathematical details that can be found in the
original research referenced in the Bibliographical and Historical Remarks section.
9.2 Lack of inherent superiority of any classifier
We now turn to the central question posed above: If we are interested solely in the
generalization performance, are there any reasons to prefer one classifier or learning
algorithm over another? If we make no prior assumptions about the nature of the
classification task, can we expect any classification method to be superior or inferior
overall? Can we even find an algorithm that is overall superior to (or inferior to)
random guessing?
9.2.1 No Free Lunch Theorem
As summarized in the No Free Lunch Theorem, the answer to these and several
related questions is no: on the criterion of generalization performance, there are
no context- or problem-independent reasons to favor one learning or classification
method over another. The apparent superiority of one algorithm or set of algorithms
is due to the nature of the problems investigated and the distribution of data. It
is an appreciation of the No Free Lunch Theorem that allows us, when confronting
practical pattern recognition problems, to focus on the aspects that matter most
— prior information, data distribution, amount of training data and cost or reward
9.2. LACK OF INHERENT SUPERIORITY OF ANY CLASSIFIER 5
functions. The Theorem also justifies a scepticism about studies that purport to
demonstrate the overall superiority of a particular learning or recognition algorithm.
When comparing algorithms we sometimes focus on generalization error for points
not in the training set D, rather than the more traditional independent identically
distributed or i.i.d. case. We do this for several reasons: First, virtually any powerful
algorithm such as the nearest-neighbor algorithm, unpruned decision trees, or neural
networks with sufficient number of hidden nodes, can learn the training set. Second,
for low-noise or low-Bayes error cases, if we use an algorithm powerful enough to learn
the training set, then the upper limit of the i.i.d. error decreases as the training set
size increases. In short, it is the off-training set error — the error on points not in off-
training
set error
the training set — that is a better measure for distinguishing algorithms. Of course,
for most applications the final performance of a fielded classifier is the full i.i.d. error.
For simplicity consider a two-category problem, where the training set D consists
of patterns xi and associated category labels yi = ±1 for i = 1, . . . , n generated by
the unknown target function to be learned, F (x), where yi = F (xi). In most cases
of interest there is a random component in F (x) and thus the same input could lead
to different categories, giving non-zero Bayes error. At first we shall assume that the
feature set is discrete; this simplifies notation and allows the use of summation and
probabilities rather than integration and probability densities. The general conclu-
sions hold in the continuous case as well, but the required technical details would
cloud our discussion.
Let H denote the (discrete) set of hypotheses, or possible sets of parameters to be
learned. A particular hypothesis h(x) ∈ H could be described by quantized weights in
a neural network, or parameters θ in a functional model, or sets of decisions in a tree,
etc. Further, P (h) is the prior probability that the algorithm will produce hypothesis
h after training; note that this is not the probability that h is correct. Next, P (h|D)
denotes the probability that the algorithm will yield hypothesis h when trained on
the data D. In deterministic learning algorithms such as the nearest-neighbor and
decision trees, P (h|D) will be everywhere zero except for a single hypothesis h. For
stochastic methods, such as neural networks trained from random initial weights, or
stochastic Boltzmann learning, P (h|D) will be a broad distribution. For a general loss
function L(·, ·) we let E = L(·, ·) be the scalar error or cost. While the natural loss
function for regression is a sum-square error, for classification we focus on zero-one
loss, and thus the generalization error is the expected value of E.
How shall we judge the generalization quality of a learning algorithm? Since we
are not given the target function, the natural measure is the expected value of the
error given D, summed over all possible targets. This scalar value can be expressed as
a weighted “inner product” between the distributions P (h|D) and P (F |D), as follows:
E [E|D] =
∑
h,F
∑
x/∈D
P (x)[1− δ(F (x), h(x))]P (h|D)P (F |D), (1)
where for the moment we assume there is no noise. The familiar Kronecker delta func-
tion, δ(·, ·), has value 1 if its two arguments match, and value 0 otherwise. Equation 1
states that the expected error rate, given a fixed training set D, is related to the sum
over all possible inputs weighted by their probabilities, P (x), as well as the “align-
ment” or “match” of the learning algorithm, P (h|D), to the actual posterior P (F |D).
The important insight provided by this equation is that without prior knowledge con-
cerning P (F |D), we can prove little about any particular learning algorithm P (h|D),
including its generalization performance.
6 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
The expected off-training set classification error when the true function is F (x)
and some candidate learning algorithm is Pk(h(x)|D) is given by
Ek(E|F, n) =
∑
x/∈D
P (x)[1− δ(F (x), h(x))]Pk(h(x)|D). (2)
With this background and the terminology of Eq. 2 we can now turn to a formal
statement of the No Free Lunch Theorem.
Theorem 9.1 (No Free Lunch) For any two learning algorithms P1(h|D) and P2(h|D),
the following are true, independent of the sampling distribution P (x) and the number
n of training points:
1. Uniformly averaged over all target functions F , E1(E|F, n)− E2(E|F, n) = 0;
2. For any fixed training set D, uniformly averaged over F , E1(E|F,D)−E2(E|F,D) =
0;
3. Uniformly averaged over all priors P (F ), E1(E|n)− E2(E|n) = 0;
4. For any fixed training set D, uniformly averaged over P (F ), E1(E|D)−E2(E|D) =
0.∗
Part 1 says that uniformly averaged over all target functions the expected error
for all learning algorithms is the same, i.e.,
∑
F
∑
D
P (D|F ) [E1(E|F, n)− E2(E|F, n)] = 0, (3)
for any two learning algorithms. In short, no matter how clever we are at choosing
a “good” learning algorithm P1(h|D), and a “bad” algorithm P2(h|D) (perhaps even
random guessing, or a constant output), if all target functions are equally likely, then
the “good” algorithm will not outperform the “bad” one. Stated more generally,
there are no i and j such that for all F (x), Ei(E|F, n) > Ej(E|F, n). Furthermore, no
matter what algorithm you use, there is at least one target function for which random
guessing is a better algorithm.
Assuming the training set can be learned by all algorithms in question, then Part 2
states that even if we know D, then averaged over all target functions no learning
algorithm yields an off-training set error error that is superior to any other, i.e.,
∑
F
[E1(E|F,D)− E2(E|F,D)] = 0. (4)
Parts 3 & 4 concern non-uniform target function distributions, and have related in-
terpretations (Problems 2 – 5). Example 1 provides an elementary illustration.
Example 1: No Free Lunch for binary data
Consider input vectors consisting of three binary features, and a particular target
function F (x), as given in the table. Suppose (deterministic) learning algorithm 1
assumes every pattern is in category ω1 unless trained otherwise, and algorithm 2
assumes every pattern is in ω2 unless trained otherwise. Thus when trained with
∗ The clever name for the Theorem was suggested by David Haussler.
9.2. LACK OF INHERENT SUPERIORITY OF ANY CLASSIFIER 7
n = 3 points in D, each algorithm returns a single hypothesis, h1 and h2, respectively.
In this case the expected errors on the off-training set data are E1(E|F,D) = 0.4 and
E2(E|F,D) = 0.6.
x F h1 h2
000 1 1 1
D 001 -1 -1 -1
010 1 1 1
011 -1 1 -1
100 1 1 -1
101 -1 1 -1
110 1 1 -1
111 1 1 -1
For this target function F (x), clearly algorithm 1 is superior to algorithm 2. But
note that the designer does not know F (x) — indeed, we assume we have no prior
information about F (x). The fact that all targets are equally likely means that D
provides no information about F (x). If we wish to compare the algorithms overall,
we therefore must average over all such possible target functions consistent with the
training data. Part 2 of Theorem 9.1 states that averaged over all possible target
functions, there is no difference in off-training set errors between the two algorithms.
For each of the 25 distinct target functions consistent with the n = 3 patterns in
D, there is exactly one other target function whose output is inverted for each of the
patterns outside the training set, and this ensures that the performances of algorithms
1 and 2 will also be inverted, so that the contributions to the formula in Part 2 cancel.
Thus indeed Part 2 of the Theorem as well as Eq. 4 are obeyed.
Figure 9.1 illustrates a result derivable from Part 1 of Theorem 9.1. Each of the
six squares represents the set of all possible classification problems; note that this is
not the standard feature space. If a learning system performs well — higher than
average generalization accuracy — over some set of problems, then it must perform
worse than average elsewhere, as shown in a). No system can perform well throughout
the full set of functions, d); to do so would violate the No Free Lunch Theorem.
In sum, all statements of the form “learning/recognition algorithm 1 is better than
algorithm 2” are ultimately statements about the relevant target functions. There
is, hence, a “conservation theorem” in generalization: for every possible learning
algorithm for binary classification the sum of performance over all possible target
functions is exactly zero. Thus we cannot achieve positive performance on some
problems without getting an equal and opposite amount of negative performance on
other problems. While we may hope that we never have to apply any particular
algorithm to certain problems, all we can do is trade performance on problems we do
not expect to encounter with those that we do expect to encounter. This, and the
other results from the No Free Lunch Theorem, stress that it is the assumptions about
the learning domains that are relevant. Another practical import of the Theorem is
that even popular and theoretically grounded algorithms will perform poorly on some
problems, ones in which the learning algorithm and the posterior happen not to be
“matched,” as governed by Eq. 1. Practitioners must be aware of this possibility, which
arises in real-world applications. Expertise limited to a small range of methods, even
powerful ones such as neural networks, will not suffice for all classification problems.
8 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
possible
learning systems
+
0+
-+
–
–
–
– –
–
–
–
–
–
–
–
–
impossible
learning systems +
–
–
–
–
—
++
+
+
+
–
–
–
-0
0 0
0
+
+ +
00
0
0
0
0
0
0
0
00 0
0
0
0
0
a) b) c)
d) e) f)
p
ro
b
le
m
s
p
a
ce
(n
o
t
fe
a
tu
re
s
p
a
ce
)
Figure 9.1: The No Free Lunch Theorem shows the generalization performance on
the off-training set data that can be achieved (top row), and the performance that
cannot be achieved (bottom row). Each square represents all possible classification
problems consistent with the training data — this is not the familiar feature space.
A + indicates that the classification algorithm has generalization higher than average,
a – indicates lower than average, and a 0 indicates average performance. The size of
a symbol indicates the amount by which the performance differs from the average.
For instance, a) shows that it is possible for an algorithm to have high accuracy on a
small set of problems so long as it has mildly poor performance on all other problems.
Likewise, b) shows that it is possible to have excellent performance throughout a large
range of problem but this will be balanced by very poor performance on a large range
of other problems. It is impossible, however, to have good performance throughout
the full range of problems, shown in d). It is also impossible to have higher than
average performance on some problems, and average performance everywhere else,
shown in e).
Experience with a broad range of techniques is the best insurance for solving arbitrary
new classification problems.
9.2.2 *Ugly Duckling Theorem
While the No Free Lunch Theorem shows that in the absence of assumptions we should
not prefer any learning or classification algorithm over another, an analogous theorem
addresses features and patterns. Roughly speaking, the Ugly Duckling Theorem states
that in the absence of assumptions there is no privileged or “best” feature represen-
tation, and that even the notion of similarity between patterns depends implicitly on
assumptions which may or may not be correct.
Since we are using discrete representations, we can use logical expressions or
“predicates” to describe a pattern, much as in Chap. ??. If we denote a binary
feature attribute by fi, then a particular pattern might be described by the predicate
“f1 AND f2,” another pattern might be described as “NOT f2,” and so on. Like-
wise we could have a predicate involving the patterns themselves, such as x1 OR x2.
Figure 9.2 shows how patterns can be represented in a Venn diagram.
Below we shall need to count predicates, and for clarity it helps to consider a
particular Venn diagram, such as that in Fig. 9.3. This is the most general Venn
diagram based on two features, since for every configuration of f1 and f2 there is
9.2. LACK OF INHERENT SUPERIORITY OF ANY CLASSIFIER 9
x2
f1
f2f3
f1
f2
f3
f1
f3
f2
a) b) c)
x1
x3
x4
x1
x2
x3
x4
x1 x2
x3x5
x6 x5
x4
Figure 9.2: Patterns xi, represented as d-tuples of binary features fi, can be placed
in Venn diagram (here d = 3); the diagram itself depends upon the classification
problem and its constraints. For instance, suppose f1 is the binary feature attribute
has legs, f2 is has right arm and f3 the attribute has right hand. Thus in part a)
pattern x1 denotes a person who has legs but neither arm nor hand; x2 a person who
has legs and an arm, but no hand; and so on. Notice that the Venn diagram expresses
the biological constraints associated with real people: it is impossible for someone to
have a right hand but no right arm. Part c) expresses different constraints, such as
the biological constraint of mutually exclusive eye colors. Thus attributes f1, f2 and
f3 might denote brown, green and blue respectively and a pattern xi describes a real
person, whom we can assume cannot have eyes that differ in color.
indeed a pattern. Here predicates can be as simple as “x1,” or more complicated,
such as “x1 OR x2 OR x4,” and so on.
x2
f1
f2
x1
x3
x4
Figure 9.3: The Venn for a problem with no constraints on two features. Thus all
four binary attribute vectors can occur.
The rank r of a predicate is the number of the simplest or indivisible elements it rank
contains. The tables below show the predicates of rank 1, 2 and 3 associated with the
Venn diagram of Fig. 9.3.∗ Not shown is the fact that there is but one predicate of
rank r = 4, the disjunction of the x1, . . . ,x4, which has the logical value True. If we
let n be the total number of regions in the Venn diagram (i.e., the number of distinct
possible patterns), then there are
(
n
r
)
predicates of rank r, as shown at the bottom of
the table.
∗ Technically speaking, we should use set operations rather than logical operations when discussing
the Venn diagram, writing x1 ∪ x2 instead of x1 OR x2. Nevertheless we use logical operations
here for consistency with the rest of the text.
10 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
rank r = 1 rank r = 2 rank r = 3
x1 f1 AND NOT f2 x1 OR x2 f1 x1 OR x2 OR x3 f1 OR f2
x2 f1 AND f2 x1 OR x3 f1 XOR f2 x1 OR x2 OR x4 f1 OR NOT f2
x3 f2 AND NOT f1 x1 OR x4 NOT f2 x1 OR x3 OR x3 NOT(f1 AND f2)
x4 NOT (f1 OR f2) x2 OR x3 f2 x2 OR x3 OR x4 f2 OR NOTf1
x2 OR x4 NOT(f1 AND f2)
x3 OR x4 NOT f1
(
4
1
)
= 4
(
4
2
)
= 6
(
4
3
)
= 4
The total number of predicates in the absence of constraints is
n∑
r=0
(
n
r
)
= (1 + 1)n = 2n, (5)
and thus for the d = 4 case of Fig. 9.3, there are 24 = 16 possible predicates (Prob-
lem 9). Note that Eq. 5 applies only to the case where there are no constraints; for
Venn diagrams that do incorporate constraints, such as those in Fig. 9.2, the formula
does not hold (Problem 10).
Now we turn to our central question: In the absence of prior information, is there
a principled reason to judge any two distinct patterns as more or less similar than two
other distinct patterns? A natural and familiar measure of similarity is the number
of features or attributes shared by two patterns, but even such an obvious measure
presents conceptual difficulties.
To appreciate such difficulties, consider first a simple example. Suppose attributes
f1 and f2 represent blind in right eye and blind in left eye, respectively. If we
base similarity on shared features, person x1 = {1, 0} (blind only in the right eye) is
maximally different from person x2 = {0, 1} (blind only in the left eye). In particular,
in this scheme x1 is more similar to a totally blind person and to a normally sighted
person than he is to x2. But this result may prove unsatisfactory; we can easily
envision many circumstances where we would consider a person blind in just the right
eye to be “similar” to one blind in just the left eye. Such people might be permitted
to drive automobiles, for instance. Further, a person blind in just one eye would differ
significantly from totally blind person who would not be able to drive.
A second, related point is that there are always multiple ways to represent vectors
(or tuples) of attributes. For instance, in the above example, we might use alter-
native features f ′1 and f
′
2 to represent blind in right eye and same in both eyes,
respectively, and then the four types of people would be represented as shown in the
tables.
f1 f2 f
′
1 f
′
2
x1 0 0 0 1
x2 0 1 0 0
x3 1 0 1 0
x4 1 1 1 1
Of course there are other representations, each more or less appropriate to the par-
ticular problem at hand. In the absence of prior information, though, there is no
principled reason to prefer one of these representations over another.
9.2. LACK OF INHERENT SUPERIORITY OF ANY CLASSIFIER 11
We must then still confront the problem of finding a principled measure the simi-
larity between two patterns, given some representation. The only plausible candidate
measure in this circumstance would be the number of predicates (rather than the
number of features) the patterns share. Consider two distinct patterns (in some rep-
resentation) xi and xj , where i �= j. Regardless of the constraints in the problem
(i.e., the Venn diagram), there are, of course, no predicates of rank r = 1 that are
shared by the two patterns. There is but one predicate of rank r = 2, i.e., xi OR xj .
A predicate of rank r = 3 must contain three patterns, two of which are xi and xj .
Since there are d patterns total, there are then
(
d−2
1
)
= d−2 predicates of rank 3 that
are shared by xi and xj . Likewise, for an arbitrary rank r, there are
(
d−2
r−2
)
predicates
shared by the two patterns, where 2 ≤ r ≤ d. The total number of predicates shared
by the two patterns is thus the sum
d∑
r−2
(
d− 2
r − 2
)
= (1 + 1)d−2 = 2d−2. (6)
Note the key result: Eq. 6 is independent of the choice of xi and xj (so long as they
are distinct). Thus we conclude that the number of predicates shared by two distinct
patterns is constant, and independent of the patterns themselves (Problem 11). We
conclude that if we judge similarity based on the number of predicates that patterns
share, then any two distinct patterns are “equally similar.” This is stated formally
as:
Theorem 9.2 (Ugly Duckling) Given that we use a finite set of predicates that en-
ables us to distinguish any two patterns under consideration, the number of predicates
shared by any two such patterns is constant and independent of the choice of those
patterns. Furthermore, if pattern similarity is based on the total number of predicates
shared by two patterns, then any two patterns are “equally similar.” ∗
In summary, then, the Ugly Duckling Theorem states something quite simple yet
important: there is no problem-independent or privileged or “best” set of features or
feature attributes. Moreover, while the above was derived using d-tuples of binary
values, it also applies to a continuous feature spaces too, if such as space is discretized
(at any resolution). The Theorem forces us to acknowledge that even the appar-
ently simple notion of similarity between patterns is fundamentally based on implicit
assumptions about the problem domain (Problem 12).
9.2.3 Minimum description length (MDL)
It is sometimes claimed that the minimum description length principle provides jus-
tification for preferring one type of classifier over another — specifically “simpler”
classifiers over “complex” ones. Briefly stated, the approach purports to find some ir-
reducible, smallest representation of all members of a category (much like a “signal”);
all variation among the individual patterns is then “noise.” The principle argues that
by simplifying recognizers appropriately, the signal can be retained while the noise is
ignored. Because the principle is so often invoked, it is important to understand what
properly derives from it, what does not, and how it relates to the No Free Lunch
∗ The Theorem gets its fanciful name from the following counter-intuitive statement: Assuming
similarity is based on the number of shared predicates, an ugly duckling A is as similar to beautiful
swan B as beautiful swan C is to B, given that these items differ from one another.
12 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
Theorem. To do so, however, we must first understand the notion of algorithmic
complexity.
Algorithmic complexity
Algorithmic complexity — also known as Kolmogorov complexity, Kolmogorov-Chaitin
complexity, descriptional complexity, shortest program length or algorithmic entropy
— seeks to quantify an inherent complexity of a binary string. (We shall assume both
classifiers and patterns are described by such strings.) Algorithmic complexity can be
explained by analogy to communication, the earliest application of information theory
(App. ??). If the sender and receiver agree upon a specification method L, such as
an encoding or compression technique, then message x can then be transmitted as y,
denoted L(y) = x or y : L(y) = x. The cost of transmission of x is the length of the
transmitted message y, that is, |y|. The least such cost is hence the minimum length
of such a message, denoted min
|y|
: L(y) = x; this minimal length is the entropy of x
under the specification or transmission method L.
Algorithmic complexity is defined by analogy to entropy, where instead of a spec-
ification method L, we consider programs running on an abstract computer, i.e., oneabstract
computer whose functions (memory, processing, etc.) are described operationally and without
regard to hardware implementation. Consider an abstract computer that takes as a
program a binary string y and outputs a string x and halts. In such a case we say
that y is an abstract encoding or description of x.
A universal description should be independent of the specification (up to some ad-
ditive constant), so that we can compare the complexities of different binary strings.
Such a method would provide a measure of the inherent information content, the
amount of data which must be transmitted in the absence of any other prior knowl-
edge. The Kolmogorov complexity of a binary string x, denoted K(x), is defined as
the size of the shortest program y, measured in bits, that without additional data
computes the string x and halts. Formally, we write
K(x) = min
|y|
[U(y) = x], (7)
where U represents an abstract universal Turing machine or Turing computer. For ourTuring
machine purposes it suffices to state that a Turing machine is “universal” in that it can imple-
ment any algorithm and compute any computable function. Kolmogorov complexity
is a measure of the incompressibility of x, and is analogous to minimal sufficient statis-
tics, the optimally compressed representation of certain properties of a distribution
(Chap. ??).
Consider the following examples. Suppose x consists solely of n 1s. This string
is actually quite “simple.” If we use some fixed number of bits k to specify a gen-
eral program containing a loop for printing a string of 1s, we need merely log2n
more bits to specify the iteration number n, the condition for halting. Thus the
Kolmogorov complexity of a string of n 1s is K(x) = O(log2n). Next consider the
transcendental number π, whose infinite sequence of seemingly random binary digits,
11.00100100001111110110101010001 . . .2, actually contains only a few bits of informa-
tion: the size of the shortest program that can produce any arbitrarily large number
of consecutive digits of π. Informally we say the algorithmic complexity of π is a
constant; formally we write K(π) = O(1), which means K(π) does not grow with in-
creasing number of desired bits. Another example is a “truly” random binary string,
which cannot be expressed as a shorter string; its algorithmic complexity is within a
9.2. LACK OF INHERENT SUPERIORITY OF ANY CLASSIFIER 13
constant factor of its length. For such a string we write K(x) = O(|x|), which means
that K(x) grows as fast as the length of x (Problem 13).
9.2.4 Minimum description length principle
We now turn to a simple, “naive” version of the minimum description length principle
and its application to pattern recognition. Given that all members of a category
share some properties, yet differ in others, the recognizer should seek to learn the
common or essential characteristics while ignoring the accidental or random ones.
Kolmogorov complexity seeks to provide an objective measure of simplicity, and thus
the description of the “essential” characteristics.
Suppose we seek to design a classifier using a training set D. The minimum
description length (MDL) principle states that we should minimize the sum of the
model’s algorithmic complexity and the description of the training data with respect
to that model, i.e.,
K(h,D) = K(h) + K(D using h). (8)
Thus we seek the model h∗ that obeys h∗ = arg min
h
K(h,D) (Problem 14). (Variations
on the naive minimum description length principle use a weighted sum of the terms
in Eq. 8.) In practice, determining the algorithmic complexity of a classifier depends
upon a chosen class of abstract computers, and this means the complexity can be
specified only up to an additive constant.
A particularly clear application of the minimum description length principle is in
the design of decision tree classifiers (Chap. ??). In this case, a model h specifies the
tree and the decisions at the nodes; thus the algorithmic complexity of the model is
proportional to the number of nodes. The complexity of the data given the model
could be expressed in terms of the entropy (in bits) of the data D, the weighted sum
of the entropies of the data at the leaf nodes. Thus if the tree is pruned based on
an entropy criterion, there is an implicit global cost criterion that is equivalent to
minimizing a measure of the general form in Eq. 8 (Computer exercise 1).
It can be shown theoretically that classifiers designed with a minimum description
length principle are guaranteed to converge to the ideal or true model in the limit
of more and more data. This is surely a very desirable property. However, such
derivations cannot prove that the principle leads to superior performance in the finite
data case; to do so would violate the No Free Lunch Theorems. Moreover, in practice
it is often difficult to compute the minimum description length, since we may not
be clever enough to find the “best” representation (Problem 17). Assume there is
some correspondence between a particular classifier and an abstract computer; in
such a case it may be quite simple to determine the length of the string y necessary
to create the classifier. But since finding the algorithmic complexity demands we
find the shortest such string, we must perform a very difficult search through possible
programs that could generate the classifier.
The minimum description length principle can be viewed from a Bayesian per-
spective. Using our current terminology, Bayes formula states
P (h|D) = P (h)P (D|h)
P (D)
(9)
for discrete hypotheses and data. The optimal hypothesis h∗ is the one yielding the
highest posterior probability, i.e.,
14 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
h∗ = arg max
h
[P (h)P (D|h)]
= arg max
h
[log2P (h) + log2P (D|h)], (10)
much as we saw in Chap. ??. We note that a string x can be communicated or repre-
sented at a cost bounded below by −log2P (x), as stated in Shannon’s optimal coding
theorem. Shannon’s theorem thus provides a link between the minimum description
length (Eq. 8) and the Bayesian approaches (Eq. 10). The minimum description
length principle states that simple models (small K(h)) are to be preferred, and thus
amounts to a bias toward “simplicity.” It is often easier in practice to specify such a
prior in terms of a description length than it is using functions of distributions (Prob-
lem 16). We shall revisit the issue of the tradeoff between simplifying the model and
fitting the data in the bias-variance dilemma in Sec. 9.3.
It is found empirically that classifiers designed using the minimum description
length principle work well in many problems. As mentioned, the principle is effectively
a method for biasing priors over models toward “simple” models. The reasons for the
many empirical success of the principle are not trivial, as we shall see in Sect. 9.2.5.
One of the greatest benefits of the principle is that it provides a computationally
clear approach to balancing model complexity and the fit of the data. In somewhat
more heuristic methods, such as pruning neural networks, it is difficult to compare
the algorithmic complexity of the network (e.g., number of units or weights) with the
entropy of the data with respect to that model.
9.2.5 Overfitting avoidance and Occam’s razor
Throughout our discussions of pattern classifiers, we have mentioned the need to avoid
overfitting by means of regularization, pruning, inclusion of penalty terms, minimizing
a description length, and so on. The No Free Lunch results throw such techniques
into question. If there are no problem-independent reasons to prefer one algorithm
over another, why is overfitting avoidance nearly universally advocated? For a given
training error, why do we generally advocate simple classifiers with fewer features and
parameters?
In fact, techniques for avoiding overfitting or minimizing description length are
not inherently beneficial; instead, such techniques amount to a preference, or “bias,”
over the forms or parameters of classifiers. They are only beneficial if they happen
to address problems for which they work. It is the match of the learning algorithm
to the problem — not the imposition of overfitting avoidance — that determines the
empirical success. There are problems for which overfitting avoidance actually leads
to worse performance. The effects of overfitting avoidance depend upon the choice of
representation too; if the feature space is mapped to a new, formally equivalent one,
overfitting avoidance has different effects (Computer exercise ??).
In light of the negative results from the No Free Lunch theorems, we might probe
more deeply into the frequent empirical “successes” of the minimum description length
principle and the more general philosophical principle of Occam’s razor. In its original
form, Occam’s razor stated merely that “entities” (or explanations) should not be
multiplied beyond necessity, but it has come to be interpreted in pattern recognition
as counselling that one should not use classifiers that are more complicated than are
necessary, where “necessary” is determined by the quality of fit to the training data.
Given the respective requisite assumptions, the No Free Lunch theorem proves that
9.3. BIAS AND VARIANCE 15
there is no benefit in “simple” classifiers (or “complex” ones, for that matter) —
simple classifiers claim neither unique nor universal validity.
The frequent empirical “successes” of Occam’s razor imply that the classes of
problems addressed so far have certain properties. What might be the reason we
explore problems that tend to favor simpler classifiers? A reasonable hypothesis is that
through evolution, we have had strong selection pressure on our pattern recognition
apparatuses to be computationally simple — require fewer neurons, less time, and
so forth — and in general such classifiers tend to be “simple.” We are more likely
to ignore problems for which Occam’s razor does not hold. Analogously, researchers
naturally develop simple algorithms before more complex ones, as for instance in
the progression from the Perceptron, to multilayer neural networks, to networks with
pruning, to networks with topology learning, to hybrid neural net/rule-based methods,
and so on — each more complex than its predecessor. Each method is found to
work on some problems, but not ones that are “too complex.” For instance the
basic Perceptron is inadequate for optical character recognition; a simple three-layer
neural network is inadequate for speaker-independent speech recognition. Hence our
design methodology itself imposes a bias toward “simple” classifiers; we generally
stop searching for a design when the classifier is “good enough.” This principle of
satisficing — creating an adequate though possibly non-optimal solution — underlies satisficing
much of practical pattern recognition as well as human cognition.
Another “justification” for Occam’s razor derives from a property we might strongly
desire or expect in a learning algorithm. If we assume that adding more training data
does not, on average, degrade the generalization accuracy of a classifier, then a version
of Occam’s razor can in fact be derived. Note, however, that such a desired property
amounts to a non-uniform prior over learning algorithms — while this property is
surely desirable, it is a premise and cannot be “proven.” Finally, the No Free Lunch
theorem implies that we cannot use training data to create a scheme by which we can
with some assurance distinguish new problems for which the classifier will generalize
well from new problems for which the classifier will generalize poorly (Problem 8).
9.3 Bias and variance
Given that there is no general best classifier unless the probability over the class of
problems is restricted, practitioners must be prepared to explore a number of methods
or models when solving any given classification problem. Below we will define two ways
to measure the “match” or “alignment” of the learning algorithm to the classification
problem: the bias and the variance. The bias measures the accuracy or quality of
the match: high bias implies a poor match. The variance measures the precision or
specificity of the match: a high variance implies a weak match. Designers can adjust
the bias and variance of classifiers, but the important bias-variance relation shows
that the two terms are not independent; in fact, for a given mean-square error, they
obey a form of “conservation law.” Naturally, though, with prior information or even
mere luck, classifiers can be created that have a different mean-square error.
9.3.1 Bias and variance for regression
Bias and variance are most easily understood in the context of regression or curve
fitting. Suppose there is a true (but unknown) function F (x) with continuous valued
output with noise, and we seek to estimate it based on n samples in a set D generated
16 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
by F (x). The regression function estimated is denoted g(x; D) and we are interested
in the dependence of this approximation on the training set D. Due to random
variations in data selection, for some data sets of finite size this approximation will
be excellent while for other data sets of the same size the approximation will be poor.
The natural measure of the effectiveness of the estimator can be expressed as its
mean-square deviation from the desired optimal. Thus we average over all training
sets D of fixed size n and find (Problem 18)
ED
[
(g(x; D)− F (x))2
]
= (ED[g(x; D)− F (x)])2︸ ︷︷ ︸
bias2
+ ED
[
(g(x; D)− ED[g(x; D)])2
]
︸ ︷︷ ︸
variance
. (11)
The first term on the right hand side is the bias (squared) — the difference betweenbias
the expected value and the true (but generally unknown) value — while the second
term is the variance. Thus a low bias means on average we accurately estimate Fvariance
from D. Further, a low variance means the estimate of F does not change much as the
training set varies. Even if an estimator is unbiased (i.e., the bias = 0 and its expected
value is equal to the true value), there can nevertheless be a large mean-square error
arising from a large variance term.
Equation 11 shows that the mean-square error can be expressed as the sum of a bias
and a variance term. The bias-variance dilemma or bias-variance trade-off is a generalbias-
variance
dilemma
phenomenon: procedures with increased flexibility to adapt to the training data (e.g.,
have more free parameters) tend to have lower bias but higher variance. Different
classes of regression functions g(x; D) — linear, quadratic, sum of Gaussians, etc. —
will have different overall errors; nevertheless, Eq. 11 will be obeyed.
Suppose for example that the true, target function F (x) is a cubic polynomial of
one variable, with noise, as illustrated in Fig. 9.4. We seek to estimate this function
based on a sampled training setD. Column a) at the left, shows a very poor “estimate”
g(x) — a fixed linear function, independent of the training data. For different training
sets sampled from F (x) with noise, g(x) is unchanged. The histogram of this mean-
square error of Eq. 11, shown at the bottom, reveals a spike at a fairly high error;
because this estimate is so poor, it has a high bias. Further, the variance of the
constant model g(x) is zero. The model in column b) is also fixed, but happens to be
a better estimate of F (x). It too has zero variance, but a lower bias than the poor
model in a). Presumably the designer imposed some prior knowledge about F (x) in
order to get this improved estimate.
The model in column c) is a cubic with trainable coefficients; it would learn F (x)
exactly if D contained infinitely many training points. Notice the fit found for every
training set is quite good. Thus the bias is low, as shown in the histogram at the
bottom. The model in d) is linear in x, but its slope and intercept are determined
from the training data. As such, the model in d) has a lower bias than the models in
a) and b).
In sum, for a given target function F (x), if a model has many parameters (generally
low bias), it will fit the data well but yield high variance. Conversely, if the model
has few parameters (generally high bias), it may not fit the data particularly well,
but this fit will not change much as for different data sets (low variance). The best
way to get low bias and low variance is the have prior information about the target
function. We can virtually never get zero bias and zero variance; to do so would mean
there is only one learning problem to be solved, in which case the answer is already
9.3. BIAS AND VARIANCE 17
x
y
F(x)
x
y
F(x)
x
y
F(x)
g(x)
g(x)
g(x)
x
y
F(x)
x
y
F(x)
x
y
F(x)
g(x)
g(x)
g(x)
x
y
F(x)
x
y
F(x)
x
y
F(x)
g(x)
g(x)
g(x)
x
y
F(x)
x
y
F(x)
x
y
F(x)
g(x)
g(x)
g(x)
D1
D2
D3
p
E
p
E
p
E
p
E
g(x) = fixed g(x) = a
0
+ a
1
x + a
0
x2 +a
3
x3
learned
g(x) = a
0
+ a
1
x
learned
g(x) = fixed
bias
va
ri
a
n
ce
va
ri
a
n
ce
va
ri
a
n
ce
va
ri
a
n
ce
bias biasbias
a) b) c) d)
Figure 9.4: The bias-variance dilemma can be illustrated in the domain of regression.
Each column represents a different model, each row a different set of n = 6 training
points, Di, randomly sampled from the true function F (x) with noise. Histograms of
the mean-square error of E ≡ ED[(g(x)− F (x))2] of Eq. 11 are shown at the bottom.
Column a) shows a very poor model: a linear g(x) whose parameters are held fixed,
independent of the training data. This model has high bias and zero variance. Column
b) shows a somewhat better model, though it too is held fixed, independent of the
training data. It has a lower bias than in a) and the same zero variance. Column
c) shows a cubic model, where the parameters are trained to best fit the training
samples in a mean-square error sense. This model has low bias, and a moderate
variance. Column d) shows a linear model that is adjusted to fit each training set;
this model has intermediate bias and variance. If these models were instead trained
with a very large number n → ∞ of points, the bias in c) would approach a small
value (which depends upon the noise), while the bias in d) would not; the variance of
all models would approach zero.
18 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
known. Furthermore, a large amount of training data will yield improved performance
so long as the model is sufficiently general to represent the target function. These
considerations of bias and variance help to clarify the reasons we seek to have as much
accurate prior information about the form of the solution, and as large a training set
as feasible; the match of the algorithm to the problem is crucial.
9.3.2 Bias and variance for classification
While the bias-variance decomposition and dilemma are simplest to understand in
the case of regression, we are most interested in their relevance to classification; here
there are a few complications. In a two-category classification problem we let the
target (discriminant) function have value 0 or +1, i.e.,
F (x) = Pr[y = 1|x] = 1− Pr[y = 0|x]. (12)
On first consideration, the mean-square error we saw for regression (Eq. 11) does not
appear to be the proper one for classification. After all, even if the mean-square error
fit is poor, we can have accurate classification, possibly even the lowest (Bayes) error.
This is because the decision rule under a zero-one loss selects the higher posterior
P (ωi|x), regardless the amount by which it is higher. Nevertheless by considering the
expected value of y, we can recast classification into the framework of regression we
saw before. To do so, we consider a discriminant function
y = F (x) + $, (13)
where $ is a zero-mean, random variable, for simplicity here assumed to be a centered
binomial distribution with variance Var[$|x] = F (x)(1− F (x)). The target function
can thus be expressed as
F (x) = E [y|x], (14)
and now the goal is to find an estimate g(x; D) which minimizes a mean-square error,
such as in Eq. 11:
ED[(g(x; D)− y)2]. (15)
In this way the regression methods of Sect. 9.3.1 can yield an estimate g(x; D) used
for classification.
For simplicity we assume equal priors, P (ω1) = P (ω2) = 0.5, and thus the Bayes
discriminant yB has threshold 1/2 and the Bayes decision boundary is the set of points
for which F (x) = 1/2. For a given training set D, if the classification error rate
Pr[g(x; D) = y] averaged over predictions at x agrees with the Bayes discriminant,
Pr[g(x; D) = y] = Pr[yB(x) �= y] = min[F (x), 1− F (x)], (16)
then indeed we have the lowest error. If not, then the prediction yields an increased
error
Pr[g(x; D)] = max[F (x), 1− F (x)] (17)
= |2F (x)− 1|+ Pr[yB(x) = y].
We average over all data sets of size n and find
9.3. BIAS AND VARIANCE 19
Pr[g(x; D) �= y] = |2F (x)− 1|Pr[g(x; D) �= yB ] + Pr[yB �= y]. (18)
Equation 18 shows that classification error rate is linearly proportional to Pr[g(x; D) �= yB ],
which can be considered a boundary error in that it represents the mis-estimation of boundary
errorthe optimal (Bayes) boundary (Problem 19).
Because of random variations in training sets, the boundary error will depend
upon p(g(x; D)), the probability density of obtaining a particular estimate of the
discriminant given D. This error is merely the area of the tail of p(g(x; D)) on
the opposite side of the Bayes discriminant value 1/2, much as we saw in Chap. ??,
Fig. ??:
Pr[g(x; D) �= yB ] =
∞∫
1/2
p(g(x; D))dg if F (x) < 1/2
1/2∫
−∞
p(g(x; D))dg if F (x) ≥ 1/2.
(19)
If we make the natural assumption that p(g(x; D)) is a Gaussian, we find (Problem 20)
Pr[g(x; D) �= yB ] = Φ
[
sgn[F (x)− 1/2]ED[g(x; D)]− 1/2√
Var[g(x; D)]
]
(20)
= Φ
[
sgn[F (x)− 1/2][ED[g(x; D)]− 1/2︸ ︷︷ ︸
boundary bias
] Var[g(x; D)]−1/2︸ ︷︷ ︸
variance
]
,
where
Φ[t] =
1√
2π
∞∫
t
e−1/2u
2
du = 1− erf[t] (21)
and erf[·] is the familiar error function (App. ??).
We have expressed this boundary error in terms of a boundary bias, in analogy boundary
biaswith the simple bias-variance relation in regression (Eq. 11). Equation 20 shows that
the effect of the variance term on the boundary error is highly nonlinear and depends
on the value of the boundary bias. Further, when the variance is small, this effect
is particularly sensitive to the sign of the bias. In regression the estimation error
is additive in bias2 and variance, whereas for classification there is a nonlinear and
multiplicative interaction. In classification the sign of the boundarybias affects the
role of variance in the error. For this reason low variance is generally important for
accurate classification while low boundarybias need not be. Or said another way, in
classification, variance generally dominates bias. In practical terms, this implies we
need not be particularly concerned if our estimation is biased, so long as the variance
is kept low. Numerous specific methods of classifier adjustment — pruning neural
networks or decision trees, varying the number of free parameters, etc. — affect the
bias and variance of a classifier; in Sect. 9.5 we shall discuss some methods applicable
to a broad range of classification methods. Much as we saw in the bias-variance
dilemma for regression, classification procedures with increased flexibility to adapt to
the training data (e.g., have more free parameters) tend to have lower bias but higher
variance.
20 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
As an illustration of boundarybias and variance in classifiers, consider a simple
two-class problem in which samples are drawn from two-dimensional Gaussian distri-
butions, each parameterized by vectors p(x|ωi) ∼ N(µi,Σi), for i = 1, 2. Here the
true distributions have diagonal covariances, as shown at the top of Fig. 9.5. We have
just a few samples from each category and estimate the parameters in three different
classes of models by maximum likelihood. Column a) at the left shows the most gen-
eral Gaussian classifiers; each component distribution can have arbitrary covariance
matrix. Column b) at the middle shows classifiers where each component Gaussian
is constrained to have a diagonal covariance. Column c) at the right shows the most
restrictive model: the covariances are equal to the identity matrix, yielding circular
Gaussian distributions. Thus the left column corresponds to very low bias, and the
right column to high bias.
Each row in Fig. 9.5 represents a different training set, randomly selected from
the true distribution (shown at the top), and the resulting classifiers. Notice that
most feature points in the high bias cases retain their classification, regardless of the
particular training set (i.e., such models have low variance), whereas the classification
of a much larger range of points varies in the low bias case (i.e., there is high variance).
While in general a lower bias comes at the expense of higher variance, the relationship
is nonlinear and multiplicative.
At the bottom of the figure, three density plots show how the location of the
decision boundary varies across many different training sets. The left-most density
plot shows a very broad distribution (high variance). The right-most plot shows a
narrow, peaked distribution (low variance). To visualize the bias, imagine taking the
spatial average of the decision boundaries obtained by running the learning algorithm
on all possible data sets. The average of such boundaries for the left-most algorithm
will be equal to the true decision boundary — this algorithm has no bias. The right-
most average will be a vertical line, and hence there will be higher error — this
algorithm has the highest bias of the three. Histograms of the generalization error
are shown along the bottom.
For a given bias, the variance will decrease as n is increased. Naturally, if we
had trained using a very large training set (n → ∞), all error histograms become
narrower and move to lower values of E. If a model is rich enough to express the
optimal decision boundary, its error histogram for the large n case will approach a
delta function at E = EB , the Bayes error.
As mentioned, to achieve the desired low generalization error it is more important
to have low variance than to have low bias. The only way to get the ideal of zero bias
and zero variance is to know the true model ahead of time (or be astoundingly lucky
and guess it), in which case no learning was needed anyway. Bias and variance can be
lowered with large training size n and accurate prior knowledge of the form of F (x).
Further, as n grows, more parameters must be added to the model, g, so the data
can be fit (reducing bias). For best classification based on a finite training set, it is
desirable to match the form of the model to that of the (unknown) true distributions;
this usually requires prior knowledge.
9.4 *Resampling for estimating statistics
When we apply some learning algorithm to a new pattern recognition problem with
unknown distribution, how can we determine the bias and variance? Figures 9.4 &
9.5 suggest a method using multiple samples, an inspiration for formal “resampling”
9.4. *RESAMPLING FOR ESTIMATING STATISTICS 21
Σi = ( ) 1 00 1Σi = ( )Σi = ( )
σ
i1 0
0 σi2
2
2
σ
i1 σi12
σi21 σi2
2
2
Bias highlow
truth
Variance lowhigh
p p p
E E EEB EB EB
D1
D2
D3
a) c)b)
bo
un
da
ry
di
st
ri
bu
ti
on
s
er
ro
r
hi
st
og
ra
m
s
x
2
x
1
x
2
x
1
x
2
x
1x
2
x
1
x
2
x
1
x
2
x
1x
2
x
1
x
2
x
1
x
2
x
1
x
2
x
1
x
2
x
1
x
2
x
1
Figure 9.5: The (boundary) bias-variance tradeoff in classification can be illustrated
with a two-dimensional Gaussian problem. The figure at the top shows the (true)
decision boundary of the Bayes classifier. The nine figures in the middle show nine
different learned decision boundaries. Each row corresponds to a different training set
of n = 8 points selected randomly from the true distributions and labeled according
to the true decision boundary. Column a) shows the decision boundaries learning
by fitting a Gaussian model with fully general covariance matrices by maximum like-
lihood. The learned boundaries differ significantly from one data set to the next;
this learning algorithm has high variance. Column b) shows the decision boundaries
resulting from fitting a Gaussian model with diagonal covariances; in this case the
decision boundaries vary less from one row to another. This learning algorithm has a
lower variance than the one at the left. Finally, column c) at the right shows decision
boundaries learning by fitting a Gaussian model with unit covariances (i.e., a linear
model); notice that the decision boundaries are nearly identical from one data set to
the next. This algorithm has low variance.
22 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
methods, which we now discuss. Later we shall turn to our ultimate goal: using
resampling and related techniques to improve classification (Sect. 9.5).
9.4.1 Jackknife
We begin with an example of how resampling can be used to yield a more informative
estimate of a general statistic. Suppose we have a set D of n data points xi (i =
1, . . . , n), sampled from a one-dimensional distribution. The familiar estimate of the
mean is, of course,
µ̂ =
1
n
n∑
i=1
xi. (22)
Likewise the estimate of the accuracy of the mean is the standard deviation, given by
σ̂2 =
1
n(n− 1)
n∑
i=1
(xi − µ̂)2. (23)
Suppose we were instead interested in the median, the point for which half of themedian
distribution is higher, half lower. Although we could determine the median explic-
itly, there does not seem to be a straightforward way to generalize Eq. 23 to give a
measure of the error of our estimate of the median. The same difficulty applies to
estimating the mode (the most frequently represented point in a data set), the 25thmode
percentile, or any of a large number of statistics other than the mean. The jackknife∗
and bootstrap (Sect. 9.4.2) are two of the most popular and theoretically grounded
resampling techniques for extending the above approach (based on Eqs. 22 & 23) to
arbitrary statistics, of which the mean is just one instance.
In resampling theory, we frequently use statistics in which a data point is elimi-
nated from the data; we denote this by means of a special subscript. For instance,
the leave-one-out mean isleave-one-
out mean
µ(i) =
1
n− 1
∑
j �=i
xj =
nx̄− xi
n− 1
, (24)
i.e., the sample average of the data set if the ith point is deleted. Next we define the
jackknife estimate of the mean to be
µ(·) =
1
n
n∑
i=1
µ(i), (25)
that is, the mean of the leave-one-out means. It is simple to prove that the traditional
estimate of the mean and the jackknife estimate of the mean are the same, i.e., µ̂ = µ(·)
(Problem 23). Likewise, the jackknife estimate of the variance of the estimate obeys
Var[µ̂] =
n− 1
n
n∑
i=1
(µ(i) − µ(·))2, (26)
and, applied to the mean, is equivalent to the traditional variance of Eq. 23 (Problem
26).
∗ The jackknife method, which also goes by the name of “leave one out,” was due to Maurice
Quenouille. The playful name was chosen by John W. Tukey to capture the impression that the
method was handy and useful in lots of ways.
9.4. *RESAMPLING FOR ESTIMATING STATISTICS 23
The benefit of expressing the variance in the form of Eq. 26 is that it can be
generalized to any other estimator θ̂, such as the median or 25th percentile or mode,
... To do so we need to compute the statistic with one data point “left out.” Thus
we let
θ̂(i) = θ̂(x1, x2, · · · , xi−1, xi+1, · · · , xn) (27)
take the place of µ(i), and let θ̂(·) take the place of µ(·) in Eqs. 25 & 26 above.
Jackknife bias estimate
The notion of bias is more general that that described in Sect. 9.3; in fact it can be
applied to the estimation of any statistic. The bias of an estimator θ is the difference bias
between its true value and its expected value, i.e.,
bias = θ − E [θ]. (28)
The jackknife method can be used estimate such a bias. The procedure is first to
sequentially delete points xi one at a time from D and compute the estimate θ̂(·).
Then the jackknife estimate of the bias is (Problem 21)
biasjack = (n− 1)(θ̂(·) − θ̂). (29)
We rearrange terms and thus see that the jackknife estimate of θ itself is
θ̃ = θ̂ − biasjack = nθ̂ − (n− 1)θ̂(·). (30)
The benefit of using Eq. 30 is that it is a quadratic function, unbiased for estimating
the true bias (Problem 25).
Jackknife variance estimate
Now we seek the jackknife estimate of the variance of an arbitrary statistic θ. First,
recall that the traditional variance is defined as:
Var[θ̂] = E [θ̂(x1, x2, · · · , xn)− E [θ̂]]2. (31)
The jackknife estimate of the variance, defined by analogy to Eq. 26, is:
Varjack[θ̂] =
n− 1
n
n∑
i=1
[θ̂(i) − θ̂(·)]2, (32)
where as before θ̂(·) =
1
n
n∑
i=1
θ̂(i).
Example 2: Jackknife estimate of bias and variance of the mode
Consider an elementary example where we are interested in the mode of the fol-
lowing n = 6 points: D = {0, 10, 10, 10, 20, 20}. It is clear from the histogram that
the most frequently represented point is θ̂ = 10. The jackknife estimate of the mode
is
24 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
θ̂(·) =
1
n
n∑
i=1
θ̂(i) =
1
6
[10 + 15 + 15 + 15 + 10 + 10] = 12.5,
where for i = 2, 3, 4 we used the fact that the mode of a distribution having two
equal peaks is the point midway between those peaks. The fact that θ̂(·) > θ̂ reveals
immediately that the jackknife estimate takes into account more of the full (skewed)
distribution than does the standard mode calculation.
x
P(x)
0 5 10 15 20 25
1
2
3
θ(.)θ
A histogram of n = 6 points whose mode is θ̂ = 10 and jackknife estimate of the mode
is θ̂(·) = 12.5. The square root of the jackknife estimate of the variance is a natural
measure of the range of probable values of the mode. This range is indicated by the
horizontal red bar.
The jackknife estimate of the bias of the estimate of the mode is given by Eq. 29:
biasjack = (n− 1)(θ̂(·) − θ̂) = 5(12.5− 10) = 12.5.
Likewise, the jackknife estimate of the variance is given by Eq. 32:
Varjack[θ̂] =
n− 1
n
n∑
i=1
(θ̂(i) − θ̂(·))2
=
5
6
[(10− 12.5)2 + 3(15− 12.5)2 + 2(10− 12.5)2] = 31.25.
The square root of this variance,
√
31.25 � 5.6, serves as an effective standard de-
viation. A red bar of twice this width, shown below the histogram, reveals that the
traditional mode lies within this tolerance to the jackknife estimate of the mode.
The jackknife resampling technique often gives us a more satisfactory estimate of
a statistic such as the mode than do traditional methods though it is more computa-
tionally complex (Problem 27).
9.4.2 Bootstrap
A “bootstrap” data set is one created by randomly selecting n points from the training
set D, with replacement. (Since D itself contains n points, there is nearly always
9.5. RESAMPLING FOR CLASSIFIER DESIGN 25
duplication of individual points in a bootstrap data set.) In bootstrap estimation,∗
this selection process is independently repeated B times to yield B bootstrap data
sets, which are treated as independent sets. The bootstrap estimate of a statistic θ,
denoted θ̂∗(·), is merely the mean of the B estimates on the individual bootstrap data
sets:
θ̂∗(·) =
1
B
B∑
b=1
θ̂∗(b), (33)
where θ̂∗(b) is the estimate on bootstrap sample b.
Bootstrap bias estimate
The bootstrap estimate of the bias is (Problem ??)
biasboot =
1
B
B∑
b=1
θ̂∗(b) − θ̂ = θ̂∗(·) − θ̂. (34)
Computer exercise 45 shows how the bootstrap can be applied to statistics that resist
computational analysis, such as the “trimmed mean,” in which the mean is calculated trimmed
meanfor a distribution in which some percentage (e.g., 5%) of the high and the low points
in a distribution have been eliminated.
Bootstrap variance estimate
The bootstrap estimate of the variance is
Varboot[θ] =
1
B
B∑
b=1
[
θ̂∗(b) − θ̂∗(·)
]2
. (35)
If the statistic θ is the mean, then in the limit of B →∞, the bootstrap estimate of
the variance is the traditional variance of the mean (Problem 22). Generally speaking,
the larger the number B of bootstrap samples, the more satisfactory is the estimate
of a statistic and its variance. One of the benefits of bootstrap estimation is that B
can be adjusted to the computational resources; if powerful computers are available
for a long time, then B can be chosen large. In contrast, a jackknife estimate requires
exactly n repetitions: fewer repetitions gives a poorer estimate that depends upon
the random points chosen; more repetitions merely duplicates information already
provided by some of the first n leave-one-out calculations.
9.5 Resampling for classifier design
The previous section addressed the use of resampling in estimating statistics, including
the accuracy of an existing classifier, but only indirectly referred to the design of
classifiers themselves. We now turn to a number of general resampling methods that
have proven effective when used in conjunction with any in a wide range of techniques
∗ “Bootstrap” comes from Rudolf Erich Raspe’s wonderful stories “The adventures of Baron Munch-
hausen,” in which the hero could pull himself up onto his horse by lifting his own bootstraps. A
different but more common usage of the term applies to starting a computer, which must first run
a program before it can run other programs.
26 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
for training classifiers. These are related to methods for estimating and comparing
classifier models that we will discuss in Sect. 9.6.
9.5.1 Bagging
The generic term arcing — adaptive reweighting and combining — refers to reusingarcing
or selecting data in order to improve classification. In Sect. 9.5.2 we shall consider
the most popular arcing procedure, AdaBoost, but first we discuss briefly one of the
simplest. Bagging — a name derived from “bootstrap aggregation” — uses multiple
versions of a training set, each created by drawing n′ < n samples from D with
replacement. Each of these bootstrap data sets is used to train a different component
classifier and the final classification decision is based on the vote of each componentcomponent
classifier classifier.∗ Traditionally the component classifiers are of the same general form —
i.e., all hidden Markov models, or all neural networks, or all decision trees — merely
the final parameter values differ among them due to their different sets of training
patterns.
A classifier/learning algorithm combination is informally called unstable if “small”instability
changes in the training data lead to significantly different classifiers and relatively
“large” changes in accuracy. As we saw in Chap. ??, decision tree classifiers trained
by a greedy algorithm can be unstable — a slight change in the position of a single
training point can lead to a radically different tree. In general, bagging improves
recognition for unstable classifiers since it effectively averages over such discontinuities.
There are no convincing theoretical derivations or simulation studies showing that
bagging will help all stable classifiers, however.
Bagging is our first encounter with multiclassifier systems, where a final overall
classifier is based on the outputs of a number of component classifiers. The global de-
cision rule in bagging — a simple vote among the component classifiers — is the most
elementary method of pooling or integrating the outputs of the component classifiers.
We shall consider multiclassifier systems again in Sect. 9.7, with particular attention
to forming a single decision rule from the outputs of the component classifiers.
9.5.2 Boosting
The goal of boosting is to improve the accuracy of any given learning algorithm. In
boosting we first create a classifier with accuracy on the training set greater than
average, and then add new component classifiers to form an ensemble whose joint
decision rule has arbitrarily high accuracy on the training set. In such a case we say
the classification performance has been “boosted.” In overview, the technique trains
successive component classifiers with a subset of the training data that is “most
informative” given the current set of component classifiers. Classification of a test
point x is based on the outputs of the component classifiers, as we shall see.
For definiteness, consider creating three component classifiers for a two-category
problem through boosting. First we randomly select a set of n1 < n patterns from
the full training set D (without replacement); call this set D1. Then we train the first
classifier, C1, with D1. Classifier C1 need only be a weak learner, i.e., have accuracyweak
learner only slightly better than chance. (Of course, this is the minimum requirement; a
weak learner could have high accuracy on the training set. In that case the benefit
∗ In Sect. 9.7 we shall come across other names for component classifiers. For the present purposes
we simply note that these are not classifiers of component features, but are instead members in an
ensemble of classifiers whose outputs are pooled so as to implement a single classification rule.
9.5. RESAMPLING FOR CLASSIFIER DESIGN 27
of boosting will be small.) Now we seek a second training set, D2, that is the “most
informative” given component classifier C1. Specifically, half of the patterns in D2
should be correctly classified by C1, half incorrectly classified by C1 (Problem 29).
Such an informative set D2 is created as follows: we flip a fair coin. If the coin is
heads, we select remaining samples from D and present them, one by one to C1 until
C1 misclassifies a pattern. We add this misclassified pattern to D2. Next we flip the
coin again. If heads, we continue through D to find another pattern misclassified by
C1 and add it to D2 as just described; if tails we find a pattern which C1 classifies
correctly. We continue until no more patterns can be added in this manner. Thus
half of the patterns in D2 are correctly classified by C1, half are not. As such D2
provides information complementary to that represented in C1. Now we train a second
component classifier C2 with D2.
Next we seek a third data set, D3, which is not well classified by the combined
system C1 and C2. We randomly select a training pattern from those remaining
in D, and classify that pattern with C1 and with C2. If C1 and C2 disagree, we
add this pattern to the third training set D3; otherwise we ignore the pattern. We
continue adding informative patterns to D3 in this way; thus D3 contains those not
well represented by the combined decisions of C1 and C2. Finally, we train the last
component classifier, C3, with the patterns in D3.
Now consider the use of the ensemble of three trained component classifiers for
classifying a test pattern x. Classification is based on the votes of the component
classifiers. Specifically, if C1 and C2 agree on the category label of x, we use that
label; if they disagree, then we use the label given by C3 (Fig. 9.6).
We skipped over a practical detail in the boosting algorithm: how to choose the
number of patterns n1 to train the first component classifier. We would like the
final system to be trained with all patterns in D of course; moreover, because the
final decision is a simple vote among the component classifiers, we would like to have
roughly equal number of patterns in each (i.e., n1 � n2 � n3 � n/3). A reasonable
first guess is to set n1 � n/3 and create the three component classifiers. If the
classification problem is very simple, however, component classifier C1 will explain
most of the data and thus n2 (and n3) will be much less than n1, and not all of the
patterns in the training set D will be used. Conversely, if the problem is extremely
difficult, then C1 will explain but little of the data, and nearly all the patterns will be
informative with respect to C1; thus n2 will be unacceptably large. Thus in practice
we may need to run the overall boosting procedure a few times, adjusting n1 in order
to use the full training set and, if possible, get roughly equal partitions of the training
set. A number of simple heuristics can be used to improve the partitioning of the
training set as well (Computer exercise ??).
The above boosting procedure can be applied recursively to the component clas-
sifiers themselves, giving a 9-component or even 27-component full classifier. In this
way, a very low training error can be achieved, even a vanishing training error if the
problem is separable.
AdaBoost
There are a number of variations on basic boosting. The most popular, AdaBoost
— from “adaptive” boosting — allows the designer to continue adding weak learners
until some desired low training error has been achieved. In AdaBoost each training
pattern receives a weight which determines its probability of being selected for a
training set for an individual component classifier. If a training pattern is accurately
28 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
n1 = 15 n2 = 8 n3 = 4
x1
x2
x1
x2
x1
x2
n = 27
x1
x2
x1
x2C1 C2 C3
co
m
p
o
n
en
t
cl
a
ss
if
ie
rs
fi
n
a
l
cl
a
ss
if
ic
a
ti
o
n
b
y
vo
ti
n
g
R 1
R 2
R 2
R 2
R 1
R 1
R 1
R 2
Figure 9.6: A two-dimensional two-category classification task is shown at the top.
The middle row shows three component (linear) classifiers Ck trained by LMS al-
gorithm (Chap. ??), where their training patterns were chosen through the basic
boosting procedure. The final classification is given by the voting of the three com-
ponent classifiers, and yields a nonlinear decision boundary, as shown at the bottom.
Given that the component classifiers are weak learners (i.e., each can learn a training
set better than chance), then the ensemble classifier will have a lower training error
on the full training set D than does any single component classifier.
classified, then its chance of being used again in a subsequent component classifier
is reduced; conversely, if the pattern is not accurately classified, then its chance of
being used again is raised. In this way, AdaBoost “focuses in” on the informative or
“difficult” patterns. Specifically, we initialize these weights across the training set to
to be uniform. On each iteration k, we draw a training set at random according to
these weights, and train component classifier Ck on the patterns selected. Next we
increase weights of training patterns misclassified by Ck and decrease weights of the
patterns correctly classified by Ck. Patterns chosen according to this new distribution
are used to train the next classifier, Ck+1, and the process is iterated.
We let the patterns and their labels in D be denoted xi and yi, respectively and let
Wk(i) be the kth (discrete) distribution over all these training samples. The AdaBoost
procedure is then:
Algorithm 1 (AdaBoost)
1 begin initialize D = {x1, y1,x2, y2, . . . ,xn, yn}, kmax,W1(i) = 1/n, i = 1, . . . , n
2 k ← 0
3 do k ← k + 1
4 Train weak learner Ck using D sampled according to distribution Wk(i)
5 Ek ← Training error of Ck measured on D using Wk(i)
6 αk ← 12 ln[(1− Ek)/Ek]
9.5. RESAMPLING FOR CLASSIFIER DESIGN 29
7 Wk+1(i)← Wk(i)Zk ×
{
e−αk if hk(xi) = yi (correctly classified)
eαk if hk(xi) �= yi (incorrectly classified)
8 until k = kmax
9 return Ck and αk for k = 1 to kmax (ensemble of classifiers with weights)
10 end
Note that in line 5 the error for classifier Ck is determined with respect to the distri-
bution Wk(i) over D on which it was trained. In line 7, Zk is simply a normalizing
constant computed to insure that Wk(i) represents a true distribution, and hk(xi) is
the category label (+1 or -1) given to pattern xi by component classifier Ck. Natu-
rally, the loop termination of line 8 could instead use the criterion of sufficiently low
training error of the ensemble classifier.
The final classification decision of a test point x is based on a discriminant function
that is merely the weighted sums of the outputs given by the component classifiers:
g(x) =
[
kmax∑
k=1
αkhk(x)
]
. (36)
The classification decision for this two-category case is then simply sgn[g(x)].
Except in pathological cases, so long as each component classifier is a weak learner,
the total training error of the ensemble can be made arbitrarily low by setting the
number of component classifiers, kmax, sufficiently high. To see this, notice that the
training error for weak learner Ck can be written as Ek = 1/2−Gk for some positive
value Gk. Thus the ensemble training error is (Problem 31):
E =
kmax∏
k=1
[
2
√
Ek(1− Ek)
]
=
kmax∏
k=1
√
1− 4G2k
≤ exp
(
−2
kmax∑
k=1
G2k
)
, (37)
as illustrated in Fig. 9.7. It is sometimes beneficial to increase kmax beyond the value
needed for zero ensemble training error since this may improve generalization. While
a large kmax could in principle lead to overfitting, simulation experiments have shown
that overfitting rarely occurs, even when kmax is extremely large.
At first glance, it appears that boosting violates the No Free Lunch Theorem in
that an ensemble classifier seems always to perform better than any single component
classifier on the full training set. After all, according to Eq. 37 the training error drops
exponentially fast with the number of component classifiers. The Theorem is not vi-
olated, however: boosting only improves classification if the component classifiers
perform better than chance, but this cannot be guaranteed a priori. If the component
classifiers cannot learn the task better than chance, then we do not have a strong
match between the problem and model, and should choose an alternate learning al-
gorithm. Moreover, the exponential reduction in error on the training set does not
insure reduction of the off-training set error or generalization, as we saw in Sect. 9.2.1.
Nevertheless, AdaBoost has proven effective in many real-world applications.
9.5.3 Learning with queries
In the previous sections we assumed there was a set of labeled training patterns D
and employed resampling methods to reuse patterns to improve classification. In
30 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
E
kmax
0.1
0.3
0.4
0.5
0.2
ensemble test errorensemble training error
weak
learn
er
weig
hted
train
ing e
rror
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 9.7: AdaBoost applied to a weak learning system can reduce the training error
E exponentially as the number of component classifiers, kmax, is increased. Because
AdaBoost “focuses on” difficult training patterns, the training error of each successive
component classifier (measured on its own weighted training set) is generally larger
than that of any previous component classifier (shown in gray). Nevertheless, so long
as the component classifiers perform better than chance (e.g., have error less than 0.5
on a two-category problem), the weighted ensemble decision of Eq. 36 insures that
the training error will decrease, as given by Eq. 37. It is often found that the test
error decreases in boosted systems as well, as shown in red.
some applications, however, the patterns are unlabeled. We shall return in Chap. ??
to the problem of learning when no labels are available but here we assume there
exists some (possibly costly) way of labeling any pattern. Our current challenge is
thus to determine which unlabeled patterns would be most informative (i.e., improve
the classifier the most) if they were labeled and used as training patterns. These
are the patterns we will present as a query to an oracle — a teacher who can label,query
oracle
without error, any pattern. This approach is called variously learning with queries,
active learning or interactive learning and is a special case of a resampling technique.
Learning with queries might be appropriate, for example, when we want to de-
sign a classifier for handwritten numerals using unlabeled pixel images scanned from
documents from a corpus too large for us to label every pattern. We could start by
randomly selecting some patterns, presenting them to an oracle, and then training the
classifier with the returned labels. We then use learning with queries to select unla-
beled patterns from our set to present to a human (the oracle) for labeling. Informally,
we would expect the most valuable patterns would be near the decision boundaries.
More generally we begin with a preliminary, weak classifier that has been developed
with a small set of labeled samples. There are two related methods for then selecting
an informative pattern, i.e., a pattern for which the current classifier is least certain.
In confidence based query selection the classifier computes discriminant functions gi(x)confidence
based query
selection
for the c categories, i = 1, . . . , c. An informative pattern x is one for which the two
largest discriminant functions have nearly the same value; such patterns lie near the
current decision boundaries. Several search heuristics can be used to find such points
efficiently (Problem 30).
The second method, voting based or committee based query selection, is similar tovoting
based query
selection
the previous method but is applicable to multiclassifier systems, that is, ones compris-
ing several component classifiers (Sect. 9.7). Each unlabeled pattern is presented to
9.5. RESAMPLING FOR CLASSIFIER DESIGN 31
each of the k component classifiers; the pattern that yields the greatest disagreement
among the k resulting category labels is considered the most informative pattern, and
is thus presented as a query to the oracle. Voting based query selection can be used
even if the component classifiers do not provide analog discriminant functions, for
instance decision trees, rule-based classifiers or simple k-nearest neighbor classifiers.
In both confidence based and voting based methods, the pattern labeled by the oracle
is then used for training the classifier in the traditional way. (We shall return in
Sect. 9.7 to training an ensemble of classifiers.)
Clearly such learning with queries does not directly exploit information about the
prior distribution of the patterns. In particular, in most problems the distributions
of query patterns will be large near the final decision boundaries (where patterns are
informative) rather than at the region of highest prior probability (where they are
typically less informative), as illustrated in Fig. 9.8. One benefit of learning with
queries is that we need not guess the form of the underlying distribution, but can
instead use non-parametric techniques, such as nearest-neighbor classification, that
allow the decision boundary to be found directly.
If there is not a large set of unlabeled samples available for queries, we can nev-
ertheless exploit learning with queries if there is a way to generate query patterns.
Suppose we have a only small set of labeled handwritten characters. Suppose too we
have image processing algorithms for altering these images to generate new, surrogate
patterns for queries to an oracle. For instance the pixel images might be rotated,
scaled, sheared, be subject to random pixel noise, or have their lines thinned. Fur-
ther, we might be able to generate new patterns “in between” two labeled patterns by
interpolating or somehow mixing them in a domain-specific way. With such generated
query patterns the classifier can explore regions of the feature space about which it is
least confident (Fig. 9.8).
9.5.4 Arcing, learning with queries, bias and variance
In Chap. ?? and many other places, we have stressed the need for training a classifier
on samples drawn from the distribution on which it will be tested. Resampling in
general, and learning with queries in particular, seem to violate this recommendation.
Why can a classifier trained on a strongly weighted distribution of data be expected
to do well — or better! — than one trained on the i.i.d. sample? Why doesn’t
resampling lead to worse performance, to the extent that the resampled distribution
differs from the i.i.d. one?
Indeed, if we were to take a model of the true distribution and train it with
a highly skewed distribution obtained by learning with queries, the final classifier
accuracy might be unacceptably low. Consider, however, two interrelated points about
resampling methods and altered distributions. The first is that resampling methods
are generally used with techniques that do not attempt to model or fit the full category
distributions. Thus even if we suspect the prior distributions for two categories are
Gaussian, we might use a non-parametric method such as nearest neighbor, radial
basis function, or RCE classifiers when using learning with queries. Thus in learning
with queries we are not fitting parameters in a model, as described in Chap. ??, but
instead are seeking decision boundaries more directly.
The second point is that as the number of component classifiers is increased,
techniques such as general boosting and AdaBoost effectively broaden that class of
implementable functions, as illustrated in Fig. 9.6. While the final classifier might
indeed be characterized as parametric, it is in an expanded space of parameters, one
32 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
x1
x2
x1
x2
x1
x2
EB = 0.02275
E = 0.05001 E = 0.02422
i.i.d. samples
active learning
Figure 9.8: Active learning can be used to create classifiers that are more accurate
than ones using i.i.d. sampling. The figure at the top shows a two-dimensional problem
with two equal circular Gaussian priors; the Bayes decision boundary is a straight line
and the Bayes error EB = 0.02275. The bottom figure on the left shows a nearest-
neighbor classifier trained with n = 30 labeled points sampled i.i.d. from the true
distributions. Note that most of these points are far from the decision boundary.
The figure at the right illustrates active learning. The first four points were sampled
near the extremes of the feature space. Subsequent query points were chosen midway
between two points already used by the classifier, one randomly selected from each of
the two categories. In this way, successive queries to the oracle “focused in” on the
true decision boundary. The final generalization error of this classifier (0.02422) is
lower than the one trained using i.i.d. samples (0.05001).
larger than that of the first component classifier.
In broad overview, resampling, boosting and related procedures are heuristic meth-
ods for adjusting the class of implementable decision functions. As such they allow
the designer to try to “match” the final classifier to the problem by indirectly adjust-
ing the bias and variance. The power of these methods is that they can be used with
an arbitrary classification technique such as the Perceptron, which would otherwise
prove extremely difficult to adjust to the complexity of an arbitrary problem.
9.6 Estimating and comparing classifiers
There are at least two reasons for wanting to know the generalization rate of a classifier
on a given problem. One is to see if the classifier performs well enough to be useful;
another is to compare its performance with that of a competing design. Estimating
the final generalization performance invariably requires making assumptions about
the classifier or the problem or both, and can fail if the assumptions are not valid.
We should stress, then, that all the following methods are heuristic. Indeed, if there
were a foolproof method for choosing which of two classifiers would generalize better
on an arbitrary new problem, we could incorporate such a method into the learning
and violate the No Free Lunch Theorem. Occasionally our assumptions are explicit
(as in parametric models), but more often than not they are implicit and difficult to
identify or relate to the final estimation (as in empirical methods).
9.6. ESTIMATING AND COMPARING CLASSIFIERS 33
9.6.1 Parametric models
One approach to estimating the generalization rate is to compute it from the as-
sumed parametric model. For example, in the two-class multivariate normal case, we
might estimate the probability of error using the Bhattacharyya or Chernoff bounds
(Chap ??), substituting estimates of the means and the covariance matrix for the
unknown parameters. However, there are three problems with this approach. First,
such an error estimate is often overly optimistic; characteristics that make the training
samples peculiar or unrepresentative will not be revealed. Second, we should always
suspect the validity of an assumed parametric model; a performance evaluation based
on the same model cannot be believed unless the evaluation is unfavorable. Finally,
in more general situations where the distributions are not simple it is very difficult to
compute the error rate exactly, even if the probabilistic structure is known completely.
9.6.2 Cross validation
In cross validation we randomly split the set of labeled training samples D into two
parts: one is used as the traditional training set for adjusting model parameters in the
classifier. The other set — the validation set — is used to estimate the generalization validation
seterror. Since our ultimate goal is low generalization error, we train the classifier until
we reach a minimum of this validation error, as sketched in Fig. 9.9. It is essential that
the validation (or the test) set not include points used for training the parameters in
the classifier — a methodological error known as “testing on the training set.”∗
E
amount of training,
parameter adjustment
training
va
lid
at
io
n
st
o
p
tr
a
in
in
g
h
er
e
Figure 9.9: In cross validation, the data set D is split into two parts. The first (e.g.,
90% of the patterns) is used as a standard training set for setting free parameters in the
classifier model; the other (e.g., 10%) is the validation set and is meant to represent the
full generalization task. For most problems, the training error decreases monotonically
during training, as shown in black. Typically, the error on the validation set decreases,
but then increases, an indication that the classifier may be overfitting the training
data. In cross validation, training or parameter adjustment is stopped at the first
minimum of the validation error.
Cross validation can be applied to virtually every classification method, where the
specific form of learning or parameter adjustment depends upon the general training
∗ A related but less obvious problem arises when a classifier undergoes a long series of refinements
guided by the results of repeated testing on the same test data. This form of “training on the test
data” often escapes attention until new test samples are obtained.
34 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
method. For example, in neural networks of a fixed topology (Chap. ??), the amount
of training is the number of epochs or presentations of the training set. Alternatively,
the number of hidden units can be set via cross validation. Likewise, the width of the
Gaussian window in Parzen windows (Chap. ??), and an optimal value of k in the
k-nearest neighbor classifier (Chap. ??) can be set by cross validation.
Cross validation is heuristic and need not (indeed cannot) give improved classifiers
in every case. Nevertheless, it is extremely simple and for many real-world problems
is found to improve generalization accuracy. There are several heuristics for choosing
the portion γ of D to be used as a validation set (0 < γ < 1). Nearly always, a
smaller portion of the data should be used as validation set (γ < 0.5) because the
validation set is used merely to set a single global property of the classifier (i.e., when
to stop adjusting parameters) rather than the large number of classifier parameters
learned using the training set. If a classifier has a large number of free parameters
or degrees of freedom, then a larger portion of D should be used as a training set,
i.e., γ should be reduced. A traditional default is to split the data with γ = 0.1,
which has proven effective in many applications. Finally, when the number of degrees
of freedom in the classifier is small compared to the number of training points, the
predicted generalization error is relatively insensitive to the choice of γ.
A simple generalization of the above method is m-fold cross validation. Here them-fold
cross
validation
training set is randomly divided into m disjoint sets of equal size n/m, where n is
again the total number of patterns in D. The classifier is trained m times, each
time with a different set held out as a validation set. The estimated performance is
the mean of these m errors. In the limit where m = n, the method is in effect the
leave-one-out approach to be discussed in Sect. 9.6.3.
We emphasize that cross validation is a heuristic and need not work on every prob-
lem. Indeed, there are problems for which anti-cross validation is effective — haltinganti-cross
validation the adjustment of parameters when the validation error is the first local maximum.
As such, in any particular problem designers must be prepared to explore different
values of γ, and possibly abandon the use of cross validation altogether if performance
cannot be improved (Computer exercise 5).
Cross validation is, at base, an empirical approach that tests the classifier experi-
mentally. Once we train a classifier using cross validation, the validation error gives
an estimate of the accuracy of the final classifier on the unknown test set. If the true
but unknown error rate of the classifier is p, and if k of the n′ independent, randomly
drawn test samples are misclassified, then k has the binomial distribution
P (k) =
(
n′
k
)
pk(1− p)n
′−k. (38)
Thus, the fraction of test samples misclassified is exactly the maximum likelihood
estimate for p (Problem 39):
p̂ =
k
n′
. (39)
The properties of this estimate for the parameter p of a binomial distribution are
well known. In particular, Fig. 9.10 shows 95% confidence intervals as a function of
p̂ and n′. For a given value of p̂, the probability is 0.95 that the true value of p lies
in the interval between the lower and upper curves marked by the number n′ of test
samples (Problem 36). These curves show that unless n′ is fairly large, the maximum
likelihood estimate must be interpreted with caution. For example, if no errors are
made on 50 test samples, with probability 0.95 the true error rate is between zero and
9.6. ESTIMATING AND COMPARING CLASSIFIERS 35
8%. The classifier would have to make no errors on more than 250 test samples to be
reasonably sure that the true error rate is below 2%.
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.1 0.3 0.5 0.7 0.9
0.1
0.3
0.5
0.7
0.9
p
p
^
10
10
15
15
20
30
50
10
025
010
001
00
0
20
30
50
10
0
25
0
Figure 9.10: The 95% confidence intervals for a given estimated error probability p̂
can be derived from a binomial distribution of Eq. 38. For each value of p̂, the true
probability has a 95% chance of lying between the curves marked by the number of
test samples n′. The larger the number of test samples, the more precise the estimate
of the true probability and hence the smaller the 95% confidence interval.
9.6.3 Jackknife and bootstrap estimation of classification ac-
curacy
A method for comparing classifiers closely related to cross validation is to use the
jackknife or bootstrap estimation procedures (Sects. 9.4.1 & 9.4.2). The application
of the jackknife approach to classification is straightforward. We estimate the accuracy
of a given algorithm by training the classifier n separate times, each time using the
training set D from which a different single training point has been deleted. This is
merely the m = n limit of m-fold cross validation. Each resulting classifier is tested
on the single deleted point and the jackknife estimate of the accuracy is then simply
the mean of these leave-one-out accuracies. Here the computational complexity may
be very high, especially for large n (Problem 28).
The jackknife, in particular, generally gives good estimates, since each of the the
n classifiers is quite similar to the classifier being tested (differing solely due to a sin-
gle training point). Likewise, the jackknife estimate of the variance of this estimate
is given by a simple generalization of Eq. 32. A particular benefit of the jackknife
approach is that it can provide measures of confidence or statistical significance in the
comparison between two classifier designs. Suppose trained classifier C1 has an accu-
racy of 80% while C2 has accuracy of 85%, as estimated by the jackknife procedure.
Is C2 really better than C1? To answer this, we calculate the jackknife estimate of
the variance of the classification accuracies and use traditional hypothesis testing to
see if C1’s apparent superiority is statistically significant (Fig. 9.11).
There are several ways to generalize the bootstrap method to the problem of es-
36 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
70 75 80 85 90 95 100
accuracy (%)
C1
C2
p
ro
b
a
b
il
it
y
a
le
a
ve
-o
n
e-
o
u
t
re
p
li
ca
ti
o
n
h
a
s
a
p
a
rt
ic
u
la
r
a
cc
u
ra
cy
Figure 9.11: Jackknife estimation can be used to compare the accuracies of classifiers.
The jackknife estimate of classifiers C1 and C2 are 80% and 85%, and full widths
(twice the square root of the jackknife estimate of the variances) are 12% and 15%,
as shown by the bars at the bottom. In this case, traditional hypothesis testing could
show that the difference is not statistically significant at some confidence level.
timating the accuracy of a classifier. One of the simplest approaches is to train B
classifiers, each with a different bootstrap data set, and test on other bootstrap data
sets. The bootstrap estimate of the classifier accuracy is simply the mean of these
bootstrap accuracies. In practice, the high computational complexity of bootstrap es-
timation of classifier accuracy is rarely worth possible improvements in that estimate.
In Sect. 9.5.1 we shall discuss bagging, a useful modification of bootstrap estimation.
9.6.4 Maximum-likelihood model comparison
Recall first the maximum-likelihood parameter estimation methods discussed in Chap. ??.
Given a model with unknown parameter vector θ, we find the value θ̂ which maxi-
mizes the probability of the training data, i.e., p(D|θ̂). Maximum-likelihood model
comparison or maximum-likelihood model selection — sometimes called ML-II — isML-II
a direct generalization of those techniques. The goal here is to choose the model that
best explains the training data, in a way that will become clear below.
We again let hi ∈ H represent a candidate hypothesis or model (assumed discrete
for simplicity), and D the training data. The posterior probability of any given model
is given by Bayes’ rule:
P (hi|D) =
P (D|hi)P (hi)
p(D)
∝ P (D|hi)P (hi), (40)
where we will rarely need the normalizing factor p(D). The data-dependent term,
P (D|hi), is the evidence for hi; the second term, P (hi), is our subjective prior overevidence
the space of hypotheses — it rates our confidence in different models even before the
data arrive. In practice, the data-dependent term dominates in Eq. 40, and hence the
priors P (hi) are often neglected in the computation. In maximum-likelihood model
comparison, we find the maximum likelihood parameters for each of the candidate
models, calculate the resulting likelihoods, and select the model with the largest such
likelihood in Eq. 40 (Fig. 9.12).
9.6. ESTIMATING AND COMPARING CLASSIFIERS 37
D
P(D|h3)
P(D|h1)
evidence
P(D|h2)
D0
Figure 9.12: The evidence (i.e., probability of generating different data sets given a
model) is shown for three models of different expressive power or complexity. Model
h1 is the most expressive, since with different values of its parameters the model can
fit a wide range of data sets. Model h3 is the most restrictive of the three. If the
actual data observed is D0, then maximum-likelihood model selection states that we
should choose h2, which has the highest evidence. Model h2 “matches” this particular
data set better than do the other two models, and should be selected.
9.6.5 Bayesian model comparison
Bayesian model comparison uses the full information over priors when computing
posterior probabilities in Eq. 40. In particular, the evidence for a particular hypothesis
is an integral,
P (D|hi) =
∫
p(D|θ, hi)p(θ|D, hi)dθ, (41)
where as before θ describes the parameters in the candidate model. It is common for
the posterior P (θ|D, hi) to be peaked at θ̂, and thus the evidence integral can often
be approximated as:
P (D|hi) � P (D|θ̂, hi)︸ ︷︷ ︸
best fit
likelihood
p(θ̂|hi)∆θ︸ ︷︷ ︸
Occam factor
. (42)
Before the data arrive, model hi has some broad range of model parameters,
denoted by ∆0θ and shown in Fig. 9.13. After the data arrive, a smaller range is
commensurate or compatible with D, denoted ∆θ. The Occam factor in Eq. 42, Occam
factor
Occam factor = p(θ̂|hi)∆θ =
∆θ
∆0θ
(43)
=
param. vol. commensurate with D
param. vol. commensurate with any data
,
is the ratio of two volumes in parameter space: 1) the volume that can account for data
D and 2) the prior volume, accessible to the model without regard to D. The Occam
factor has magnitude less than 1.0; it is simply the factor by which the hypothesis
space collapses by the presence of data. The more the training data, the smaller the
range of parameters that are commensurate with it, and thus the greater this collapse
in the parameter space and the larger the Occam factor (Fig. 9.13).
38 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
θ
p(θ|D,hi)
p(θ|hi)
θ̂
∆θ
∆0θ
Figure 9.13: In the absence of training data, a particular model h has available a
large range of possible values of its parameters, denoted ∆0θ. In the presence of a
particular training set D, a smaller range is available. The Occam factor, ∆θ/∆0θ,
measures the fractional decrease in the volume of the model’s parameter space due
to the presence of training data D. In practice, the Occam factor can be calculated
fairly easily if the evidence is approximated as a k-dimensional Gaussian, centered on
the maximum-likelihood value θ̂.
Naturally, once the posteriors for different models have been calculated by Eq. 42 &
40, we select the single one having the highest such posterior. (Ironically, the Bayesian
model selection procedure is itself not truly Bayesian, since a Bayesian procedure
would average over all possible models when making a decision.)
The evidence for hi, i.e., P (D|hi), was ignored in a maximum-likelihood setting
of parameters θ̂; nevertheless it is the central term in our comparison of models. As
mentioned, in practice the evidence term in Eq. 40 dominates the prior term, and it
is traditional to ignore such priors, which are often highly subjective or problematic
anyway (Problem 38, Computer exercise 7). This procedure represents an inherent
bias towards simple models (small ∆θ); models that are overly complex (large ∆θ) are
automatically self-penalizing where “overly complex” is a data-dependent concept.
In the general case, the full integral of Eq. 41 is too difficult to calculate ana-
lytically or even numerically. Nevertheless, if θ is k-dimensional and the posterior
can be assumed to be a Gaussian, then the Occam factor can be calculated directly
(Problem 37), yielding:
P (D|hi) � P (D|θ̂, hi)︸ ︷︷ ︸
best fit
likelihood
p(θ̂|hi)(2π)k/2|H|−1/2︸ ︷︷ ︸
Occam factor
. (44)
where
H =
∂2lnp(θ|D, hi)
∂θ2
(45)
is a Hessian matrix — a matrix of second-order derivatives — and measures how
“peaked” the posterior is around the value θ̂. Note that this Gaussian approximation
does not rely on the fact that the underlying model of the distribution of the data
in feature space is or is not Gaussian. Rather, it is based on the assumption that
the evidence distribution arises from a large number of independent uncorrelated
processes and is governed by the Law of Large Numbers. The integration inherent
9.6. ESTIMATING AND COMPARING CLASSIFIERS 39
in Bayesian methods is simplified using this Gaussian approximation to the evidence.
Since calculating the needed Hessian via differentiation is nearly always simpler than
a high-dimensional numerical integration, the Bayesian method of model selection
is not at a severe computational disadvantage relative to its maximum likelihood
counterpart.
There may be a problem due to degeneracies in a model — several parameters could
be relabeled and leave the classification rule (and hence the likelihood) unchanged.
The resulting degeneracy leads, in essence, to an “overcounting” which alters the
effective volume in parameter space. Degeneracies are especially common in neu-
ral network models where the parameterization comprises many equivalent weights
(Chap. ??). For such cases, we must multiply the right hand side of Eq. 42 by the
degeneracy of θ̂ in order to scale the Occam factor, and thereby obtain the proper
estimate of the evidence (Problem 42).
Bayesian model selection and the No Free Lunch Theorem
There seems to be a fundamental contradiction between two of the deepest ideas
in the foundation of statistical pattern recognition. On the one hand, the No Free
Lunch Theorem states that in the absence of prior information about the problem,
there is no reason to prefer one classification algorithm over another. On the other
hand, Bayesian model selection is theoretically well founded and seems to show how
to reliably choose the better of two algorithms.
Consider two “composite” algorithms — algorithm A and algorithm B — each
of which employs two others (algorithm 1 and algorithm 2). For any problem, algo-
rithm A uses Bayesian model selection and applies the “better” of algorithm 1 and
algorithm 2. Algorithm B uses anti-Bayesian model selection and applies the “worse”
of algorithm 1 and algorithm 2. It appears that algorithm A will reliably outperform
algorithm B throughout the full class of problems — in contradiction with Part 1 of
the No Free Lunch Theorem.
What is the resolution of this apparent contradiction? In Bayesian model selection
we ignore the prior over the space of models, H, effectively assuming it is uniform.
This assumption therefore does not take into account how those models correspond to
underlying target functions, i.e., mappings from input to category labels. Accordingly,
Bayesian model selection usually corresponds to a non-uniform prior over target func-
tions. Moreover, depending on the arbitrary choice of model, the precise non-uniform
prior will vary. In fact, this arbitrariness is very well-known in statistics, and good
practitioners rarely apply the principle of indifference, assuming a uniform prior over principle of
indifferencemodels, as Bayesian model selection requires. Indeed, there are many “paradoxes”
described in the statistics literature that arise from not being careful to have the prior
over models be tailored to the choice of models (Problem 38). The No Free Lunch
Theorem allows that for some particular non-uniform prior there may be a learning
algorithm that gives better than chance — or even optimal — results. Apparently
Bayesian model selection corresponds to non-uniform priors that seem to match many
important real-world problems.
9.6.6 The problem-average error rate
The examples we have given thus far suggest that the problem with having only a
small number of samples is that the resulting classifier will not perform well on new
data — it will not generalize well. Thus, we expect the error rate to be a function of
40 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
the number n of training samples, typically decreasing to some minimum value as n
approaches infinity. To investigate this analytically, we must carry out the following
familiar steps:
1. Estimate the unknown parameters from samples.
2. Use these estimates to determine the classifier.
3. Calculate the error rate for the resulting classifier.
In general this analysis is very complicated. The answer depends on everything — on
the particular training patterns, on the way they are used to determine the classifier,
and on the unknown, underlying probability structure. However, by using histogram
approximations to the unknown probability densities and averaging appropriately, it
is possible to draw some illuminating conclusions.
Consider a case in which two categories have equal prior probabilities. Suppose
that we partition the feature space into some number m of disjoint cells C1, ..., Cm. If
the conditional densities p(x|ω1) and p(x|ω2) do not vary appreciably within any cell,
then instead of needing to know the actual value of x, we need only know into which
cell x falls. This reduces the problem to the discrete case. Let pi = P (x ∈ Ci|ω1)
and qi = P (x ∈ Ci|ω2). Then, since we have assumed that P (ω1) = P (ω2) = 1/2, the
vectors p = (p1, ..., pm)t and q = (q1, ..., qm)t determine the probability structure of
the problem. If x falls in Ci, the Bayes decision rule is to decide ωi if pi > qi. The
resulting Bayes error rate is given by
P (E|p,q) = 1
2
m∑
i=1
min[pi, qi] (46)
When the parameters p and q are unknown and must be estimated from a set
of training patterns, the resulting error rate will be larger than the Bayes rate. The
exact error probability will depend on the set of training patterns and the way in
which they are used to obtain the classifier. Suppose that half of the samples are
labeled ω1 and half are labeled ω2, with nij being the number that fall in Ci and
are labeled ωj . Suppose further that we design the classifier by using the maximum
likelihood estimates p̂i = 2ni1/n and q̂i = 2ni2/n as if they were the true values.
Then a new feature vector falling in Ci will be assigned to ω1 if ni1 > ni2. With all of
these assumptions, it follows that the probability of error for the resulting classifier
is given by
P (E|p,q,D) = 1
2
∑
ni1>ni2
qi +
1
2
∑
ni1≤ni2
pi. (47)
To evaluate this probability of error, we need to know the true conditional proba-
bilities p and q, and the set of training patterns, or at least the numbers nij . Different
sets of n randomly chosen patterns will yield different values for P (E|p,q,D). We
can use the fact that the numbers nij have a multinomial distribution to average over
all of the possible sets of n random samples and obtain an average probability of error
P (E|p,q, n). Roughly speaking, this is the typical error rate one should expect for
n samples. However, evaluation of this average error rate still requires knowing the
underlying problem, i.e., the values for p and q. If p and q are quite different, the
average error rate will be near zero, while if p and q are quite similar, it will be near
0.5.
9.6. ESTIMATING AND COMPARING CLASSIFIERS 41
A sweeping way to eliminate this dependence of the answer upon the problem
is to average the answer over all possible problems. That is, we assume some prior
distribution for the unknown parameters p and q, and average P (E|p,q, n) with
respect to p and q. The resulting problem-average probability of error P̄ (E|m,n)
will depend only on the number m of cells, the number n of samples, and the prior
distributions.
Of course, choosing the prior distributions is a delicate matter. By favoring easy
problems, we can make P̄ approach zero, and by favoring hard problems we can make
P̄ approach 0.5. We would like to choose a prior distribution corresponding to the
class of problems we typically encounter, but there is no obvious way to do that. A
bold approach is merely to assume that problems are “uniformly distributed,” i.e.,
that the vectors p and q are distributed uniformly over the simplexes
pi ≥ 0,
m∑
i=1
pi = 1, (48)
qi ≥ 0,
m∑
i=1
qi = 1.
Note that this uniform distribution over the space of p and q does not correspond to
some purported uniform distribution over possible distributions or target functions,
the issue pointed out in Sect. 9.6.5.
Figure 9.14 summarizes simulation experiments and shows curves of P̄ as a func-
tion number of cells for fixed numbers of training patterns. With an infinite number of
training patterns the maximum likelihood estimates are perfect, and P̄ is the average
of the Bayes error rate over all problems. The corresponding curve for P̄ (E|m,∞)
decreases rapidly from 0.5 at m = 1 to the asymptotic value of 0.25 as m approaches
infinity. The fact that P̄ = 0.5 if m = 1 is not surprising, since if there is only one
cell the decision must be based solely on the prior probabilities. The fact that P̄
approaches 0.25 as m approaches infinity is aesthetically pleasing, since this value is
halfway between the extremes of 0.0 and 0.5. The fact that the problem-average error
rate is so high merely shows that many hopelessly difficult classification problems
are included in this average. Clearly, it would be rash indeed to conclude that the
“average” pattern recognition problem will have this error rate.
However, the most interesting feature of these curves is that for every curve in-
volving a finite number of samples there is an optimal number of cells. This is directly
related to the fact that with a finite number of samples the performance will worsen
if too many features are used. In this case it is clear why there exists an optimal
number of cells for any given n and m. At first, increasing the number of cells makes
it easier to distinguish between the distributions represented by the vectors p and q,
thereby allowing improved performance. However, if the number of cells becomes too
large, there will not be enough training patterns to fill them. Eventually, the number
of patterns in most cells will be zero, and we must return to using just the ineffective a
priori probabilities for classification. Thus, for any finite n, P̄ (E|m,n) must approach
0.5 as m approaches infinity.
The value of m for which P̄ (E|m,n) is minimum is quite small. For n = 500
samples, it is somewhere around m = 200 cells. Suppose that we were to form the
cells by dividing each feature axis into l intervals. Then with d features we would
have m = ld cells. If l = 2, which is extremely crude quantization, this implies that
using more than four or five binary features will lead to worse rather than better
42 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
25
10
20
50 100
200
500
1000
1 10 100 1000
0.25
0.30
0.35
0.40
0.45
0.50
Number of cells, m
Number of
training patterns, n
P(E|m,n)
∞
Figure 9.14: The probability of error E on a two-category problem for a given number
of samples, n, can be estimated by splitting the feature space into m cells of equal size
and classifying a test point by according to the label of the most frequently represented
category in the cell. The graphs show the average error of a large number of random
problems having the given n and m indicated.
performance. This is a very pessimistic result, but then so is the statement that the
average error rate is 0.25. These numerical values are a consequence of the prior
distribution chosen for the problems, and are of no significance when one is facing
a particular problem. The main thing to be learned from this analysis is that the
performance of a classifier certainly does depend on the number of training patterns,
and that if this number is fixed, increasing the number of features beyond a certain
point raises the variance unacceptably, and will be counterproductive.
9.6.7 Predicting final performance from learning curves
Training on very large data sets can be computationally intensive, requiring days,
weeks or even months on powerful machines. If we are exploring and comparing several
different classification techniques, the total training time needed may be unacceptably
long. What we seek, then, is a method to compare classifiers without the need of
training all of them fully on the complete data set. If we can determine the most
promising model quickly and efficiently, we need then only train this model fully.
One method is to use a classifier’s performance on a relatively small training set
to predict its performance on the ultimate large training set. Such performance is
revealed in a type of learning curve in which the test error is plotted versus the size
of the training set. Figure 9.15 shows the error rate on an independent test set after
the classifier has been fully trained on n′ ≤ n points in the training set. (Note that
in this form of learning curve the training error decreases monotonically and does not
show “overtraining” evident in curves such as Fig. 9.9.)
For many real-world problems, such learning curves decay monotonically and can
be adequately described by a power-law function of the form
Etest = a + b/n
′α (49)
9.6. ESTIMATING AND COMPARING CLASSIFIERS 43
2000 4000 6000 8000 10000
0.2
0.4
0.6
0.8
1
n’
Etest
Figure 9.15: The test error for three classifiers, each fully trained on the given number
n′ of training patterns, decreases in a typical monotonic power-law function. Notice
that the rank order of the classifiers trained on n′ = 500 points differs from that for
n′ = 10000 points and the asymptotic case.
where a, b and α ≥ 1 depend upon the task and the classifier. In the limit of very
large n′, the training error equals the test error, since both the training and test sets
represent the full problem space. Thus we also model the training error as a power-law
function, having the same asymptotic error,
Etrain = a− c/n′β . (50)
If the classifier is sufficiently powerful, this asymptotic error, a, is equal to the Bayes
error. Furthermore, such a powerful classifier can learn perfectly the small training
sets and thus the training error (measured on the n′ points) will vanish at small n′,
as shown in Fig. 9.16.
2000 4000 6000 8000 10000
0.2
0.4
0.6
0.8
n’
E
Etest
Etrain
a
Figure 9.16: Test and training error of a classifier fully trained on data subsets of
different size n′ selected randomly from the full setD. At low n′, the classifier can learn
the category labels of the points perfectly, and thus the training error vanishes there.
In the limit n′ → ∞, both training and test errors approach the same asymptotic
value, a. If the classifier is sufficiently powerful and the training data is sampled i.i.d.,
then a is the Bayes error rate, EB .
Now we seek to estimate the asymptotic error, a, from the training and test errors
on small and intermediate size training sets. From Eqs. 49 & 50 we find:
Etest + Etrain = 2a +
b
n′α
− c
n′β
(51)
Etest − Etrain =
b
nα
+
c
nβ
.
44 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
If we make the assumption of α = β and b = c, then Eq. 51 reduces to
Etest + Etrain = 2a (52)
Etest − Etrain =
2b
n′α
.
Given this assumption, it is a simple matter to measure the training and test errors
for small and intermediate values of n′, plot them on a log-log scale and estimate a,
as shown in Fig. 9.17. Even if the approximations α = β and b = c do not hold in
practice, the difference Etest −Etrain nevertheless still forms a straight line on a log-
log plot and the sum, s = b+c, can be found from the height of the log[Etest+Etrain]
curve. The weighted sum cEtest + bEtrain will be a straight line for some empirically
set values of b and c, constrained to obey b + c = s, enabling a to be estimated
(Problem 41). Once a has been estimated for each in the set of candidate classifiers,
the one with the lowest a is chosen and must be trained on the full training set D.
log[n’]
log[E]
log[b+c]≈log[2b]
log[Etest -Etrain]
log[Etest +Etrain]≈log[2a]
1
α
Figure 9.17: If the test and training errors versus training set size obey the power-law
functions of Eqs. 49 & 50, then the log of the sum and log of the difference of these
errors are straight lines on a log-log plot. The estimate of the asymptotic error rate
a is then simply related to the height of the log[Etest + Etrain] line, as shown.
9.6.8 The capacity of a separating plane
Consider the partitioning of a d-dimensional feature space by a hyperplane wtx+w0 =
0, as might be trained by the Perceptron algorithm (Chap. ??). Suppose that we are
given n sample points in general position, that is, with no subset of d+1 points falling
in a (d− 1)-dimensional subspace. Assume each point is labeled either ω1 or ω2. Of
the 2n possible dichotomies of n points in d dimensions, a certain fraction f(n, d)
are said to be linear dichotomies. These are the labellings for which there exists a
hyperplane separating the points labeled ω1 from the points labeled ω2. It can be
shown (Problem 40) that this fraction is given by
f(n, d) =
1 n ≤ d + 1
2
2n
d∑
i=0
(
n−1
i
)
n > d + 1,
(53)
as plotted in Fig. 9.18 for several values of d.
To understand the issue more fully, consider the one-dimensional case with four
points; according to Eq. 53, we have f(n = 4, d = 1) = 0.5. The table shows
9.7. COMBINING CLASSIFIERS 45
schematically all sixteen of the equally likely labels for four patterns along a line.
(For instance, 0010 indicates that the labels are assigned ω1ω1ω2ω1.) The × marks
those arrangements that are linearly separable, i.e., in which a single point decision
boundary can separate all ω1 patterns from all ω2 patterns. Indeed as given by Eq. 53,
8 of the 16 — half — are linearly separable.
labels lin. sep.? labels lin. sep.?
0000 × 1000 ×
0001 × 1001
0010 1010
0011 × 1011
0100 1100 ×
0101 1101
0110 1110 ×
0111 × 1111 ×
Note from Fig. 9.18 that all dichotomies of d + 1 or fewer points are linear. This
means that a hyperplane is not overconstrained by the requirement of correctly clas-
sifying d + 1 or fewer points. In fact, if d is large it is not until n is a sizable fraction
of 2(d + 1) that the problem begins to become difficult. At n = 2(d + 1), which is
sometimes called the capacity of a hyperplane, half of the possible dichotomies are still capacity
linear. Thus, a linear discriminant is not effectively overdetermined until the number
of samples is several times as large as the dimensionality of the feature space or subset
of the problems. This is often expressed as: “generalization begins only after learning
ends.” Alternatively, we cannot expect a linear classifier to “match” a problem, on
average, if the dimension of the feature space is greater than n/2− 1.
1 2 3 4 5
0.2
0.4
0.6
0.8
1.0
d=1
5
25
∞
n/(d+1)
f(n,d)
Figure 9.18: The fraction of dichotomies of n points in d dimensions that are linear,
as given by Eq. 53.
9.7 Combining classifiers
We have already mentioned classifiers whose decision is based on the outputs of com-
ponent classifiers (Sects. 9.5.1 & 9.5.2). Such full classifiers are variously called mixture
of expert models, ensemble classifiers, modular classifiers or occasionally pooled clas- mixture
of expertsifiers. Such classifiers are particularly useful if each of its component classifiers is
highly trained — i.e., an “expert” — in a different region of the feature space. We
46 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
first consider the case where each component classifier provides probability estimates.
Later, in Sect. 9.7.2 we consider the case where component classifiers provide rank
order information or one-of-c outputs.
9.7.1 Component classifiers with discriminant functions
We assume that each pattern is produced by a mixture model, in which first somemixture
model fundamental process or function indexed by r (where 1 ≤ r ≤ k) is randomly chosen
according to distribution P (r|x,θ00) where θ
0
0 is a parameter vector. Next, the selected
process r emits an output y (e.g., a category label) according to P (y|x,θ0r), where the
parameter vector θ0r describes the state of the process. (The superscript 0 indicates
the properties of the generating model. Below, terms without this superscript refer
to the parameters in a classifier.) The overall probability of producing output y is
then the sum over all the processes according to:
P (y|x,Θ0) =
k∑
r=1
P (r|x,η0)P (y|x,θ0), (54)
where Θ0 = [θ00,θ
0
1, . . . ,θ
0
k]
t represents the vector of all relevant parameters. Equa-
tion 54 describes a mixture density, which could be discrete or continuous (Chap. ??).mixture
density Figure 9.19 shows the basic architecture of an ensemble classifier whose task is
to classify a test pattern x into one of c categories; this architecture matches the
assumed mixture model. A test pattern x is presented to each of the k component
classifiers, each of which emits c scalar discriminant values, one for each category. The
c discriminant values from component classifier r are grouped and marked g(x,θr) in
the figure, with
c∑
j=1
grj = 1 for all r. (55)
All discriminant values from component classifier r are multiplied by a scalar weight
wr, governed by the gating subsystem, which has a parameter vector θ0. Below wegating
subsystem shall use the conditional mean of the mixture density, which can be calculated from
Eq. 54
µ = E [y|x,Θ] =
k∑
r=1
wrµr (56)
where µr is the conditional mean associated with P (y|x,θ
0
r).
The mixture-of-experts architecture is trained so that each component classifier
models a corresponding process in the mixture model, and the gating subsystem
models the mixing parameters P (r|x,θ00) in Eq. 54. The goal is to find parameters
that maximize the log-likelihood for n training patterns x1, . . .xn in set D:
l(D,Θ) =
n∑
i=1
ln
(
k∑
r=1
P (r|xi,θ0)P (yi|xi,θr)
)
. (57)
A straightforward approach is to use gradient descent on the parameters, where the
derivatives are (Problem 43)
9.7. COMBINING CLASSIFIERS 47
component
classifier 1
component
classifier k…
gating
subsystem
pooling
system
x x x
x
component
classifier 2
g1(x, q1) g2(x, q2) gk(x, qk)
w1
wk
w2
q0
q1 q2 qk
g
Figure 9.19: The mixture of experts architecture consists of k component classifiers
or “experts,” each of which has trainable parameters θi, i = 1, . . . , k. For each input
pattern x, each component classifier i gives estimates of the category membership
gir = P (ωr|x,θi). The outputs are weighted by the gating subsystem governed by
parameter vector θ0, and pooled for ultimate classification.
∂l(D,Θ)
∂µr
=
n∑
i=1
P (r|yi,xi) ∂
∂µr
ln[P (yi|xi,θr)] for r = 1, . . . k (58)
and
∂l(D,Θ)
∂gr
=
n∑
i=1
(P (r|yi,xi)− wir). (59)
Here (P (r|yi,xi) is the posterior probability of process r conditional on the in-
put and output being xi and yi, respectively. Moreover, wir is the prior probabil-
ity P (r|xi) that process r is chosen given the input is xi. Gradient descent ac-
cording to Eq. 59 moves the prior probabilities to the posterior probabilities. The
Expectation-Maximization (EM) algorithm can be used to train this architecture as
well (Chap. ??).
The final decision rule is simply to choose the category corresponding to the max-
imum discriminant value after the pooling system. An alternative, winner-take-all winner-
take-allmethod is to use the decision of the single component classifier that is “most confi-
dent,” i.e., has the largest single discriminant value grj . While the winner-take-all
method is provably sub-optimal, it nevertheless is simple and can work well if the
component classifiers are experts in separate regions of the input space.
We have skipped over a problem: how many component classifiers should be used?
Of course, if we have prior information about the number of component processes that
generated the mixture density, this should guide our choice of k. In the absence of
such information, we may have to explore different values of k, thereby tailoring the
bias and variance of the full ensemble classifier. Typically, if the true number of com-
ponents in the mixture density is k∗, a mixture-of-experts more than k∗ component
48 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
classifiers will generalize better than one with fewer than k∗ component classifiers be-
cause the extra component classifiers learn to duplicate one another and hence become
redundant.
9.7.2 Component classifiers without discriminant functions
Occasionally we seek to form an ensemble classifier from highly trained component
classifiers, some of which might not themselves compute discriminant functions. For
instance, we might have four component classifiers — a k-nearest-neighbor classifier,
a decision tree, a neural network, and a rule-based system — all addressing the same
problem. While a neural network would provide analog values for each of the c
categories, the rule-based system would give only a single category label (i.e., a one-
of-c representation) and the k-nearest neighbor classifier would give only rank order
of the categories.
In order to integrate the information from the component classifiers we must con-
vert the their outputs into discriminant values obeying the constraint of Eq. 55 so
we can use the framework of Fig. 9.19. The simplest heuristics to this end are the
following:
Analog If the outputs of a component classifier are analog values g̃i, we can use the
softmax transformation,softmax
gi =
eg̃i
c∑
j=1
eg̃i
. (60)
to convert them to values gi.
Rank order If the output is a rank order list, we assume the discriminant function
is linearly proportional to the rank order of the item on the list. Of course, the
resulting gi should then be properly normalized, and thus sum to 1.0.
One-of-c If the output is a one-of-c representation, in which a single category is
identified, we let gj = 1 for the j corresponding to the chosen category, and 0
otherwise.
The table gives a simple illustration of these heuristics.
Analog value Rank order One-of-c
g̃i gi g̃i gi g̃i gi
0.4 0.158 3rd 4/21 = 0.194 0 0
0.6 0.193 6th 1/21 = 0.048 1 1.0
0.9 0.260 5th 2/21 = 0.095 0 0
0.3 0.143 1st 6/21 = 0.286 0 0
0.2 0.129 2nd 5/21 = 0.238 0 0
0.1 0.111 4th 3/21 = 0.143 0 0
Once the outputs of the component classifiers have been converted to effective
discriminant functions in this way, the component classifiers are themselves held fixed,
but the gating network is trained as described in Eq. 59. This method is particularly
useful when several highly trained component classifiers are pooled to form a single
decision.
9.7. SUMMARY 49
Summary
The No Free Lunch Theorem states that in the absence of prior information about
the problem there are no reasons to prefer one learning algorithm or classifier model
over another. Given that a finite set of feature values are used to distinguish the
patterns under consideration, the Ugly Duckling Theorem states that the number of
predicates shared by any two different patterns is constant, and does not depend upon
the choice of the two objects. Together, these theorems highlight the need for insight
into proper features and matching the algorithm to the data distribution — there
is no problem independent “best” learning or pattern recognition system nor feature
representation. In short, formal theory and algorithms taken alone are not enough;
pattern classification is an empirical subject.
Two ways to describe the match between classifier and problem are the bias and
variance. The bias measures the accuracy or quality of the match (high bias implies
a poor match) and the variance measures the precision or specificity of the match (a
high variance implies a weak match). The bias-variance dilemma states that learning
procedures with increased flexibility to adapt to the training data (e.g., have more
free parameters) tend to have lower bias but higher variance. In classification there
is a non-linear relationship between bias and variance, and low variance tends to be
more important for classification than low bias. If classifier models can be expressed
as binary strings, the minimum description length principle states that the best model
is the one with the minimum sum of such a model description and the training data
with respect to that model. This general principle can be extended to cover model-
specific heuristics such as weight decay and pruning in neural networks, regularization
in specific models, and so on.
The basic insight underlying resampling techniques — such as the bootstrap, jack-
knife, boosting, and bagging — is that multiple data sets selected from a given data
set enable the value and ranges of arbitrary statistics to be computed. In classifica-
tion, boosting techniques such as AdaBoost adjust the match of full classifier to the
problem (and thus the bias and variance) even for an arbitrary basic classification
method. In learning with queries, the classifier system presents query patterns to an
oracle for labeling. Such learning is most efficient if informative patterns — ones for
which the classifier is least certain — are presented as queries.
There are a number of methods for estimating the final accuracy of classifiers and
thus comparing them. Each is based on assumptions, for example that the parametric
model is known, or that the form of its learning curve is known. Cross validation,
jackknife and bootstrap methods are closely related techniques that use subsets of
the training data to estimate classifier accuracy. Maximum likelihood (ML-II) and
Bayesian methods — extensions of methods for setting parameters — can be used
to compare and choose among models. A key term in Bayesian model selection is
the Occam factor, which describes how the allowable volume in parameter space
shrinks due to constraints imposed by the training data. The method penalizes “overly
complex” models, where such complexity is a data-dependent property.
There are a number of methods for combining the outputs of separate component
or “expert” classifiers, such as linear weighting, winner-takes-all, and so on. Overall
classification is generally better when the decision rules of the component classifiers
differ and provide complementary information.
50 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
Bibliographical and Historical Remarks
The No Free Lunch Theorem appears in [109] as well as Wolpert’s collection of con-
tributions on the foundations of the theory of generalization [108]. Schaffer’s “conser-
vation law in generalization,” is a reformulation of one of the Parts of the Theorem,
and was the inspiration for Fig. 9.1 [81]. The Ugly Duckling Theorem was proven in
[104], which also explores some of its philosophical implications [77].
The foundational work on Kolmogorov complexity appears in [57, 58, 91, 92], but
a short elementary overview [14] and Chaitin’s [15] and particularly Li and Vitányi’s
citeLiVitanyi:97 books are far more accessible. Barron and Cover were the first to use
a minimum description length (MDL) principle to estimate densities [7]. There are
several versions of MDL [78, 79], such as the Akaike Information Criterion (AIC) [1, 2]
and the Bayes Information Criterion (BIC) [84] (which differ from MDL by relative
weighting of model penalty). Likewise, the Network Information Criterion (NIC) can
be used to compare neural networks of the same architecture [71]. More generally,
neural network pruning and general regularization methods can be cast as “minimum
description” principles, but with different measures for model and fit of the data [65].
Convincing theoretical and philosophical justifications of Occam’s razor have been
elusive. Karl Popper has argued that Occam’s razor is without operational value,
since there is no clear criterion or measure of simplicity [74], a point echoed by other
philosophers [90]. It is worth pointing out alternatives to Occam’s razor, which Isaac
Newton cast in Principia as “Natura enim simplex est, et rerum causis superfluis
non luxuriat,” or “for nature indeed is simple, and does not luxuriate in superfluous
causes” [72]. The first alternative stems from Epicurus (342?–270?BC), who in a
letter to Pythocles stated what we now call the principle of multiple explanations or
principle of indifference: if several theories are consistent with the data, retain all
such theories [29]. The second is a restatement of Bayes approach: the probability of
a model or hypothesis being true is proportional to the designer’s prior belief in the
hypothesis multiplied by the conditional probability of the data given the hypothesis in
question. Occam’s razor, or here favoring “simplicity” in classifiers, can be motivated
by considering the cost (difficulty) of designing the classifier and the principle of
bounded rationality — that we often settle for an adequate but not necessarily the
optimal solution [87]. An empirical study showing that simple classifiers often work
well can be found in [45].
The basic bias-variance decomposition and bias-variance dilemma [37] in regression
appear in many statistics books [41, 16]. Geman et al. give a very clear presentation in
the context of neural networks, but their discussion of classification is only indirectly
related to their mathematical derivations for regression [35]. Our presentation for
classification (zero-one loss) is based on Friedman’s important paper [32]; the bias-
variance decomposition has been explored in other non-quadratic cost functions as
well [42].
Quenouille introduced the term jackknife in 1956 [76]. The theoretical founda-
tions of resampling techniques are presented in Efron’s clear book [28], and practical
guides to their use include [36, 25]. Papers on bootstrap techniques for error estima-
tion include [48]. Breiman has been particularly active in introducing and exploring
resampling methods for estimation and classifier design, such as bagging [11] and gen-
eral arcing [13]. AdaBoost [31] builds upon Schapire’s analysis of the strength of weak
learnability [82] and Freund’s early work in the theory of learning [30]. Boosting in
multicategory problems is a bit more subtle than in two-category problems we dis-
cussed [83]. Angluin’s early work on queries for concept learning [3] was generalized to
9.7. PROBLEMS 51
active learning by Cohn and many others [18, 20] and is fundamental to some efforts
in collecting large databases [93, 95, 94, 99].
Cross validation was introduced by Cover [23], and has been used extensively in
conjunction with classification methods such as neural network. Estimates of error
under different conditions include [34, 110, 103] and an excellent paper, which derives
the size of test set needed for accurate estimation of classification accuracy is [39].
Bowyer and Phillip’s book covers empirical evaluation techniques in computer vision
[10], many of which apply to more general classification domains.
The roots of maximum likelihood model selection stem from Bayes himself, but one
of the earlier technical presentations is [38]. Interest in Bayesian model selection was
revived in a series of papers by MacKay, whose primary interest was in applying the
method to neural networks and interpolation [66, 69, 68, 67]. These model selection
methods have subtle relationships to minimum description length (MDL) [78] and
so-called maximum entropy approaches — topics that would take us a bit beyond our
central concerns. Cortes and her colleagues pioneered the analysis of learning curves
for estimating the final quality of a classifier [22, 21]. No rate of convergence results
can be made in the arbitrary case for finding the Bayes error, however [6]. Hughes
[46] first carried out the required computations and obtained in Fig. 9.14.
Extensive books on techniques for combining general classifiers include [55, 56] and
for combining neural nets in particular include [86, 9]. Perrone and Cooper described
the benefits that arise when expert classifiers disagree [73]. Dasarathy’s book [24] has
a nice mixture of theory (focusing more on sensor fusion than multiclassifier systems
per se) and a collection of important original papers, including [43, 61, 96]. The simple
heuristics for converting 1-of-c and rank order outputs to numerical values enabling
integration were discussed in [63]. The hierarchical mixture of experts architecture and
learning algorithm was first described in [51, 52]. A specific hierarchical multiclassifier
technique is stacked generalization [107, 88, 89, 12], where for instance Gaussian kernel
estimates at one level are pooled by yet other Gaussian kernels at a higher level.
We have skipped over a great deal of work from the formal field of computational
learning theory. Such work is generally preoccupied with convergence properties,
asymptotics, and computational complexity, and usually relies on simplified or general
models. Anthony and Biggs’ short, clear and elegant book is an excellent introduction
to the field [5]; broader texts include [49, 70, 53]. Perhaps the work from the field most
useful for pattern recognition practitioners comes from weak learnability and boosting,
mentioned above. The Probably approximately correct (PAC) framework, introduced
by Valiant [98], has been very influential in computation learning theory, but has had
only minor influence on the development of practical pattern recognition systems.
A somewhat broader formulation, Probably almost Bayes (PAB), is described in [4].
The work by Vapnik and Chervonenkis on structural risk minimization [102], and later
Vapnik-Chervonenkis (VC) theory [100, 101], derives (among other things) expected
error bounds; it too has proven influential to the theory community. Alas, the bounds
derived are somewhat loose in practice [19, 106].
Problems
⊕
Section 9.2
1. One of the “conservations laws” for generalization states that the positive gen-
eralization performance of an algorithm in some learning situations must be offset
52 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
by negative performance elsewhere. Consider a very simple learning algorithm that
seems to contradict this law. For each test pattern, the prediction of the majority
learning algorithm is merely the category most prevalent in the training data.
(a) Show that averaged over all two-category problems of a given number of features
that the off-training set error is 0.5.
(b) Repeat (a) but for the minority learning algorithm, which always predicts the
category label of the category least prevalent in the training data.
(c) Use your answers from (a) & (b) to illustrate Part 2 of the No Free Lunch
Theorem (Theorem 9.1).
2. Prove Part 1 of Theorem 9.1, i.e., that uniformly averaged over all target functions
F , E1(E|F, n)− E2(E|F, n) = 0. Summarize and interpret this result in words.
3. Prove Part 2 of Theorem 9.1, i.e., for any fixed training set D, uniformly averaged
over F , E1(E|F,D)− E2(E|F,D) = 0. Summarize and interpret this result in words.
4. Prove Part 3 of Theorem 9.1, i.e., uniformly averaged over all priors P (F ),
E1(E|n)− E2(E|n) = 0. Summarize and interpret this result in words.
5. Prove Part 4 of Theorem 9.1, i.e., for any fixed training set D, uniformly averaged
over P (F ), E1(E|D)− E2(E|D) = 0. Summarize and interpret this result in words.
6. Suppose you call an algorithm better if it performs slightly better than average
over most problems, but very poorly on a small number of problems. Explain why
the NFL Theorem does not preclude the existence of algorithms “better” in this way.
7. Show by simple counterexamples that the averaging in the different Parts of the
No Free Lunch Theorem (Theorem 9.1) must be “uniformly.” For instance imagine
that the sampling distribution is a Dirac delta distribution centered on a single tar-
get function, and algorithm 1 guesses the target function exactly while algorithm 2
disagrees with algorithm 1 on every prediction.
(a) Part 1
(b) Part 2
(c) Part 3
(d) Part 4
8. State how the No Free Lunch theorems imply that you cannot use training data to
distinguish between new problems for which you generalize well from those for which
you generalize poorly. Argue by reductio ad absurdum: that if you could distinguish
such problems, then the No Free Lunch Theorem would be violated.
9. Prove the relation
n∑
r=0
(
n
r
)
= (1 + 1)n = 2n of Eq. 5 two ways:
(a) State the polynomial expansion of (x + y)n as a summation of coefficients and
powers of x and y. Then, make a simple substitution for x and y.
(b) Prove the relation by induction. Let K(n) =
n∑
r=0
(
n
r
)
. First confirm that the
relation is valid for n = 1, i.e., that K(1) = 21. Now prove that K(n + 1) =
2K(n) for arbitrary n.
9.7. PROBLEMS 53
10. Consider the number of different Venn diagrams for k binary features f1, . . . , fk.
(Figure 9.2 shows several of these configurations for the k = 3 case.)
(a) How many functionally different Venn diagrams exist for the k = 2 case? Sketch
all of them. For each case, state how many different regions exists, i.e., how many
different patterns can be described.
(b) Repeat part (a) for the k = 3 case.
(c) How many functionally different Venn diagrams exist for the arbitrary k case?
11. While the text outlined a proof of the Ugly Duckling Theorem (Theorem 9.2)
this problem asks you to fill in some of the details and explain some of its implications.
(a) The discussion in the text assumed the classification problem had no constraints,
and thus could be described by the most general Venn diagram, in which all
predicates of a given rank r were present. How do the derivations change, if at
all, if we know that there are constraints provided by the problem, and thus not
all predicates of a given rank are possible, as in Fig. 9.2 (b) & (c)?
(b) Someone sees two cars, A and B, made by the same manufacturer in the same
model year, both are four-door and have same engine type, but they differ solely
in that one is red the other green. Car C is made by a different manufacturer,
has a different engine, is two-door and is blue. Explain in as much detail as
possible why, even in this seemingly clear case, that in fact there are no prior
reasons to view cars A and B as any “more similar” than cars B and C.
12. Suppose we describe patterns by means of predicates of a particular rank r∗.
Show the Ugly Duckling Theorem (Theorem 9.2) applies to any single level r∗, and
thus for all predicates up to an arbitrary maximum level.
13. Make some simple assumptions and state, using O(·) notation, the Kolmogorov
complexity of the following binary strings:
(a) 010110111011110 . . .︸ ︷︷ ︸
n
(b) 000 . . . 00100 . . . 000︸ ︷︷ ︸
n
(c) e = 10.10110111111000010 . . .2
(d) 2e = 101.01101111110000101 . . .2
(e) The binary digits of π, but where every 100th digit is changed to the numeral
1.
(f) The binary digits of π, but where every nth digit is changed to the numeral 1.
14. Recall the notation from our discussion of the No Free Lunch Theorem and of
Kolmogorov complexity. Suppose we use a learning algorithm with uniform P (h|D).
In that case K(h,D) = K(D) in Eq. 8. Explain and interpret this result.
15. Consider two binary strings x1 and x2. Explain why the Kolmogorov complexity
of the pair obeys K(x1, x2) ≤ K(x1) + K(x2) + c for some positive constant c.
16. Case where MDL is easier than imposing probabilities. xxx
54 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
17. The Berry paradox is related to famous liar’s paradox (“this statement is false”), Berry
paradoxas well as a number of paradoxes in set theory explored by Bertrand Russell and
Kurt Gödel. The Berry paradox shows indirectly how the notion of Kolmogorov
complexity can be difficult and subtle. Consider describing positive integers by means
of sentences, for instance “the number of fingers on a human hand,” or “the number
of primes less than a million.” Explain why the definition “the least number that
cannot be defined in less than twenty words” is paradoxical, and informally how this
relates to the difficulty in computing Kolmogorov complexity.
⊕
Section 9.3
18. Expand the left hand side of Eq. 11 to get the right hand side, which expresses
the mean-square error as a sum of a bias2 and variance. Can bias ever be negative?
Can variance ever be negative?
19. Fill in the steps leading to Eq. 18, i.e.,
Pr[g(x; D) �= y] = |2F (x)− 1|Pr[g(x; D) �= yB ] + Pr[yB �= y]
where the target function is F (x), g(x; D) is the computed discriminant value, yB is
the Bayes discriminant value.
20. Assume that the probability of a obtaining a particular discriminant value for
pattern x for a training algorithm trained with data D, denoted p(g(x; D)) is a
Gaussian. Use this Gaussian assumption and Eq. 19 to derive Eq. 20.
21. Derive the jackknife estimate of the bias in Eq. 29.
22. Prove that in the limit of B →∞, the bootstrap estimate of the variance of the
mean is the same as the standard estimate of the variance of the mean.
⊕
Section 9.4
23. Prove that Eq. 24 for the average of the leave one out means, µ(·), is equivalent
to Eq. 22 for the sample mean, µ̂.
24. We say that an estimator is consistent if it converges to the true value in theconsistent
limit of infinite data. Prove that the standard mean of Eq. 22 is not consistent for
the distribution p(x) ∼ tan−1(x− a) for any finite real constant a.
25. Prove that the jackknife estimate of an arbitrary statistic θ given in Eq. 30 is
unbiased for estimating the true bias.
26. Verify that Eq. 26 for the jackknife estimate of the variance of the mean is
formally equivalent to the variance implied by the traditional estimate given in Eq. 23.
27. Consider n points in one dimension. Use O(·) notation to express the computa-
tional complexity associated with each of the following estimations.
(a) The jackknife estimate of the mean.
(b) The jackknife estimate of the median.
(c) The jackknife estimate of the standard deviation.
(d) The bootstrap estimate of the mean.
(e) The bootstrap estimate of the median.
(f) The bootstrap estimate of the standard deviation.
9.7. PROBLEMS 55
⊕
Section 9.5
28. What is the computational complexity of full jackknife estimate of accuracy and
variance for an unpruned nearest-neighbor classifier (Chap. ??)?
29. In standard boosting applied to a two-category problem, we must create a data
set that is “most informative” with respect to the current state of the classifier. Why
does this imply that half of its patterns should be classified correctly, rather than none
of them? In a c category problem, what portion of the patterns should be misclassified
in a “most informative” set?
30. In active learning, learning can be speeded by creating patterns that are “infor-
mative,” i.e., those for which the two largest discriminants are approximately equal.
Consider the two-category case where for any point x in feature space, discriminant
values g1 and g2 are returned by the classifier. Write pseudocode that takes two points
— x1 classified as ω1 and x2 classified as ω2 — and rapidly finds a new point x3 that
is “near” the current decision boundary, and hence is “informative.” Assume only
that the discriminant functions are monotonic along the line linking x1 and x2.
31. Consider AdaBoost with an arbitrary number of component classifiers.
(a) State clearly any assumptions you make, and derive Eq. 37 for the ensemble
training error of the full boosted system.
(b) Recall that the training error for a weak learner applied to a two-category prob-
lem can be written Ek = 1/2 − Gk for some positive value Gk. The training
error for the first component classifier is E1 = 0.25. Suppose that Gk = 0.05 for
all k = 1 to kmax. Plot the upper bound on the ensemble test error given by
Eq. 37, such as shown in Fig. 9.7.
(c) Suppose that Gk decreases as a function of k. Specifically, repeat part (b) with
the assumption Gk = 0.05/k for k = 1 to kmax.
⊕
Section 9.6
32. The No Free Lunch Theorem implies that if all problems are equally likely, then
cross validation must fail as often as it succeeds. Show this as follows: Consider algo-
rithm 1 to be standard cross validation, and algorithm 2 to be anti-cross validation,
which advocates choosing the model that does worst on a validation set. Argue that
if cross validation were better than anti-cross validation overall, the No Free Lunch
Theorem would be violated.
33. Suppose we believe that the data for a pattern classification task from one
category comes either from a uniform distribution p(x) ∼ U(xl, xu) or from a normal
distribution, p(x) ∼ N(µ, σ2), but we have no reason to prefer one over the other.
Our sample data is D = {.2, .5, .4, .3, .9, .7, .6}.
(a) Find the maximum likelihood values of xl, and xu for the uniform model.
(b) Find the maximum likelihood values of µ and σ for the Gaussian model.
(c) Use maximum likelihood model selection to decide which model should be pre-
ferred.
34. Suppose we believe that the data for a pattern classification task from one
category comes either from a uniform distribution bounded below by 0, i.e., p(x) ∼
56 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING
U(0, xu) or from a normal distribution, p(x) ∼ N(µ, σ2), but we have no reason to
prefer one over the other. Our sample data is D = {.2, .5, .4, .3, .9, .7, .6}.
(a) Find the maximum likelihood values of xu for the uniform model.
(b) Find the maximum likelihood values of µ and σ for the Gaussian model.
(c) Use maximum likelihood model selection to decide which model should be pre-
ferred.
(d) State qualitatively the difference between your solution here and that to Prob-
lem 33, without necessarily having to solve that problem. In particular, what
are the implications from the fact that the two candidate models have different
numbers of parameters?
35. Consider three candidate one-dimensional distributions each parameterized by
an unknown value for its “center”:
• Gaussian: p(x) ∼ N(µ, 1)
• Triangle: p(x) ∼ T (µ,1) =
{
1− |x− µ| for |x− µ| < 1 0 otherwise • Uniform: p(x) ∼ U(µ− 1, µ + 1). We are given the data D = {−0.9,−0.1, 0., 0.1, 0.9}, and thus, clearly the maximum likelihood solution µ̂ = 0 applies to each model. (a) Use maximum likelihood model selection to determine the best model for this data. State clearly any assumptions you make. (b) Suppose we are sure for each model that the center must lie in the range −1 ≤ µ ≤ 1. Calculate the Occam factor for each model and the data given. (c) Use Bayesian model selection to determine the “best” model given D. 36. Use Eq. 38 and generate curves of the form shown in Fig. 9.10. Prove analytically that the curves are symmetric with respect to the interchange p̂ → (1 − p̂) and p→ (1− p). Explain the reasons for this symmetry. 37. Let model hi be described by a k-dimensional parameter vector θ. State your assumptions and show that the Occam factor can be written as p(θ̂|hi)(2π)k/2|H|−1/2, as given in Eq. 44, where the Hessian H is a matrix of second-order derivatives defined in Eq. 45. 38. Bertrand’s paradox shows how the notion of “uniformly distributed” models canBertrand’s paradox be problematic, and leads us to question the principle of indifference (cf., Computer exercise 7). Consider the following problem: Given a circle, find the probability that a “randomly selected” chord has length greater than the side of an inscribed equilateral triangle. Here are three possible solutions to this problem and their justifications, illustrated in the figure: 9.7. PROBLEMS 57 1. By definition, a chord strikes the circle at two points. We can arbitrarily rotate the figure so as to place one of those points at the top. The other point is equally likely to strike any other point on the circle. As shown in the figure at the left, one-third of such points (red) will yield a chord with length greater than that of the side of an inscribed equilateral triangle. Thus the probability a chord is longer than the length of the side of an inscribed equilateral triangle is P = 1/3. 2. A chord is uniquely determined by the location of its midpoint. Any such mid- point that lies in the circular disk whose radius is half that of the full circle will yield a chord with length greater than that of an inscribed equilateral triangle. Since the area of this red disk is one-fourth that of the full circle, the probability is P = 1/4. 3. We can arbitrarily rotate the circle such that the midpoint of a chord lies on a vertical line. If the midpoint lies closer to the center than half the radius of the circle, the chord will be longer than the side of the inscribed equilateral triangle. Thus the probability is P = 1/2. P=1/3 P=1/4 P=1/2 Explain why there is little or no reason to prefer one of the solution methods over another, and thus the solution to the problem itself is ill-defined. Use your answer to reconcile Bayesian model selection with the No Free Lunch Theorem (Theorem 9.1). 39. If k of n′ independent, randomly chosen test patterns are misclassified, then as given in Eq. 38 k has a binomial distribution P (k) = ( n′ k ) pk(1− p)n ′−k Prove that the maximum likelihood estimate for p is then p̂ = k/n′, as given in Eq. 39. 40. Derive the relation for f(n, d), the fraction of dichotomies of n randomly chosen points in d dimensions that are linearly separable, given by Eq. 53. Explain why f(n, d) = 1 for n ≤ d + 1. 41. Write pseudocode for an algorithm to determine the large n′ limit of the test error given the assumption of a power-law decrease in error described by Eq. 52 and illustrated in Fig. 9.17. 42. Suppose a standard three-layer neural network having I input units, H hid- den units, single bias unit and 2 output units is trained on a two-category problem (Chap. ??). What is the degeneracy of the final assignment of weights? That is, how many ways can the weights be relabeled with the decision rule being unchanged? Explain how this degeneracy would need to be incorporated into a Bayesian model selection. ⊕ Section 9.7 58 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING 43. Let xi and yi denote input and output vectors, respectively, and r index com- ponent processes (1 ≤ r ≤ k) in a mixture model. Use Bayes theorem, P (r|yi,xi) = P (r|x i)P (yi|xi, r) k∑ q=1 P (q|xi)P (yi|xi, q) to derive the derivatives Eqs. 58 & 59 used for gradient descent learning in the mixture of experts model. 44. Suppose a mixture of experts classifier has k Gaussians component classifiers of arbitrary mean and covariance in d dimensions, N(µ,Σ). Derive the specific learning rules for the parameters of each component classifier and for the gating subsystem, special cases of Eqs. 58 & 59. Computer exercises Several exercises will make use of the following three-dimensional data sampled from four categories, denoted ωi. ω1 ω2 ω3 ω4 sample x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 1 2.5 3.4 7.9 4.2 4.9 11.3 2.9 15.5 4.6 16.9 12.4 0.2 2 4.3 4.4 7.1 11.7 5.3 10.5 3.6 13.9 9.8 12.1 16.8 2.1 3 7.1 0.8 6.3 8.4 11.1 6.6 10.3 6.1 12.3 13.7 12.1 5.5 4 1.4 -0.2 2.5 8.2 10.4 4.9 8.2 5.5 7.1 11.9 13.4 3.4 5 3.9 4.3 3.4 5.3 7.7 8.8 13.3 4.7 11.7 14.5 15.5 2.8 6 3.2 6.8 5.1 7.9 4.5 9.5 6.6 8.1 16.7 15.6 14.9 4.4 7 7.3 6.5 7.1 10.7 6.9 10.9 12.2 5.1 5.9 16.2 12.3 3.2 8 -0.7 3.1 8.1 9.6 9.7 7.3 15.6 3.3 10.7 12.2 16.3 3.2 9 2.8 5.9 2.2 8.2 11.2 6.3 4.6 10.1 13.8 14.5 12.9 -0.9 10 6.1 7.6 4.3 5.3 10.1 4.9 9.1 4.4 8.9 15.8 15.6 4.5 ⊕ Section 9.2 1. Consider the use of the minimum description length principle for the design of a binary decision tree classifier (Chap. ??). Each question at a node is of the form “is xi > θ?” (or alternatively “is xi < θ?”). Specify each such question with 5 bits: two bits specify the feature queried (x1, x2, or x3), a single bit specifies whether the comparison is > or <, and four bits specify each θ as an integer 0 ≤ θ ≤ 16. Assume the Kolmogorov complexity of the classifier is, up to an additive constant, the sum of the bits of all questions. Assume too that the Kolmogorov complexity of the data given the tree classifier is merely the entropy of the data at the leaves, also measured in bits. (a) Train your tree with the data from the four-category problem in the table above. Starting at the root, grow your tree a single node at a time, continuing until each node is as pure as possible. Plot as a function of the total number of nodes the Kolmogorov complexity of 1) the classifier, 2) the data with respect to the classifier, and 3) their sum (Eq. 8). Show the tree (including the questions at its nodes) having the minimum description length. 9.7. COMPUTER EXERCISES 59 (b) The minimum description length principle gives a principled method for com- paring classifiers in which the resolution of parameters (e.g., weights, thresholds, etc.) can be altered. Repeat part (a) but using only three bits to specify each threshold θ in the nodes. (c) Assume that the additive constants for the Kolmogorov complexities of your above classifiers are equal. Which of all the classifiers has the minimum descrip- tion length? ⊕ Section 9.3 2. Illustrate the bias-variance decomposition and the bias-variance dilemma for regression through simulations. Let the target function be F (x) = x2 with Gaussian noise of variance 0.1. First, randomly generate 100 data sets, each of size n = 10, by selecting a value of x uniformly in the range −1 ≤ x ≤ 1 and then applying F (x) with noise. Train any free parameters ai (by minimum square error criterion) in each of the regression functions in parts (a) – (d), one data set at a time. Then make a histogram of the sum-square error of Eq. 11 (cf. Fig. 9.4). For each model use your results to estimate the bias and the variance. (a) g(x) = 0.5 (b) g(x) = 1.0 (c) g(x) = a0 + a1x (d) g(x) = a0 + a1x + a2x2 + a3x3 (e) Repeat parts (a) – (d) for 100 data sets of size n = 100. (f) Summarize all your above results, with special consideration of the bias-variance decomposition and dilemma, and the effect of the size of the data set. ⊕ Section 9.4 2. The trimmed mean of a distribution is merely the sample mean of the distribution trimmed meanfrom which some portion α (e.g., 0.1) of the highest and of the lowest points have been deleted. The trimmed mean is, of course, less sensitive to the presence of outliers than is the traditional sample mean. (a) Show how in the limit α → 0.5, the trimmed mean of a distribution is the median. (b) Let the data D be the x3 values of the 10 patterns in category ω2 in the table above. Write a program to determine the jackknife estimate of the median of D, and the jackknife estimate of the variance of this estimate. (c) Repeat part (b) but for the α = 0.1 trimmed mean and its variance. (d) Repeat part (b) but for the α = 0.2 trimmed mean and its variance. (e) Repeat parts (b) – (d) but where D has an additional (“outlier”) point at x3 = 20. 60 CHAPTER 9. ALGORITHM-INDEPENDENT MACHINE LEARNING (f) Interpret your results, with special attention to the sensitivity of the trimmed mean to outliers. ⊕ Section 9.5 3. Write a program to implement the AdaBoost procedure (Algorithm 1) with com- ponent classifiers whose linear discriminants are trained by the basic LMS algorithm (Algorithm ?? from Chap. ??). (a) Apply your system to the problem of discriminating the ten points in ω1 from the ten points in ω2 in the table above. Plot your training error as a function of the number of component classifiers. Be sure the graph extends to a kmax sufficiently high that the training error vanishes. (b) Define a “super-category” consisting of all the patterns in ω1 and ω2 in the table, and another super-category for the ω3 and ω4 patterns. Repeat part (a) for discriminating these super-categories. (c) Compare and interpret your graphs in (a) & (b), paying particular attention to the relative difficulties of the classification problems. 4. Explore the value of active learning in a two-dimensional two-category problem in which the priors are Gaussians, p(x) ∼ N(µi,Σi) with µ1 = ( +5 +5 ) ,µ2 = (−5 −5 ) , Σ1 = Σ2 = ( 20 0 0 20 ) and P (ω1) = P (ω2) = 0.5. Throughout this problem, restrict data to be in the domain −10 ≤ xi ≤ +10, for i = 1, 2. (a) State by inspection the Bayes classifier. This will be the decision used by the oracle in part (c). (b) Generate a training set of 100 points, 50 labeled ω1 sampled according to p(x) ∼ N(µ1,Σ1) and likewise 50 patterns according to p(x) ∼ N(µ2,Σ2). Train a nearest-neighbor classifier (Chap. ??) using your data, and plot the decision boundary in two dimensions. (c) Now assume there is an oracle, which can label any query pattern according to your answer in part (a), which we exploit through a particular form of active learning. To begin the learning, choose 10 points according to a uniform distri- bution in the domain −10 ≤ xi ≤ +10, for i = 1, 2. Apply labels to these points according to the oracle to get D1 and D2, for each category. Now generate new query points as follows. Randomly choose a point from D1 and a point from D2; create a query point midway between these two points. Label the point according to the oracle and add it to the appropriate Dj . Continue until the total number of labeled points is 100. Now create a nearest-neighbor classifier using all points, and plot the decision boundary in two dimensions. (d) Compare qualitatively your classifiers from parts (a), (b) & (c), and discuss your results. ⊕ Section 9.6 5. Explore a case where cross validation need not yield and improved classifier. Throughout, the classifier will be k-nearest-neighbor (Chap. ??), where k will be set by cross validation. two-category problem in two dimensions, with uniform prior distributions throughout the range 0 ≤ xi ≤ 1 for i = 1, 2. 9.7. COMPUTER EXERCISES 61 (a) First form a third, test set Dtest of 20 points — 10 points in ω1 and 10 in ω2 — randomly chosen according to a uniform distribution. (b) Next generate 100 points — 50 patterns in each category. Set γ = 0.1 and split this set into a training set Dtrain (90 points) and validation set Dval (10 points). (c) Now form a k-nearest-neighbor classifier in which k is increased until the first minimum the validation error is found. (Restrict k to odd values, to avoid ties.) Now determine the error of your classifier using the test set. (d) Repeat part (c), but instead find the k that is the first maximum of the validation error. (e) Repeat parts (c) & (d) five times, noting the test error in all ten cases. (f) Discuss your results, in particular how they depend or do not depend upon the fact that the data were all uniformly distributed. 6. Consider three candidate one-dimensional distributions each parameterized by an unknown value for its “center”: • Gaussian: p(x) ∼ N(µ, σ2) • Triangle: p(x) ∼ T (µ,1) = { 1− |x− µ| for |x− µ| < 1 0 otherwise • Uniform: p(x) ∼ U(µ− 2, µ + 2) . Suppose we are sure that for each model the center must lie in the range −1 < µ < 1, and for the Gaussian that 0 ≤ σ2 ≤ 1. Suppose too that we are given the data D = {−.9,−.1, 0., .1, .9}. Clearly, the maximum likelihood solution µ̂ = 0 applies to each model. (a) Estimate the Occam factor in each case. (b) Use Bayesian model selection to choose the best of these models. 7. Problem 38 describes Bertrand’s paradox, which involves the probability that a circle’s chord “randomly chosen” will have length greater than that of an inscribed equilateral triangle. (a) Write a program to generate chords according to the logic of solution (1) in Problem 38. 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Index abstract computer, see computer, ab- stract abstract encoding of a string, 12 active learning, see learning, active AdaBoost Algorithm, 28 algorithmic complexity, 12–13 algorithmic entropy, see algorithmic com- plexity anti-cross validation, see cross valida- tion, anti arcing, 26 bagging, 26 Bayes formula, 13 Bayes rule model, 36 Berry paradox, 54 Bertrand’s paradox, 56 bias, 16 boundary, 19 estimator, 23 preference, 14 bias and variance, 15–20 classification, 18 regression, 15 bias-variance dilemma, 16 Boltzmann network, 5 boosting, 26–29 boostrap aggregation, see bagging bootstrap, 24–25 bias estimate, 25 variance estimate, 25 boundary bias, see bias, boundary boundary error, see error, boundary capacity hyperplane, 45 charge physical, 3 classifier complex, 11 component, 26 expert, 4 simple, 3, 11 complexity descriptional, see algorithmic com- plexity Kolmogorov, see algorithmic com- plexity component classifier, see classifier, com- ponent compression technique, 12 computer abstract, 12 confidence based query selection, see query, selection, confidence based conservation law, 3 consistency, 54 cost, 5, see error cross validation anti, 34 m-fold, 34 cross-validation, 4 decision boundary smooth, 3 decision tree, 5 distribution prior, 5 energy physical, 3 entropy, 12 maximum, 51 error Bayes, 4 boundary, 19 generalization, 4 off-training set, 5 71 72 INDEX validation, 33 error rate estimate, 32–36 estimator consistent, 54 evidence for model, 36 explanations multiple, see indifference, princi- ple gating subsystem, 46 Hessian, 38 Gaussian approximation, 38 hypothesis, 5 i.i.d., 5 incompressibility, 12 indifference principle, 39, 50 information theory, 12 instability, 26 jackknife, 22–24 bias estimate, 23 variance estimate, 23 Kronecker delta, 5 Law of Large Numbers Hessian calculation, 38 learner weak, 26 learning active, 30 algorithm best, 3 majority, 52 minority, 52 curve, 42–44 with queries, 29–31 leave-one-out mean, 22 loss function, 5 maximum entropy, see entropy, maxi- mum MDL, see minimum description length, see minimum description length mean trimmed, 25, see trimmed mean median, 22 minimum description length, 11–14, 51 principle, 13–14 Bayes relation, 13 mixture density, 46 mixture model, 46 mixture of experts, 45 mode, 22 model selection, 39 maximum likelihood, 36 momentum physical, 3 nearest neighbor, 5 No Free Lunch Theorem, 4–8 Occam factor, 37 Occam’s razor, 3, 14–15 optimal coding theorem, 14 oracle, 30 overfitting, 14 penalty, 14 Perceptron, 32 π, algorithmic complexity, 12 posterior model, 36 predicate, 8 principle of indifference, see indiffer- ence, principle prior, 4 subjective, 36 prior distribution, 5 pruning, 14 query, 30 selection confidence based, 30 voting based, 30 random guess, 4 rank, 9 regularize, 14 resampling, 4 satisficing, 15 Shannon coding theorem, 14 softmax, 48 specification method, 12 statistic INDEX 73 sufficient, 12 string random, 12 trimmed mean, see mean, trimmed, 59 Turing machine, 12 Ugly Duckling Theorem, 8–11 universal Turing machine, see Turing machine validation error, see error, validation set, 33 variance, 16 voting based query selection, see query, selection, voting based weak learner, see learner, weak winner-take-all, 47 Contents 10 Unsupervised Learning and Clustering 3 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 10.2 Mixture Densities and Identifiability . . . . . . . . . . . . . . . . . . . 4 10.3 Maximum-Likelihood Estimates . . . . . . . . . . . . . . . . . . . . . . 6 10.4 Application to Normal Mixtures . . . . . . . . . . . . . . . . . . . . . 7 10.4.1 Case 1: Unknown Mean Vectors . . . . . . . . . . . . . . . . . 8 Example 1: Mixture of two 1D Gaussians . . . . . . . . . . . . . . . . 9 10.4.2 Case 2: All Parameters Unknown . . . . . . . . . . . . . . . . . 11 10.4.3 K-means clustering . . . . . . . . . . . . . . . . . . . . . . . . . 13 Algorithm 1: K-means . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 10.4.4 *Fuzzy k-means clustering . . . . . . . . . . . . . . . . . . . . . 14 Algorithm 2: Fuzzy k-means . . . . . . . . . . . . . . . . . . . . . . . . 15 10.5 Unsupervised Bayesian Learning . . . . . . . . . . . . . . . . . . . . . 17 10.5.1 The Bayes Classifier . . . . . . . . . . . . . . . . . . . . . . . . 17 10.5.2 Learning the Parameter Vector . . . . . . . . . . . . . . . . . . 18 Example 2: Unsupervised learning of Gaussian data . . . . . . . . . . . 21 10.5.3 Decision-Directed Approximation . . . . . . . . . . . . . . . . . 23 10.6 *Data Description and Clustering . . . . . . . . . . . . . . . . . . . . . 24 10.6.1 Similarity Measures . . . . . . . . . . . . . . . . . . . . . . . . 25 10.7 Criterion Functions for Clustering . . . . . . . . . . . . . . . . . . . . 29 10.7.1 The Sum-of-Squared-Error Criterion . . . . . . . . . . . . . . . 29 10.7.2 Related Minimum Variance Criteria . . . . . . . . . . . . . . . 30 10.7.3 Scattering Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 31 Example 3: Clustering criteria . . . . . . . . . . . . . . . . . . . . . . 33 10.8 *Iterative Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Algorithm 3: Basic minimum-squared-error . . . . . . . . . . . . . . . 36 10.9 Hierarchical Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 10.9.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 10.9.2 Agglomerative Hierarchical Clustering . . . . . . . . . . . . . . 39 Algorithm 4: Agglomerative hierarchical . . . . . . . . . . . . . . . . . 39 10.9.3 Stepwise-Optimal Hierarchical Clustering . . . . . . . . . . . . 41 Algorithm 5: Stepwise optimal hierarchical clustering . . . . . . . . . . 42 10.9.4 Hierarchical Clustering and Induced Metrics . . . . . . . . . . . 43 10.10*The Problem of Validity . . . . . . . . . . . . . . . . . . . . . . . . . 43 10.11Competitive Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Algorithm 6: Competitive learning . . . . . . . . . . . . . . . . . . . . 47 10.11.1Unknown number of clusters . . . . . . . . . . . . . . . . . . . 48 Algorithm 7: leader-follower . . . . . . . . . . . . . . . . . . . . . . . . 48 1 2 CONTENTS 10.11.2Adaptive Resonance . . . . . . . . . . . . . . . . . . . . . . . . 49 10.12*Graph Theoretic Methods . . . . . . . . . . . . . . . . . . . . . . . . 51 10.13Component analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 10.13.1Principal component analysis (PCA) . . . . . . . . . . . . . . . 53 10.13.2Non-linear component analysis . . . . . . . . . . . . . . . . . . 54 10.13.3 *Independent component analysis (ICA) . . . . . . . . . . . . . 55 10.14Low-Dimensional Representations and Multidimensional Scaling (MDS) 58 10.14.1Self-organizing feature maps . . . . . . . . . . . . . . . . . . . . 61 10.14.2Clustering and Dimensionality Reduction . . . . . . . . . . . . 65 Algorithm 8: Hierarchical dimensionality reduction . . . . . . . . . . . 66 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Bibliographical and Historical Remarks . . . . . . . . . . . . . . . . . . . . 68 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Computer exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Chapter 10 Unsupervised Learning and Clustering 10.1 Introduction Until now we have assumed that the training samples used to design a classifier werelabeled by their category membership. Procedures that use labeled samples are said to be supervised. Now we shall investigate a number of unsupervised procedures, which use unlabeled samples. That is, we shall see what can be done when all one has is a collection of samples without being told their category. One might wonder why anyone is interested in such an unpromising problem, and whether or not it is possible even in principle to learn anything of value from un- labeled samples. There are at least five basic reasons for interest in unsupervised procedures. First, collecting and labeling a large set of sample patterns can be sur- prisingly costly. For instance, recorded speech is virtually free, but accurately labeling the speech — marking what word or phoneme is being uttered at each instant — can be very expensive and time consuming. If a classifier can be crudely designed on a small set of labeled samples, and then “tuned up” by allowing it to run without supervision on a large, unlabeled set, much time and trouble can be saved. Second, one might wish to proceed in the reverse direction: train with large amounts of (less expensive) unlabeled data, and only then use supervision to label the groupings found. This may be appropriate for large “data mining” applications where the contents of a large database are not known beforehand. Third, in many applications the charac- teristics of the patterns can change slowly with time, for example in automated food classification as the seasons change. If these changes can be tracked by a classifier running in an unsupervised mode, improved performance can be achieved. Fourth, we can use unsupervised methods to find features, that will then be useful for cate- gorization. There are unsupervised methods that represent a form of data-dependent “smart preprocessing” or “smart feature extraction.” Lastly, in the early stages of an investigation it may be valuable to gain some insight into the nature or structure of the data. The discovery of distinct subclasses or similarities among patterns or of major departures from expected characteristics may suggest we significantly alter our 3 4 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING approach to designing the classifier. The answer to the question of whether or not it is possible in principle to learn anything from unlabeled data depends upon the assumptions one is willing to accept — theorems can not be proved without premises. We shall begin with the very restric- tive assumption that the functional forms for the underlying probability densities are known, and that the only thing that must be learned is the value of an unknown pa- rameter vector. Interestingly enough, the formal solution to this problem will turn out to be almost identical to the solution for the problem of supervised learning given in Chap. ??. Unfortunately, in the unsupervised case the solution suffers from the usual problems associated with parametric assumptions without providing any of the bene- fits of computational simplicity. This will lead us to various attempts to reformulate the problem as one of partitioning the data into subgroups or clusters. While some of the resulting clustering procedures have no known significant theoretical properties, they are still among the more useful tools for pattern recognition problems. 10.2 Mixture Densities and Identifiability We begin by assuming that we know the complete probability structure for the prob- lem with the sole exception of the values of some parameters. To be more specific, we make the following assumptions: 1. The samples come from a known number c of classes. 2. The prior probabilities P (ωj) for each class are known, j = 1, . . . , c. 3. The forms for the class-conditional probability densities p(x|ωj ,θj) are known, j = 1, . . . , c. 4. The values for the c parameter vectors θ1, . . . ,θc are unknown. 5. The category labels are unknown. Samples are assumed to be obtained by selecting a state of nature ωj with prob- ability P (ωj) and then selecting an x according to the probability law p(x|ωj ,θj). Thus, the probability density function for the samples is given by p(x|θ) = c∑ j=1 p(x|ωj ,θj)P (ωj), (1) where θ = (θ1, . . . ,θc). For obvious reasons, a density function of this form is called a mixture density. The conditional densities p(x|ωj ,θj) are called the component densities, and the prior probabilities P (ωj) are called the mixing parameters. Thecomponent densities mixing parameters mixing parameters can also be included among the unknown parameters, but for the moment we shall assume that only θ is unknown. Our basic goal will be to use samples drawn from this mixture density to estimate the unknown parameter vector θ. Once we know θ we can decompose the mixture into its components and use a Bayesian classifier on the derived densities, if indeed classification is our final goal. Before seeking explicit solutions to this problem, how- ever, let us ask whether or not it is possible in principle to recover θ from the mixture. Suppose that we had an unlimited number of samples, and that we used one of the nonparametric methods of Chap. ?? to determine the value of p(x|θ) for every x. If 10.2. MIXTURE DENSITIES AND IDENTIFIABILITY 5 there is only one value of θ that will produce the observed values for p(x|θ), then a solution is at least possible in principle. However, if several different values of θ can produce the same values for p(x|θ), then there is no hope of obtaining a unique solution. These considerations lead us to the following definition: a density p(x|θ) is said to be identifiable if θ �= θ′ implies that there exists an x such that p(x|θ) �= p(x|θ′). Or put another way, a density p(x|θ) is not identifiable if we cannot recover a unique θ, even from an infinite amount of data. In the discouraging situation where we cannot infer any of the individual parameters (i.e., components of θ), the density is completely unidentifiable.∗ Note that the identifiability of θ is a property of the complete unidentifi- ability model, irrespective of any procedure we might use to determine its value. As one might expect, the study of unsupervised learning is greatly simplified if we restrict ourselves to identifiable mixtures. Fortunately, most mixtures of commonly encountered density functions are identifiable, as are most complex or high-dimensional density functions encountered in real-world problems. Mixtures of discrete distributions are not always so obliging. As a simple example consider the case where x is binary and P (x|θ) is the mixture P (x|θ) = 1 2 θx1 (1− θ1)1−x + 1 2 θx2 (1− θ2)1−x = { 1 2 (θ1 + θ2) if x = 1 1− 1 2 (θ1 + θ2) if x = 0. Suppose, for example, that we know for our data that P (x = 1|θ) = 0.6, and hence that P (x = 0|θ) = 0.4. Then we know the function P (x|θ), but we cannot determine θ, and hence cannot extract the component distributions. The most we can say is that θ1+θ2 = 1.2. Thus, here we have a case in which the mixture distribution is com- pletely unidentifiable, and hence a case for which unsupervised learning is impossible in principle. Related situations may permit us to determine one or some parameters, but not all (Problem 3). This kind of problem commonly occurs with discrete distributions. If there are too many components in the mixture, there may be more unknowns than independent equations, and identifiability can be a serious problem. For the continuous case, the problems are less severe, although certain minor difficulties can arise due to the possibility of special cases. Thus, while it can be shown that mixtures of normal densities are usually identifiable, the parameters in the simple mixture density p(x|θ) = P (ω1)√ 2π exp [ −1 2 (x− θ1)2 ] + P (ω2)√ 2π exp [ −1 2 (x− θ2)2 ] (2) cannot be uniquely identified if P (ω1) = P (ω2), for then θ1 and θ2 can be interchanged without affecting p(x|θ). To avoid such irritations, we shall acknowledge that identi- fiability can be a problem, but shall henceforth assume that the mixture densities we are working with are identifiable. ∗ Technically speaking, a distribution is not identifiable if we cannot determine the parameters without bias. We might guess their correct values, but such a guess would have to be biased in some way. 6 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING 10.3 Maximum-Likelihood Estimates Suppose now that we are given a set D = {x1, . . . ,xn} of n unlabeled samples drawn independently from the mixture density p(x|θ) = c∑ j=1 p(x|ωj ,θj)P (ωj), (1) where the full parameter vector θ is fixed but unknown. The likelihood of the observed samples is, by definition, the joint density p(D|θ) ≡ n∏ k=1 p(xk|θ). (3) The maximum-likelihood estimate θ̂ is that value of θ that maximizes p(D|θ). If we assume that p(D|θ) is a differentiable function of θ, then we can derive some interesting necessary conditions for θ̂. Let l be the logarithm of the likelihood, and let ∇θi l be the gradient of l with respect to θi. Then l = n∑ k=1 ln p(xk|θ) (4) and ∇θi l = n∑ k=1 1 p(xk|θ) ∇θi c∑ j=1 p(xk|ωj ,θj)P (ωj) . (5) If we assume that the elements of θi and θj are functionally independent if i �= j, and if we introduce the posterior probability P (ωi|xk,θ) = p(xk|ωi,θi)P (ωi) p(xk|θ) , (6) we see that the gradient of the log-likelihood can be written in the interesting form ∇θi l = n∑ k=1 P (ωi|xk,θ)∇θi ln p(xk|ωi,θi). (7) Since the gradient must vanish at the value of θi that maximizes l, the maximum- likelihood estimate θ̂i must satisfy the conditions n∑ k=1 P (ωi|xk, θ̂)∇θi ln p(xk|ωi, θ̂i) = 0, i = 1, . . . , c. (8) Among the solutions to these equations for θ̂i we may find the maximum-likelihood solution. It is not hard to generalize these results to include the prior probabilities P (ωi) among the unknown quantities. In this case the search for the maximum value of p(D|θ) extends over θ and P (ωi), subject to the constraints P (ωi) ≥ 0 i = 1, . . . , c (9) 10.4. APPLICATION TO NORMAL MIXTURES 7 and c∑ i=1 P (ωi) = 1. (10) Let P̂ (ωi) be the maximum-likelihood estimate for P (ωi), and let θ̂i be the maximum- likelihood estimate for θi. It can be shown (Problem ??) that if the likelihood function is differentiable and if P̂ (ωi) �= 0 for any i, then P̂ (ωi) and θ̂i must satisfy P̂ (ωi) = 1 n n∑ k=1 P̂ (ωi|xk, θ̂) (11) and n∑ k=1 P̂ (ωi|xk, θ̂)∇θi ln p(xk|ωi, θ̂i) = 0, (12) where P̂ (ωi|xk, θ̂) = p(xk|ωi, θ̂i)P̂ (ωi) c∑ j=1 p(xk|ωj , θ̂j)P̂ (ωj) . (13) These equations have the following interpretation. Equation 11 states that the maximum-likelihood estimate of the probability of a category is the average over the entire data set of the estimate derived from each sample — each sample is weighted equally. Equation 13 is ultimately related to Bayes Theorem, but notice that in estimating the probability for class ωi, the numerator on the right-hand side depends on θ̂i and not the full θ̂ directly. While Eq. 12 is a bit subtle, we can understand it clearly in the trivial n = 1 case. Since P̂ �= 0, this case states merely that the probability density is maximized as a function of θi — surely what is needed for the maximum-likelihood solution. 10.4 Application to Normal Mixtures It is enlightening to see how these general results apply to the case where the compo- nent densities are multivariate normal, p(x|ωi,θi) ∼ N(µi,Σi). The following table illustrates a few of the different cases that can arise depending upon which parameters are known (×) and which are unknown (?): Case µi Σi P (ωi) c 1 ? × × × 2 ? ? ? × 3 ? ? ? ? Case 1 is the simplest, and will be considered in detail because of its pedagogical value. Case 2 is more realistic, though somewhat more involved. Case 3 represents the problem we face on encountering a completely unknown set of data; unfortunately, it cannot be solved by maximum-likelihood methods. We shall postpone discussion of what can be done when the number of classes is unknown until Sect. ??. 8 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING 10.4.1 Case 1: Unknown Mean Vectors If the only unknown quantities are the mean vectors µi, then of course θi consists of the components of µi. Equation 8 can then be used to obtain necessary conditions on the maximum-likelihood estimate for µi. Since the likelihood is ln p(x|ωi,µi) = −ln [ (2π)d/2|Σi|1/2 ] − 1 2 (x− µi)tΣ−1i (x− µi), (14) its derivative is ∇µ i ln p(x|ωi,µi) = Σ −1 i (x− µi). (15) Thus according to Eq. 8, the maximum-likelihood estimate µ̂i must satisfy n∑ k=1 P (ωi|xk, µ̂)Σ−1i (xk − µ̂i) = 0, where µ̂ = (µ̂1, . . . , µ̂c). (16) After multiplying by Σi and rearranging terms, we obtain the solution: µ̂i = n∑ k=1 P (ωi|xk, µ̂)xk n∑ k=1 P (ωi|xk, µ̂) . (17) This equation is intuitively very satisfying. It shows that the maximum-likelihood estimate for µi is merely a weighted average of the samples; the weight for the kth sample is an estimate of how likely it is that xk belongs to the ith class. If P (ωi|xk, µ̂) happened to be 1.0 for some of the samples and 0.0 for the rest, then µ̂i would be the mean of those samples estimated to belong to the ith class. More generally, suppose that µ̂i is sufficiently close to the true value of µi that P (ωi|xk, µ̂) is essentially the true posterior probability for ωi. If we think of P (ωi|xk, µ̂) as the fraction of those samples having value xk that come from the ith class, then we see that Eq. 17 essentially gives µ̂i as the average of the samples coming from the ith class. Unfortunately, Eq. 17 does not give µ̂i explicitly, and if we substitute P (ωi|xk, µ̂) = p(xk|ωi, µ̂i)P (ωi) c∑ j=1 p(xk|ωj , µ̂j)P (ωj) with p(x|ωi, µ̂i) ∼ N(µ̂i,Σi), we obtain a tangled snarl of coupled simultaneous nonlinear equations. These equations usually do not have a unique solution, and we must test the solutions we get to find the one that actually maximizes the likelihood. If we have some way of obtaining fairly good initial estimates µ̂i(0) for the unknown means, Eq. 17 suggests the following iterative scheme for improving the estimates: µ̂i(j + 1) = n∑ k=1 P (ωi|xk, µ̂(j))xk n∑ k=1 P (ωi|xk, µ̂(j)) (18) This is basically a gradient ascent or hill-climbing procedure for maximizing the log- likelihood function. If the overlap between component densities is small, then the 10.4. APPLICATION TO NORMAL MIXTURES 9 coupling between classes will be small and convergence will be fast. However, when convergence does occur, all that we can be sure of is that the gradient is zero. Like all hill-climbing procedures, this one carries no guarantee of yielding the global maximum (Computer exercise 19). Note too that if the model is mis-specified (for instance we assume the “wrong” number of clusters) then the log-likelihood can actually decrease (Computer exercise 21). Example 1: Mixtures of two 1D Gaussians To illustrate the kind of behavior that can occur, consider the simple two-component one-dimensional normal mixture: p(x|µ1, µ2) = 1 3 √ 2π exp [ −1 2 (x− µ1)2 ] ︸ ︷︷ ︸ ω1 + 2 3 √ 2π exp [ −1 2 (x− µ2)2 ] ︸ ︷︷ ︸ ω2 , where ωi denotes a Gaussian component. The 25 samples shown in the table were drawn sequentially from this mixture with µ1 = −2 and µ2 = 2. Let us use these samples to compute the log-likelihood function l(µ1, µ2) = n∑ k=1 ln p(xk|µ1, µ2) for various values of µ1 and µ2. The bottom figure shows how l varies with µ1 and µ2. The maximum value of l occurs at µ̂1 = −2.130 and µ̂2 = 1.668, which is in the rough vicinity of the true values µ1 = −2 and µ2 = 2. However, l reaches another peak of comparable height at µ̂1 = 2.085 and µ̂2 = −1.257. Roughly speaking, this solution corresponds to interchanging µ1 and µ2. Note that had the prior probabilities been equal, interchanging µ1 and µ2 would have produced no change in the log-likelihood function. Thus, as we mentioned before, when the mixture density is not identifiable, the maximum-likelihood solution is not unique. k xk ω1 ω2 k xk ω1 ω2 k xk ω1 ω2 1 0.608 × 9 0.262 × 17 -3.458 × 2 -1.590 × 10 1.072 × 18 0.257 × 3 0.235 × 11 -1.773 × 19 2.569 × 4 3.949 × 12 0.537 × 20 1.415 × 5 -2.249 × 13 3.240 × 21 1.410 × 6 2.704 × 14 2.400 × 22 -2.653 × 7 -2.473 × 15 -2.499 × 23 1.396 × 8 0.672 × 16 2.608 × 24 3.286 × 25 -0.712 × Additional insight into the nature of these multiple solutions can be obtained by examining the resulting estimates for the mixture density. The figure at the top shows the true (source) mixture density and the estimates obtained by using the two maximum-likelihood estimates as if they were the true parameter values. The 25 sample values are shown as a scatter of points along the abscissa — ω1 points in 10 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING black, ω2 points in red. Note that the peaks of both the true mixture density and the maximum-likelihood solutions are located so as to encompass two major groups of data points. The estimate corresponding to the smaller local maximum of the log- likelihood function has a mirror-image shape, but its peaks also encompass reasonable groups of data points. To the eye, neither of these solutions is clearly superior, and both are interesting. -5 -2.5 0 2.5 5 -4 -2 0 2 4 -150 -100 -50 -4 -3 -2 -1 0 1 2 3 4 x source density l(µ1, µ2) µ1 µ2 -56.7 -52.2 µa µb p(x|µa) p(x|µb) (Above) The source mixture density used to generate sample data, and two maximum- likelihood estimates based on the data in the table. (Bottom) Log-likelihood of a mixture model consisting of two univariate Gaussians as a function of their means, for the data in the table. Trajectories for the iterative maximum-likelihood estimation of the means of a two-Gaussian mixture model based on the data are shown as red lines. Two local optima (with log-likelihoods -52.2 and -56.7) correspond to the two density estimates shown above. If Eq. 18 is used to determine solutions to Eq. 17 iteratively, the results depend on the starting values µ̂1(0) and µ̂2(0). The bottom figure shows trajectories from two different starting points. Although not shown, if µ̂1(0) = µ̂2(0), convergence to a saddle point occurs in one step. This is not a coincidence; it happens for the simple reason that for this starting point P (ωi|xk, µ̂i(0), µ̂i(0)) = P (ωi). In such a case Eq. 18 yields the mean of all of the samples for µ̂1 and µ̂2 for all successive iterations. Clearly, this is a general phenomenon, and such saddle-point solutions can be expected if the starting point does not bias the search away from a symmetric 10.4. APPLICATION TO NORMAL MIXTURES 11 answer. 10.4.2 Case 2: All Parameters Unknown If µi, Σi, and P (ωi) are all unknown, and if no constraints are placed on the covariance matrix, then the maximum-likelihood principle yields useless singular solutions. The reason for this can be appreciated from the following simple example in one dimension. Let p(x|µ, σ2) be the two-component normal mixture: p(x|µ, σ2) = 1 2 √ 2πσ exp [ − 1 2 (x− µ σ )2] + 1 2 √ 2π exp [ − 1 2 x2 ] . The likelihood function for n samples drawn from this probability density is merely the product of the n densities p(xk|µ, σ2). Suppose that we let µ = x1, the value of the first sample. In this situation the density is p(x|µ, σ2) = 1 2 √ 2πσ + 1 2 √ 2π exp [ −1 2 x2 ] . Clearly, for the rest of the samples p(xk|µ, σ2) ≥ 1 2 √ 2π exp [ −1 2 x2k ] , so that p(x1, . . . , xn|µ, σ2) ≥ { 1 σ + exp [ −1 2 x21 ]} 1 (2 √ 2π)n exp [ −1 2 n∑ k=2 x2k ] . Thus, the first term at the right shows that by letting σ approach zero we can make the likelihood arbitrarily large, and the maximum-likelihood solution is singular. Ordinarily, singular solutions are of no interest, and we are forced to conclude that the maximum-likelihood principle fails for this class of normal mixtures. However, it is an empirical fact that meaningful solutions can still be obtained if we restrict our attention to the largest of the finite local maxima of the likelihood function. Assuming that the likelihood function is well behaved at such maxima, we can use Eqs. 11 – 13 to obtain estimates for µi, Σi, and P (ωi). When we include the elements of Σi in the elements of the parameter vector θi, we must remember that only half of the off-diagonal elements are independent. In addition, it turns out to be much more convenient to let the independent elements of Σ−1i rather than Σi be the unknown parameters. With these observations, the actual differentiation of ln p(xk|ωi,θi) = ln |Σ−1i |1/2 (2π)d/2 − 1 2 (xk − µi)tΣ−1i (xk − µi) 12 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING with respect to the elements of µi and Σ −1 i is relatively routine. Let xp(k) be the pth element of xk, µp(i) be the pth element of µi, σpq(i) be the pqth element of Σi, and σpq(i) be the pqth element of Σ−1i . Then differentiation gives ∇µ i ln p(xk|ωi,θi) = Σ−1i (xk − µi) and ∂ln p(xk|ωi,θi) ∂σpq(i) = ( 1− δpq 2 )[ σpq(i)− (xp(k)− µp(i))(xq(k)− µq(i)) ] , where δpq is the Kronecker delta. We substitute these results in Eq. 12 and perform a small amount of algebraic manipulation (Problem 16) and thereby obtain the following equations for the local-maximum-likelihood estimate µ̂i, Σ̂i, and P̂ (ωi): P̂ (ωi) = 1 n n∑ k=1 P̂ (ωi|xk, θ̂) (19) µ̂i = n∑ k=1 P̂ (ωi|xk, θ̂)xk n∑ k=1 P̂ (ωi|xk, θ̂) (20) Σ̂i = n∑ k=1 P̂ (ωi|xk, θ̂)(xk − µi)(xk − µi)t n∑ k=1 P̂ (ωi|xk, θ̂) (21) where P̂ (ωi|xk, θ̂) = p(xk|ωi, θ̂i)P̂ (ωi) c∑ j=1 p(xk|ωj , θ̂j)P̂ (ωj) = |Σ̂i|−1/2 exp [ − 1 2 (xk − µ̂i)tΣ̂−1i (xk − µ̂i) ] P̂ (ωi) c∑ j=1 |Σ̂j |−1/2 exp [ − 1 2 (xk − µ̂j)tΣ̂−1j (xk − µ̂j) ] P̂ (ωj) . (22) While the notation may make these equations appear to be rather formidable, their interpretation is actually quite simple. In the extreme case where P̂ (ωi|xk, θ̂) is 1.0 when xk is from Class ωi and 0.0 otherwise, P̂ (ωi) is the fraction of samples from ωi, µ̂i is the mean of those samples, and Σ̂i is the corresponding sample covariance matrix. More generally, P̂ (ωi|xk, θ̂) is between 0.0 and 1.0, and all of the samples play some role in the estimates. However, the estimates are basically still frequency ratios, sample means, and sample covariance matrices. The problems involved in solving these implicit equations are similar to the prob- lems discussed in Sect. ??, with the additional complication of having to avoid singular solutions. Of the various techniques that can be used to obtain a solution, the most obvious approach is to use initial estimates to evaluate Eq. 22 for P̂ (ωi|xk, θ̂) and then 10.4. APPLICATION TO NORMAL MIXTURES 13 to use Eqs. 19 – 21 to update these estimates. If the initial estimates are very good, having perhaps been obtained from a fairly large set of labeled samples, convergence can be quite rapid. However, the results do depend upon the starting point, and the problem of multiple solutions is always present. Furthermore, the repeated computa- tion and inversion of the sample covariance matrices can be quite time consuming. Considerable simplification can be obtained if it is possible to assume that the covariance matrices are diagonal. This has the added virtue of reducing the number of unknown parameters, which is very important when the number of samples is not large. If this assumption is too strong, it still may be possible to obtain some simplification by assuming that the c covariance matrices are equal, which also may eliminate the problem of singular solutions (Problem 16). 10.4.3 K-means clustering Of the various techniques that can be used to simplify the computation and acceler- ate convergence, we shall briefly consider one elementary, approximate method. From Eq. 22, it is clear that the probability P̂ (ωi|xk, θ̂) is large when the squared Maha- lanobis distance (xk − µ̂i)tΣ̂−1i (xk − µ̂i) is small. Suppose that we merely compute the squared Euclidean distance ‖xk − µ̂i‖2, find the mean µ̂m nearest to xk, and approximate P̂ (ωi|xk, θ̂) as P̂ (ωi|xk, θ̂) ≈ { 1 if i = m 0 otherwise. (23) Then the iterative application of Eq. 20 leads to the following procedure for finding µ̂1, . . . , µ̂c. (Although the algorithm is historically referred to as k-means clustering, we retain the notation c, our symbol for the number of clusters.) Algorithm 1 (K-means clustering) 1 begin initialize n, c,µ1,µ2, . . . ,µc 2 do classify n samples according to nearest µi 3 recompute µi 4 until no change in µi 5 return µ1,µ2, . . . ,µc 6 end The computational complexity of the algorithm is O(ndcT ) where d the number of features and T the number of iterations (Problem 15). In practice, the number of iterations is generally much less than the number of samples. This is typical of a class of procedures that are known as clustering procedures or algorithms. Later on we shall place it in the class of iterative optimization procedures, since the means tend to move so as to minimize a squared-error criterion function. For the moment we view it merely as an approximate way to obtain maximum-likelihood estimates for the means. The values obtained can be accepted as the answer, or can be used as starting points for the more exact computations. It is interesting to see how this procedure behaves on the example data we saw in Example 1. Figure 10.1 shows the sequence of values for µ̂1 and µ̂2 obtained for several different starting points. Since interchanging µ̂1 and µ̂2 merely interchanges the labels assigned to the data, the trajectories are symmetric about the line µ̂1 = µ̂2. The trajectories lead either to the point µ̂1 = −2.176, µ̂2 = 1.684 or to its symmetric 14 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING -6 -4 -2 2 4 6 -4 -2 2 4 µ µ Figure 10.1: The k-means clustering procedure is a form of stochastic hill climbing in the log-likelihood function. The contours represent equal log-likelihood values for the one-dimensional data in Example 1. The dots indicate parameter values after different iterations of the k-means algorithm. Six of the starting points shown lead to local maxima, whereas two (i.e., µ1(0) = µ2(0)) lead to a saddle point near µ = 0. image. This is close to the solution found by the maximum-likelihood method (viz., µ̂1 = −2.130 and µ̂2 = 1.688), and the trajectories show a general resemblance to those shown in Example 1. In general, when the overlap between the component densities is small the maximum-likelihood approach and the k-means procedure can be expected to give similar results. Figure 10.2 shows a two-dimensional example, with the assumption of c = 3 clus- ters. The three initial cluster centers, chosen randomly from the training points, and their associated Voronoi tesselation, are shown in pink. According to the algorithm, the points in each of the three Voronoi cells are used to calculate new cluster centers (dark pink), and so on. Here, after the third iteration the algorithm has converged (red). Because the k-means algorithm is very simple and works well in practice, it is a staple of clustering methods. 10.4.4 *Fuzzy k-means clustering In every iteration of the classical k-means procedure, each data point is assumed to be in exactly one cluster, as implied by Eq. 23 and by lines 2 & 3 of Algorithm 1. We can relax this condition and assume that each sample xj has some graded or “fuzzy” cluster membership µi(xj) in cluster ωi, where 0 ≤ µi(xj) ≤ 1. At root, these “memberships” are equivalent to the probabilities P̂ (ωi|xj , θ̂) given in Eq. 22, and thus we use this symbol. In the resulting fuzzy k-means clustering algorithm we seek a minimum of a global cost function L = c∑ i=1 n∑ j=1 [P̂ (ωi|xj , θ̂)]b||xj − µi||2, (24) where b > 1 is a free parameter chosen to adjust the “blending” of different clusters.
10.4. APPLICATION TO NORMAL MIXTURES 15
x1
x2
1
2
3
Figure 10.2: Trajectories for the means of the k-means clustering procedure applied to
two-dimensional data. The final Voronoi tesselation (for classification) is also shown
— the means correspond to the “centers” of the Voronoi cells.
If b is set to 0, this criterion function is merely a sum-of-squared errors criterion we
shall see again in Eq. 49. The probabilities of cluster membership for each point are
normalized as
c∑
i=1
P̂ (ωi|xj) = 1, j = 1, . . . , n. (25)
At the solution, i.e., the minimum of L, we have
∂L/∂µi = 0 and ∂L/∂P̂j = 0, (26)
and these lead (Problem 14) to the conditions
µj =
n∑
j=1
[P (ωi|xj)]bxj
n∑
j=1
[P (ωi|xj)]b
(27)
and
P (ωi|xj) =
(1/dij)1/(b−1)
c∑
r=1
(1/drj)1/(b−1)
, dij = ||xj − µi||2. (28)
In general, the criterion is minimized when the cluster centers µj are near those
points that have high estimated probability of being in cluster j. Since Eqs 27 & 28
rarely have analytic solutions, the cluster means and point probabilities are estimated
iteratively according to the following algorithm:
Algorithm 2 (Fuzzy k-means clustering)
16 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
x1
x2
1
2
3
4
Figure 10.3: At each iteration of the fuzzy k-means clustering algorithm, the prob-
ability of category memberships for each point are adjusted according to Eqs. 27 &
28 (here b = 2). While most points have non-negligible memberships in two or three
clusters, we nevertheless draw the boundary of a Voronoi tesselation to illustrate the
progress of the algorithm. After four iterations, the algorithm has converged and the
red cluster centers and associated Voronoi tesselation would be used for assigning new
points to clusters.
1 begin initialize n,µ1, . . . ,µc, P (ωi | xj), i = 1 . . . , c; j = 1, . . . , n
2 normalize proabilities of cluster memberships by Eq. 25
3 do classify n samples according to nearest µi
4 recompute µi by Eq. 27
5 recompute P (ωi | xj) by Eq. 28
6 until no change in µi and P (ωi | xj)
7 return µ1,µ2, . . . ,µc
8 end
Figure 10.3 illustrates the algorithm. At early iterations the means lie near the center
of the full data set because each point has a non-negligible “membership” (i.e., prob-
ability) in each cluster. At later iterations the means separate and each membership
tends toward the value 1.0 or 0.0. Clearly, the classical k-means algorithm is just of
special case where the memberships for all points obey
P (ωi|xj) =
{
1 if ‖xj − µi‖ < ‖xj − µi′‖ for all i′ �= i
0 otherwise,
(29)
as given by Eq. 17.
While it may seem that such graded membership might improve convergence of
k-means over its classical counterpart, in practice there are several drawbacks to the
fuzzy method. One is that according to Eq. 25 the probability of “membership” of
a point xj in a cluster i depends implicitly on the number of clusters, and when
the number of clusters is specified incorrectly, serious problems may arise (Computer
exercise 4).
10.5. UNSUPERVISED BAYESIAN LEARNING 17
10.5 Unsupervised Bayesian Learning
10.5.1 The Bayes Classifier
As we saw in Chap. ??, maximum-likelihood methods do not assume the parameter
vector θ to be random — it is just unknown. In such methods, prior knowledge about
the likely values for θ is not directly relevant, although in practice such knowledge
may be used in choosing good starting points for hill-climbing procedures. In this
section, however, we shall take a Bayesian approach to unsupervised learning. That
is, we shall assume that θ is a random variable with a known prior distribution p(θ),
and we shall use the samples to compute the posterior density p(θ|D). Interestingly
enough, the analysis will closely parallel the analysis of supervised Bayesian learning
(Sect. ??.??), showing that the two problems are formally very similar.
We begin with an explicit statement of our basic assumptions. We assume that
1. The number of classes c is known.
2. The prior probabilities P (ωj) for each class are known, j = 1, . . . , c.
3. The forms for the class-conditional probability densities p(x|ωj ,θj) are known,
j = 1, . . . , c, but the full parameter vector θ = (θ1, . . . ,θc) is not known.
4. Part of our knowledge about θ is contained in a known prior density p(θ).
5. The rest of our knowledge about θ is contained in a set D of n samples x1, . . . ,xn
drawn independently from the familiar mixture density
p(x|θ) =
c∑
j=1
p(x|ωj ,θj)P (ωj). (30)
At this point we could go directly to the calculation of p(θ|D). However, let us
first see how this density is used to determine the Bayes classifier. Suppose that a
state of nature is selected with probability P (ωi) and a feature vector x is selected
according to the probability law p(x|ωi,θi). To derive the Bayes classifier we must use
all of the information at our disposal to compute the posterior probability P (ωi|x).
We exhibit the role of the samples explicitly by writing this as P (ωi|x,D). By Bayes’
formula, we have
P (ωi|x,D) =
p(x|ωi,D)P (ωi|D)
c∑
j=1
p(x|ωj ,D)P (ωj |D)
. (31)
Since the selection of the state of nature ωi was done independently of the previously
drawn samples, P (ωi|D) = P (ωi), and we obtain
P (ωi|x,D) =
p(x|ωi,D)P (ωi)
c∑
j=1
p(x|ωj ,D)P (ωj)
. (32)
Central to the Bayesian approach is the introduction of the unknown parameter
vector θ via
18 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
p(x|ωi,D) =
∫
p(x,θ|ωi,D) dθ
=
∫
p(x|θ, ωi,D)p(θ|ωi,D) dθ. (33)
Since the selection of x is independent of the samples, we have p(x|θ, ωi,D) =
p(x|ωi,θi). Similarly, since knowledge of the state of nature when x is selected tells
us nothing about the distribution of θ, we have p(θ|ωi,D) = p(θ|D), and thus
P (x|ωi,D) =
∫
p(x|ωi,θi)p(θ|D) dθ. (34)
That is, our best estimate of p(x|ωi) is obtained by averaging p(x|ωi,θi) over θi.
Whether or not this is a good estimate depends on the nature of p(θ|D), and thus
our attention turns at last to that density.
10.5.2 Learning the Parameter Vector
We can use Bayes’ formula to write
p(θ|D) = p(D|θ)p(θ)∫
p(D|θ)p(θ) dθ
, (35)
where the independence of the samples yields the likelihood
p(D|θ) =
n∏
k=1
p(xk|θ). (36)
Alternatively, letting Dn denote the set of n samples, we can write Eq. 35 in the
recursive form
p(θ|Dn) = p(xn|θ)p(θ|D
n−1)∫
p(xn|θ)p(θ|Dn−1) dθ
. (37)
These are the basic equations for unsupervised Bayesian learning. Equation 35
emphasizes the relation between the Bayesian and the maximum-likelihood solutions.
If p(θ) is essentially uniform over the region where p(D|θ) peaks, then p(θ|D) peaks
at the same place. If the only significant peak occurs at θ = θ̂, and if the peak is very
sharp, then Eqs. 32 & 34 yield
p(x|ωi,D) ≈ p(x|ωi, θ̂) (38)
and
P (ωi|x,D) ≈
p(x|ωi, θ̂i)P (ωi)
c∑
j=1
p(x|ωj , θ̂j)P (ωj)
. (39)
That is, these conditions justify the use of the maximum-likelihood estimate as if it
were the true value of θ in designing the Bayes classifier.
As we saw in Sect. ??.??, in the limit of large amounts of data, maximum-likelihood
and the Bayes methods will agree (or nearly agree). While many small sample size
10.5. UNSUPERVISED BAYESIAN LEARNING 19
θ
θ̂
p(D|θ)
Figure 10.4: In a highly skewed or multiple peak posterior distribution such as illus-
trated here, the maximum-likelihood solution θ̂ will yield a density very different from
a Bayesian solution, which requires the integration over the full range of parameter
space θ.
problems they will agree, there exist small problems where the approximations are
poor (Fig. 10.4). As we saw in the analogous case in supervised learning whether one
chooses to use the maximum-likelihood or the Bayes method depends not only on how
confident one is of the prior distributions, but also on computational considerations;
maximum-likelihood techniques are often easier to implement than Bayesian ones.
Of course, if p(θ) has been obtained by supervised learning using a large set of
labeled samples, it will be far from uniform, and it will have a dominant influence on
p(θ|Dn) when n is small. Equation 37 shows how the observation of an additional
unlabeled sample modifies our opinion about the true value of θ, and emphasizes the
ideas of updating and learning. If the mixture density p(x|θ) is identifiable, then
each additional sample tends to sharpen p(θ|Dn), and under fairly general conditions
p(θ|Dn) can be shown to converge (in probability) to a Dirac delta function centered
at the true value of θ (Problem 8). Thus, even though we do not know the categories
of the samples, identifiability assures us that we can learn the unknown parameter
vector θ, and thereby learn the component densities p(x|ωi,θ).
This, then, is the formal Bayesian solution to the problem of unsupervised learning.
In retrospect, the fact that unsupervised learning of the parameters of a mixture
density is so similar to supervised learning of the parameters of a component density
is not at all surprising. Indeed, if the component density is itself a mixture, there
would appear to be no essential difference between the two problems.
There are, however, some significant differences between supervised and unsuper-
vised learning. One of the major differences concerns the issue of identifiability. With
supervised learning, the lack of identifiability merely means that instead of obtaining
a unique parameter vector we obtain an equivalence class of parameter vectors. Be-
cause all of these yield the same component density, lack of identifiability presents no
theoretical difficulty. A lack of identifiability is much more serious in unsupervised
learning. When θ cannot be determined uniquely, the mixture cannot be decomposed
into its true components. Thus, while p(x|Dn) may still converge to p(x), p(x|ωi,Dn)
given by Eq. 34 will not in general converge to p(x|ωi), and a theoretical barrier to
learning exists. It is here that a few labeled training samples would be valuable: for
“decomposing” the mixture into its components.
Another serious problem for unsupervised learning is computational complexity.
With supervised learning, the possibility of finding sufficient statistics allows solutions
20 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
that are analytically pleasing and computationally feasible. With unsupervised learn-
ing, there is no way to avoid the fact that the samples are obtained from a mixture
density,
p(x|θ) =
c∑
j=1
p(x|ωj ,θj)P (ωj), (1)
and this gives us little hope of every finding simple exact solutions for p(θ|D). Such
solutions are tied to the existence of a simple sufficient statistic (Sect. ??.??), and the
factorization theorem requires the ability to factor p(D|θ) as
p(D|θ) = g(s,θ)h(D). (40)
But from Eqs. 36 & 1, we see that the likelihood can be written as
p(D|θ) =
n∏
k=1
[ c∑
j=1
p(xk|ωj ,θj)P (ωj)
]
. (41)
Thus, p(D|θ) is the sum of cn products of component densities. Each term in this
sum can be interpreted as the joint probability of obtaining the samples x1, . . . ,xn
bearing a particular labeling, with the sum extending over all of the ways that the
samples could be labeled. Clearly, this results in a thorough mixture of θ and the x’s,
and no simple factoring should be expected. An exception to this statement arises
if the component densities do not overlap, so that as θ varies only one term in the
mixture density is non-zero. In that case, p(D|θ) is the product of the n nonzero
terms, and may possess a simple sufficient statistic. However, since that case allows
the class of any sample to be determined, it actually reduces the problem to one of
supervised learning, and thus is not a significant exception.
Another way to compare supervised and unsupervised learning is to substitute the
mixture density for p(xn|θ) in Eq. 37 and obtain
p(θ|Dn) =
c∑
j=1
p(xn|ωj ,θj)P (ωj)
c∑
j=1
∫
p(xn|ωj ,θj)P (ωj)p(θ|Dn−1) dθ
p(θ|Dn−1). (42)
If we consider the special case where P (ω1) = 1 and all the other prior probabilities
are zero, corresponding to the supervised case in which all samples come from Class
ω1, then Eq. 42 simplifies to
p(θ|Dn) = p(xn|ω1,θ1)∫
p(xn|ω1,θ1)p(θ|Dn−1) dθ
p(θ|Dn−1). (43)
Let us compare Eqs. 42 & 43 to see how observing an additional sample changes
our estimate of θ. In each case we can ignore the normalizing denominator, which is
independent of θ. Thus, the only significant difference is that in the supervised case
we multiply the “prior” density for θ by the component density p(xn|ω1,θ1), while
in the unsupervised case we multiply it by the mixture density
c∑
j=1
p(xn|ωj ,θj)P (ωj).
Assuming that the sample really did come from Class ω1, we see that the effect of
not knowing this category membership in the unsupervised case is to diminish the
influence of xn on changing θ. Since xn could have come from any of the c classes, we
10.5. UNSUPERVISED BAYESIAN LEARNING 21
cannot use it with full effectiveness in changing the component(s) of θ associated with
any one category. Rather, we must distributed its effect over the various categories
in accordance with the probability that it arose from each category.
Example 2: Unsupervised learning of Gaussian data
As an example, consider the one-dimensional, two-component mixture with p(x|ω1) ∼
N(µ, 1), p(x|ω2, θ) ∼ N(θ, 1), where µ, P (ω1) and P (ω2) are known. Here we have
p(x|θ) = P (ω1)√
2π
exp
[
−1
2
(x− µ)2
]
︸ ︷︷ ︸
ω1
+
P (ω2)√
2π
exp
[
−1
2
(x− θ)2
]
︸ ︷︷ ︸
ω2
,
and we seek the mean of the second component.
Viewed as a function of x, this mixture density is a superposition of two normal
densities — one peaking at x = µ and the other peaking at x = θ. Viewed as a
function of θ, p(x|θ) has a single peak at θ = x. Suppose that the prior density p(θ)
is uniform from a to b. Then after one observation (x = x1) we have
p(θ|x1) = αp(x1|θ)p(θ)
=
α′{P (ω1)exp[− 12 (x1 − µ)
2]+
P (ω2)exp[− 12 (x1 − θ)
2]} a ≤ θ ≤ b
0 otherwise
,
where α and α′ are normalizing constants that are independent of θ. If the sample
x1 is in the range a ≤ x ≤ b, then p(θ|x1) peaks at θ = x1, of course. Otherwise it
peaks either at θ = a if x1 < a or at θ = b if x1 > b. Note that the additive constant
exp [−(1/2)(x1 − µ)2] is large if x1 is near µ, and thus the peak of p(θ|x1) is less
pronounced if x1 is near µ. This corresponds to the fact that if x1 is near µ, it is
more likely to have come from the p(x|ω1) component, and hence its influence on our
estimate for θ is diminished.
With the addition of a second sample x2, p(θ|x1) changes to
p(θ|x1, x2) = βp(x2|θ)p(θ|x1)
=
β′{P (ω1)P (ω1)exp [− 12 (x1 − µ)
2 − 1
2
(x2 − µ)2]
+[P (ω1)P (ω2)exp [− 12 (x1 − µ)
2 − 1
2
(x2 − θ)2]
+[P (ω2)P (ω1)exp [− 12 (x1 − θ)
2 − 1
2
(x2 − µ)2]
+[P (ω2)P (ω2)exp [− 12 (x1 − θ)
2 − 1
2
(x2 − θ)2]}
a ≤ θ ≤ b
0 otherwise.
Unfortunately, the primary thing we learn from this expression is that p(θ|Dn) is
already complicated when n = 2. The four terms in the sum correspond to the
four ways in which the samples could have been drawn from the two component
populations. With n samples there will be 2n terms, and no simple sufficient statistics
can be found to facilitate understanding or to simplify computations.
It is possible to use the relation
22 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
p(θ|Dn) = p(xn|θ)p(θ|D
n−1)∫
p(xn|θ)p(θ|Dn−1) dθ
and numerical integration to obtain an approximate numerical solution for p(θ|Dn).
This was done for the data in Example 1 using the values µ = 2, P (ω1) = 1/3, and
P (ω2) = 2/3. A prior density p(θ) uniform from −4 to +4 encompasses the data in
the table. When this was used to start the recursive computation of p(θ|Dn), the
results shown in the figure. As n goes to infinity we can confidently expect p(θ|Dn)
to approach an impulse centered at θ = 2. This graph gives some idea of the rate of
convergence.
-4 -2 2 4
.25
.5
.75
1
1.25
1.5
n = 0
n = 1
θ
p(θ | x1, …, xn)
n = 4
n = 9
n = 9
n = 16
n = 25
0
n = 0
n = 1
θ
p(θ | x1, …, xn)
n = 4
n = 16
n = 25
-4 -2 2 4
.25
.5
.75
1
1.25
1.5
0
In unsupervised Bayesian learning of the parameter θ, the density becomes more
peaked as the number of samples increases. The top figures uses a wide uniform prior
p(θ) = 1/8,−4 ≤ θ ≤ 4 while the bottom figure uses a narrower one, p(θ) = 1/2, 1 ≤
θ ≤ 3. Despite these different prior distributions, after all 25 samples have been used,
the posterior densities are virtually identical in the two cases — the information in
the samples overwhelms the prior information.
One of the main differences between the Bayesian and the maximum-likelihood
approaches to unsupervised learning appears in the presence of the prior density p(θ).
The figure shows how p(θ|Dn) changes when p(θ) is assumed to be uniform from 1 to
3, corresponding to more certain initial knowledge about θ. The results of this change
are most pronounced when n is small. It is here (just as in the classification analog
of Chap. ??) that the differences between the Bayesian and the maximum-likelihood
10.5. UNSUPERVISED BAYESIAN LEARNING 23
solutions are most significant. As n increases, the importance of prior knowledge
diminishes, and in the particular case the curves for n = 25 are virtually identical. In
general, one would expect the difference to be small when the number of unlabeled
samples is several times the effective number of labeled samples used to determine
p(θ).
10.5.3 Decision-Directed Approximation
Although the problem of unsupervised learning can be stated as merely the problem of
estimating parameters of a mixture density, neither the maximum-likelihood nor the
Bayesian approach yields analytically simple results. Exact solutions for even the sim-
plest nontrivial examples lead to computational requirements that grow exponentially
with the number of samples (Problem ??). The problem of unsupervised learning is
too important to abandon just because exact solutions are hard to find, however, and
numerous procedures for obtaining approximate solutions have been suggested.
Since the important difference between supervised and unsupervised learning is
the presence or absence of labels for the samples, an obvious approach to unsuper-
vised learning is to use the prior information to design a classifier and to use the
decisions of this classifier to label the samples. This is called the decision-directed
approach to unsupervised learning, and it is subject to many variations. It can be
applied sequentially on-line by updating the classifier each time an unlabeled sample
is classified. Alternatively, it can be applied in parallel (batch mode) by waiting un-
til all n samples are classified before updating the classifier. If desired, this process
can be repeated until no changes occur in the way the samples are labeled. Various
heuristics can be introduced to make the extent of any corrections depend upon the
confidence of the classification decision.
There are some obvious dangers associated with the decision-directed approach.
If the initial classifier is not reasonably good, or if an unfortunate sequence of samples
is encountered, the errors in classifying the unlabeled samples can drive the classifier
the wrong way, resulting in a solution corresponding roughly to one of the lesser
peaks of the likelihood function. Even if the initial classifier is optimal, in general
the resulting labeling will not be the same as the true class membership; the act of
classification will exclude samples from the tails of the desired distribution, and will
include samples from the tails of the other distributions. Thus, if there is significant
overlap between the component densities, one can expect biased estimates and less
than optimal results.
Despite these drawbacks, the simplicity of decision-directed procedures makes the
Bayesian approach computationally feasible, and a flawed solution is often better than
none. If conditions are favorable, performance that is nearly optimal can be achieved
at far less computational expense. In practice it is found that most of these procedures
work well if the parametric assumptions are valid, if there is little overlap between
the component densities, and if the initial classifier design is at least roughly correct
(Computer exercise 7).
24 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
10.6 *Data Description and Clustering
Let us reconsider our original problem of learning something of use from a set of
unlabeled samples. Viewed geometrically, these samples may form clouds of points in
a d-dimensional space. Suppose that we knew that these points came from a single
normal distribution. Then the most we could learn form the data would be contained
in the sufficient statistics — the sample mean and the sample covariance matrix.
In essence, these statistics constitute a compact description of the data. The sample
mean locates the center of gravity of the cloud; it can be thought of as the single point
m that best represents all of the data in the sense of minimizing the sum of squared
distances from m to the samples. The sample covariance matrix describes the amount
the data scatters along various directions. If the data points are actually normally
distributed, then the cloud has a simple hyperellipsoidal shape, and the sample mean
tends to fall in the region where the samples are most densely concentrated.
Of course, if the samples are not normally distributed, these statistics can give
a very misleading description of the data. Figure 10.5 shows four different data sets
that all have the same mean and covariance matrix. Obviously, second-order statistics
are incapable of revealing all of the structure in an arbitrary set of data.
Figure 10.5: These four data sets have identical statistics up to second-order, i.e., the
same mean µ and covariance Σ. In such cases it is important to include in the model
more parameters to represent the structure more completely.
If we assume that the samples come from a mixture of c normal distributions,
we can approximate a greater variety of situations. In essence, this corresponds to
assuming that the samples fall in hyperellipsoidally shaped clouds of various sizes
and orientations. If the number of component densities is sufficiently high, we can
approximate virtually any density function as a mixture model in this way, and use the
parameters of the mixture to describe the data. Alas, we have seen that the problem
of estimating the parameters of a mixture density is not trivial. Furthermore, in
situations where we have relatively little prior knowledge about the nature of the
data, the assumption of particular parametric forms may lead to poor or meaningless
results. Instead of finding structure in the data, we would be imposing structure on
10.6. *DATA DESCRIPTION AND CLUSTERING 25
it.
One alternative is to use one of the nonparametric methods described in Chap. ??
to estimate the unknown mixture density. If accurate, the resulting estimate is cer-
tainly a complete description of what we can learn from the data. Regions of high
local density, which might correspond to significant subclasses in the population, can
be found from the peaks or modes of the estimated density.
If the goal is to find subclasses, a more direct alternative is to use a clustering
procedure. Roughly speaking, clustering procedures yield a data description in terms clustering
procedureof clusters or groups of data points that possess strong internal similarities. Formal
clustering procedures use a criterion function, such as the sum of the squared dis-
tances from the cluster centers, and seek the grouping that extremizes the criterion
function. Because even this can lead to unmanageable computational problems, other
procedures have been proposed that are intuitively appealing but that lead to solu-
tions having few if any established properties. Their use is usually justified on the
ground that they are easy to apply and often yield interesting results that may guide
the application of more rigorous procedures.
10.6.1 Similarity Measures
Once we describe the clustering problem as one of finding natural groupings in a set of
data, we are obliged to define what we mean by a natural grouping. In what sense are
we to say that the samples in one cluster are more like one another than like samples
in other clusters? This question actually involves two separate issues:
• How should one measure the similarity between samples?
• How should one evaluate a partitioning of a set of samples into clusters?
In this section we address the first of these issues.
The most obvious measure of the similarity (or dissimilarity) between two samples
is the distance between them. One way to begin a clustering investigation is to define
a suitable distance function and compute the matrix of distances between all pairs
of samples. If distance is a good measure of dissimilarity, then one would expect the
distance between samples in the same cluster to be significantly less than the distance
between samples in different clusters.
Suppose for the moment that we say that two samples belong to the same cluster
if the Euclidean distance between them is less than some threshold distance d0. It is
immediately obvious that the choice of d0 is very important. If d0 is very large, all
of the samples will be assigned to one cluster. If d0 is very small, each sample will
form an isolated, singleton cluster. To obtain “natural” clusters, d0 will have to be
greater than the typical within-cluster distances and less than typical between-cluster
distances (Fig. 10.6).
Less obvious perhaps is the fact that the results of clustering depend on the choice
of Euclidean distance as a measure of dissimilarity. That particular choice is generally
justified if the feature space is isotropic and the data is spread roughly evenly along
all directions. Clusters defined by Euclidean distance will be invariant to translations
or rotations in feature space — rigid-body motions of the data points. However, they
will not be invariant to linear transformations in general, or to other transformations
that distort the distance relationships. Thus, as Fig. 10.7 illustrates, a simple scaling
of the coordinate axes can result in a different grouping of the data into clusters. Of
course, this is of no concern for problems in which arbitrary rescaling is an unnatural
26 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
.2 .4 .6 .8 1
0
.2
.4
.6
.8
1
.2 .4 .6 .8 1
0
.2
.4
.6
.8
.2 .4 .6 .8 1
0
.2
.4
.6
.8
1
d0 = .3
x1 x1 x1
x2 x2 x2
1
d0 = .1 d0 = .03
Figure 10.6: The distance threshold affects the number and size of clusters. Lines are
drawn between points closer than a distance d0 apart for three different values of d0
— the smaller the value of d0, the smaller and more numerous the clusters.
or meaningless transformation. However, if clusters are to mean anything, they should
be invariant to transformations natural to the problem.
One way to achieve invariance is to normalize the data prior to clustering. For
example, to obtain invariance to displacement and scale changes, one might translate
and scale the axes so that all of the features have zero mean and unit variance —
standardize the data. To obtain invariance to rotation, one might rotate the axes so
that they coincide with the eigenvectors of the sample covariance matrix. This trans-
formation to principal components (Sect. 10.13.1) can be preceded and/or followed by
normalization for scale.
However, we should not conclude that this kind of normalization is necessarily
desirable. Consider, for example, the matter of translating and whitening — scaling
the axes so that each feature has zero mean and unit variance. The rationale usually
given for this normalization is that it prevents certain features from dominating dis-
tance calculations merely because they have large numerical values, much as we saw
in networks trained with backpropagation (Sect. ??.??). Subtracting the mean and
dividing by the standard deviation is an appropriate normalization if this spread of
values is due to normal random variation; however, it can be quite inappropriate if the
spread is due to the presence of subclasses (Fig. ??). Thus, this routine normalization
may be less than helpful in the cases of greatest interest.∗ Section ?? describes other
ways to obtain invariance to scaling.
Instead of scaling axes, we can change the metric in interesting ways. For instance,
one broad class of distance metrics is of the form
d(x,x′) =
(
d∑
k=1
|xk − x′k|q
)1/q
, (44)
where q ≥ 1 is a selectable parameter — the general Minkowski metric we consideredMinkowski
metric in Chap. ??. Setting q = 2 gives the familiar Euclidean metric while setting q = 1
the Manhattan or city block metric — the sum of the absolute distances along each
city block
metric
of the d coordinate axes. Note that only q = 2 is invariant to an arbitrary rotation or
∗ In backpropagation, one of the goals for such preprocessing and scaling of data was to increase
learning speed; in contrast, such preprocessing does not significantly affect the speed of these
clustering algorithms.
10.6. *DATA DESCRIPTION AND CLUSTERING 27
.2 .4 .6 .8 1
0
.2
.4
.6
.8
1
.25 .5 .75 1 1.25 1.5 1.75 2
0
.1
.2
.3
.4
.5
.1 .2 .3 .4 .5
0
.2
.4
.6
.8
1
1.2
1.4
1.6
.5 0
0 2( )
2 0
0 .5( )
x2
x2
x2
x1
x1
x1
Figure 10.7: Scaling axes affects the clusters in a minimum distance cluster method.
The original data and minimum-distance clusters are shown in the upper left — points
in one cluster are shown in red, the other gray. When the vertical axis is expanded
by a factor of 2.0 and the horizontal axis shrunk by a factor of 0.5, the clustering is
altered (as shown at the right). Alternatively, if the vertical axis is shrunk by a factor
of 0.5 and the horizontal axis expanded by a factor of 2.0, smaller more numerous
clusters result (shown at the bottom). In both these scaled cases, the clusters differ
from the original.
translation in feature space. Another alternative is to use some kind of metric based
on the data itself, such as the Mahalanobis distance.
More generally, one can abandon the use of distance altogether and introduce a
nonmetric similarity function s(x,x′) to compare two vectors x and x′. Convention- similarity
functionally, this is a symmetric functions whose value is large when x and x′ are somehow
“similar.” For example, when the angle between two vectors is a meaningful measure
of their similarity, then the normalized inner product
s(x,x′) =
xtx′
‖x‖ ‖x′‖
(45)
28 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
x1x1
x2x2
Figure 10.8: If the data fall into well-separated clusters (left), normalization by a
whitening transform for the full data may reduce the separation, and hence be unde-
sirable (right). Such a whitening normalization may be appropriate if the full data
set arises from a single fundamental process (with noise), but inappropriate if there
are several different processes, as shown here.
may be an appropriate similarity function. This measure, which is the cosine of the
angle between x and x′, is invariant to rotation and dilation, though it is not invariant
to translation and general linear transformations.
When the features are binary valued (0 or 1), this similarity functions has a simple
non-geometrical interpretation in terms of shared features or shared attributes. Let
us say that a sample x possesses the ith attribute if xi = 1. Then xtx′ is merely the
number of attributes possessed by both x and x′, and ‖x‖ ‖x′‖ = (xtxx′tx′)1/2 is the
geometric mean of the number of attributes possessed by x and the number possessed
by x′. Thus, s(x,x′) is a measure of the relative possession of common attributes.
Some simple variations are
s(x,x′) =
xtx′
d
, (46)
the fraction of attributes shared, and
s(x,x′) =
xtx′
xtx + x′tx′ − xtx′
, (47)
the ratio of the number of shared attributes to the number possessed by x or x′. This
latter measure (sometimes known as the Tanimoto coefficient or Tanimoto distance) isTanimoto
distance frequently encountered in the fields of information retrieval and biological taxonomy.
Related measures of similarity arise in other applications, the variety of measures
testifying to the diversity of problem domains (Computer exercise ??).
Fundamental issues in measurement theory are involved in the use of any distance
or similarity function. The calculation of the similarity between two vectors always
involves combining the values of their components. Yet in many pattern recognition
applications the components of the feature vector measure seemingly noncomparable
quantities, such as meters and kilograms. Recall our example of classifying fish: how
can one compare the lightness of the skin to the length or weight of the fish? Should
the comparison depend on whether the length is measured in meters or inches? How
does one treat vectors whose components have a mixture of nominal, ordinal, interval
and ratio scales? Ultimately, there are rarely clear methodological answers to these
questions. When a user selects a particular similarity function or normalizes the data
in a particular way, information is introduced that gives the procedure meaning. We
have given examples of some alternatives that have proved to be useful. (Competitive
10.7. CRITERION FUNCTIONS FOR CLUSTERING 29
learning, discussed in Sect. 10.11, is a popular decision directed clustering algorithm.)
Beyond that we can do little more than alert the unwary to these pitfalls of clustering.
Amidst all this discussion of clustering, we must not lose sight of the fact that
often the clusters found will later be labeled (e.g., by resorting to a teacher or small
number of labeled samples), and that the clusters can then be used for classification.
In that case, the same similarity (or metric) should be used for classification as was
used for forming the clusters (Computer exercise 8).
10.7 Criterion Functions for Clustering
We have just considered the first major issue in clustering: how to measure “similar-
ity.” Now we turn to the second major issue: the criterion function to be optimized.
Suppose that we have a set D of n samples x1, . . . ,xn that we want to partition
into exactly c disjoint subsets D1, . . . ,Dc. Each subset is to represent a cluster, with
samples in the same cluster being somehow more similar than samples in different
clusters. One way to make this into a well-defined problem is to define a criterion
function that measures the clustering quality of any partition of the data. Then the
problem is one of finding the partition that extremizes the criterion function. In this
section we examine the characteristics of several basically similar criterion functions,
postponing until later the question of how to find an optimal partition.
10.7.1 The Sum-of-Squared-Error Criterion
The simplest and most widely used criterion function for clustering is the sum-of-
squared-error criterion. Let ni be the number of samples in Di and let mi be the
mean of those samples,
mi =
1
ni
∑
x∈Di
x. (48)
Then the sum-of-squared errors is defined by
Je =
c∑
i=1
∑
x∈Di
‖x−mi‖2. (49)
This criterion function has a simple interpretation: for a given cluster Di, the
mean vector mi is the best representative of the samples in Di in the sense that it
minimizes the sum of the squared lengths of the “error” vectors x−mi in Di. Thus,
Je measures the total squared error incurred in representing the n samples x1, . . . ,xn
by the c cluster centers m1, . . . ,mc. The value of Je depends on how the samples are
grouped into clusters and the number of clusters; the optimal partitioning is defined
as one that minimizes Je. Clusterings of this type are often called minimum variance minimum
variancepartitions.
What kind of clustering problems are well suited to a sum-of-squared-error crite-
rion? Basically, Je is an appropriate criterion when the clusters form compact clouds
that are rather well separated from one another. A less obvious problem arises when
there are great differences in the number of samples in different clusters. In that case
it can happen that a partition that splits a large cluster is favored over one that main-
tains the integrity of the natural clusters, as illustrated in Fig. 10.9. This situation
frequently arises because of the presence of “outliers” or “wild shots,” and brings up
30 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
the problem of interpreting and evaluating the results of clustering. Since little can
be said about that problem, we shall merely observe that if additional considerations
render the results of minimizing Je unsatisfactory, then these considerations should
be used, if possible, in formulating a better criterion function.
Je = large
Je = small
Figure 10.9: When two natural groupings have very different numbers of points, the
clusters minimizing a sum-squared-error criterion (Eq. 49) may not reveal the true
underlying structure. Here the criterion is smaller for the two clusters at the bottom
than at the more natural clustering at the top.
10.7.2 Related Minimum Variance Criteria
By some simple algebraic manipulation (Problem 19) we can eliminate the mean
vectors from the expression for Je and obtain the equivalent expression
Je =
1
2
c∑
i=1
nis̄i, (50)
where
s̄i =
1
n2
∑
x∈Di
∑
x′∈Di
‖x− x′‖2. (51)
Equation 51 leads us to interpret s̄i as the average squared distance between points in
the ith cluster, and emphasizes the fact that the sum-of-squared-error criterion uses
Euclidean distance as the measure of similarity. It also suggests an obvious way of
obtaining other criterion functions. For example, one can replace s̄i by the average,
the median, or perhaps the maximum distance between points in a cluster. More
generally, one can introduce an appropriate similarity function s(x,x′) and replace s̄i
by functions such as
s̄i =
1
n2i
∑
x∈Di
∑
x′∈Di
s(x,x′) (52)
or
s̄i = min
x,x′∈Di
s(x,x′). (53)
10.7. CRITERION FUNCTIONS FOR CLUSTERING 31
Table 10.1: Mean vectors and scatter matrices used in clustering criteria.
Depend on
cluster
center?
Yes No
Mean vector for
the ith cluster
× mi =
1
ni
∑
x∈Di
x (54)
Total mean vector × m =
1
n
∑
D
x =
1
n
c∑
i=1
nimi (55)
Scatter matrix for
the ith cluster
× Si =
∑
x∈Di
(x−mi)(x−mi)t (56)
Within-cluster
scatter matrix
× SW =
c∑
i=1
Si (57)
Between-cluster
scatter matrix
× SB =
c∑
i=1
ni(mi −m)(mi −m)t (58)
Total scatter matrix × ST =
∑
x∈D
(x−m)(x−m)t (59)
As in Chap. ??, we define an optimal partition as one that extremizes the crite-
rion function. This creates a well-defined problem, and the hope is that its solution
discloses the intrinsic structure of the data.
10.7.3 Scattering Criteria
The scatter matrices
Another interesting class of criterion functions can be derived from the scatter matri-
ces used in multiple discriminant analysis. The following definitions directly parallel
those given in Chapt. ??.
As before, it follows from these definitions that the total scatter matrix is the sum
of the within-cluster scatter matrix and the between-cluster scatter matrix:
ST = SW + SB . (60)
Note that the total scatter matrix does not depend on how the set of samples is par-
titioned into clusters; it depends only on the total set of samples. The within-cluster
and between-cluster scatter matrices taken separately do depend on the partitioning,
of course. Roughly speaking, there is an exchange between these two matrices, the
between-cluster scatter going up as the within-cluster scatter goes down. This is for-
tunate, since by trying to minimize the within-cluster scatter we will also tend to
maximize the between-cluster scatter.
To be more precise in talking about the amount of within-cluster or between-
cluster scatter, we need a scalar measure of the “size” of a scatter matrix. The two
measures that we shall consider are the trace and the determinant. In the univariate
case, these two measures are equivalent, and we can define an optimal partition as one
32 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
that minimizes SW or maximizes SB . In the multivariate case things are somewhat
more complicated, and a number of related but distinct optimality criteria have been
suggested.
The Trace Criterion
Perhaps the simplest scalar measure of a scatter matrix is its trace — the sum of its
diagonal elements. Roughly speaking, the trace measures the square of the scattering
radius, since it is proportional to the sum of the variances in the coordinate directions.
Thus, an obvious criterion function to minimize is the trace of SW . In fact, this
criterion is nothing more or less than the sum-of-squared-error criterion, since the
definitions of scatter matrices (Eqs. 56 & 57) yield
tr SW =
c∑
i=1
tr Si =
c∑
i=1
∑
x∈Di
‖x−mi‖2 = Je. (61)
Since trST = trSW+trSB and trST is independent of how the samples are partitioned,
we see that no new results are obtained by trying to maximize trSB . However, it is
comforting to know that in seeking to minimize the within-cluster criterion Je = trSW
we are also maximizing the between-cluster criterion
trSB =
c∑
i=1
ni‖mi −m‖2. (62)
The Determinant Criterion
In Sect. ?? we used the determinant of the scatter matrix to obtain a scalar measure
of scatter. Roughly speaking, the determinant measures the square of the scattering
volume, since it is proportional to the product of the variances in the directions of
the principal axes. Since SB will be singular if the number of clusters is less than or
equal to the dimensionality, |SB | is obviously a poor choice for a criterion function.
Furthermore, SB may become singular, and will certainly be so if n − c is less than
the dimensionality d (Problem 27). However, if we assume that SW is nonsingular,
we are led to consider the determinant criterion function
Jd = |SW | =
∣∣∣∣∣
c∑
i=1
Si
∣∣∣∣∣. (63)
The partition that minimizes Jd is often similar to the one that minimizes Je,
but the two need not be the same, as shown in Example 3. We observed before that
the minimum-squared-error partition might change if the axes are scaled, though this
does not happen with Jd (Problem 26). Thus Jd is to be favored under conditions
where there may be unknown or irrelevant linear transformations of the data.
Invariant Criteria
It is not particularly hard to show that the eigenvalues λ1, . . . , λd of S
−1
W SB are invari-
ant under nonsingular linear transformations of the data (Problem ??). Indeed, these
eigenvalues are the basic linear invariants of the scatter matrices. Their numerical
values measure the ratio of between-cluster to within-cluster scatter in the direction
of the eigenvectors, and partitions that yield large values are usually desirable. Of
10.7. CRITERION FUNCTIONS FOR CLUSTERING 33
course, as we pointed out in Sect. ??, the fact that the rank of SB can not exceed
c−1 means that no more than c−1 of these eigenvalues can be nonzero. Nevertheless,
good partitions are ones for which the nonzero eigenvalues are large.
One can invent a great variety of invariant clustering criteria by composing appro-
priate functions of these eigenvalues. Some of these follow naturally from standard
matrix operations. For example, since the trace of a matrix is the sum of its eigen-
values, one might elect to maximize the criterion function
trS−1W SB =
d∑
i=1
λi. (64)
By using the relation ST = SW + SB , one can derive the following invariant relatives
of [trSW and |SW | (Problem 25):
Jf = trS
−1
T SW =
d∑
i=1
1
1 + λi
(65)
and
|SW |
|ST |
=
d∏
i=1
1
1 + λi
. (66)
Since all of these criterion functions are invariant to linear transformations, the
same is true of the partitions that extremize them. In the special case of two clusters,
only one eigenvalue is nonzero, and all of these criteria yield the same clustering.
However, when the samples are partitioned into more than two clusters, the optimal
partitions, though often similar, need not be the same, as shown in Example 3.
Example 3: Clustering criteria
We can gain some intuition by considering these criteria applied to the following
data set.
sample x1 x2 sample x1 x2
1 -1.82 0.24 11 0.41 0.91
2 -0.38 -0.39 12 1.70 0.48
3 -0.13 0.16 13 0.92 -0.49
4 -1.17 0.44 14 2.41 0.32
5 -0.92 0.16 15 1.48 -0.23
6 -1.69 -0.01 16 -0.34 1.88
7 0.33 -0.17 17 0.83 0.23
8 -0.71 -0.21 18 0.62 0.81
9 1.27 -0.39 19 -1.42 -0.51
10 -0.16 -0.23 20 0.67 -0.55
All of the clusterings seem reasonable, and there is no strong argument to favor one
over the others. For the case c = 2, the clusters minimizing the Je indeed tend to favor
clusters of roughly equal numbers of points, as illustrated in Fig. 10.9; in contrast,
Jd favors one large and one fairly small cluster. Since the full data set happens to
be spread horizontally more than vertically, the eigenvalue in the horizontal direction
is greater than that in the vertical direction. As such, the clusters are “stretched”
34 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
c = 2 c = 3
Je
Jd
Jf
The clusters found by minimizing a criterion depends upon the criterion function
as well as the assumed number of clusters. The sum-of-squared-error criterion Je
(Eq. 49), the determinant criterion Jd (Eq. 63) and the more subtle trace criterion
Jf (Eq. 65) were applied to the 20 points in the table with the assumption of c = 2
and c = 3 clusters. (Each point in the table is shown, with bounding boxes defined
by −1.8 < x < 2.5 and −0.6 < y < 1.9.)
horizontally somewhat. In general, the differences between the cluster criteria become
less pronounced for large numbers of clusters. For the c = 3 case, for instance, the
clusters depend only mildly upon the cluster criterion — indeed, two of the clusterings
are identical.
With regard to the criterion function involving ST , note that ST does not depend
on how the samples are partitioned into clusters. Thus, the clusterings that minimize
|SW |/|ST | are exactly the same as the ones that minimize |SW |. If we rotate and scale
the axes so that ST becomes the identity matrix, we see that minimizing tr[S
−1
T SW ]
is equivalent to minimizing the sum-of-squared-error criterion trSW after performing
this normalization. Clearly, this criterion suffers from the very defects that we warned
about in Sect. ??, and it is probably the least desirable of these criteria.
One final warning about invariant criteria is in order. If different apparent clusters
can be obtained by scaling the axes or by applying any other linear transformation,
then all of these groupings will be exposed by invariant procedures. Thus, invariant
criterion functions are more likely to possess multiple local extrema, and are corre-
spondingly more difficult to optimize.
The variety of the criterion functions we have discussed and the somewhat subtle
differences between them should not be allowed to obscure their essential similarity. In
every case the underlying model is that the samples form c fairly well separated clouds
of points. The within-cluster scatter matrix SW is used to measure the compactness
of these clouds, and the basic goal is to find the most compact grouping. While this
approach has proved useful for many problems, it is not universally applicable. For
example, it will not extract a very dense cluster embedded in the center of a diffuse
cluster, or separate intertwined line-like clusters. For such cases one must devise other
10.8. *ITERATIVE OPTIMIZATION 35
criterion functions that are better matched to the structure present or being sought.
10.8 *Iterative Optimization
Once a criterion function has been selected, clustering becomes a well-defined problem
in discrete optimization: find those partitions of the set of samples that extremize the
criterion function. Since the sample set is finite, there are only a finite number of
possible partitions. Thus, in theory the clustering problem can always be solved
by exhaustive enumeration. However, the computational complexity renders such an
approach unthinkable for all but the simplest problems; there are approximately cn/c!
ways of partitioning a set of n elements into c subsets, and this exponential growth
with n is overwhelming (Problem 17). For example an exhaustive search for the best
set of 5 clusters in 100 samples would require considering more than 1067 partitionings.
Simply put, in most applications an exhaustive search is completely infeasible.
The approach most frequently used in seeking optimal partitions is iterative op-
timization. The basic idea is to find some reasonable initial partition and to “move”
samples from one group to another if such a move will improve the value of the cri-
terion function. Like hill-climbing procedures in general, these approaches guarantee
local but not global optimization. Different starting points can lead to different solu-
tions, and one never knows whether or not the best solution has been found. Despite
these limitations, the fact that the computational requirements are bearable makes
this approach attractive.
Let us consider the use of iterative improvement to minimize the sum-of-squared-
error criterion Je, written as
Je =
c∑
i=1
Ji, (67)
where an effective error per cluster is defined to be
Ji =
∑
x∈Di
‖x−mi‖2 (68)
and the mean of each cluster is, as before,
mi =
1
ni
∑
x∈Di
x. (48)
Suppose that a sample x̂ currently in cluster Di is tentatively moved to Dj . Then mj
changes to
m∗j = mj +
x̂−mj
nj + 1
(69)
and Jj increases to
J∗j =
∑
x∈Di
‖x−m∗j‖2 + ‖x̂−m∗j‖2
=
( ∑
x∈Di
‖x−mj −
x̂−mj
nj + 1
‖2
)
+ ‖ nj
nj + 1
(x̂−mj)‖2
36 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
= Jj +
nj
nj + 1
‖x̂−mj‖2. (70)
Under the assumption that ni �= 1 (singleton clusters should not be destroyed), a
similar calculation (Problem 29) shows that mi changes to
m∗i = m−
x̂−mi
ni − 1
(71)
and Ji decreases to
J∗i = Ji −
ni
ni − 1
‖x̂−mi‖2. (72)
These equations greatly simplify the computation of the change in the criterion
function. The transfer of x̂ from Di to Dj is advantageous if the decrease in Ji is
greater than the increase in Jj . This is the case if
ni
ni − 1
‖x̂−mi‖2 >
nj
nj + 1
‖x̂−mj‖2, (73)
which typically happens whenever x̂ is closer to mj than mi. If reassignment is
profitable, the greatest decrease in sum of squared error is obtained by selecting the
cluster for which nj/(nj + 1)‖x̂ − mj‖2 is minimum. This leads to the following
clustering procedure:
Algorithm 3 (Basic iterative minimum-squared-error clustering)
1 begin initialize n, c,m1,m2, . . . ,mc
2 do randomly select a sample x̂;
3 i← arg min
i′
‖mi′ − x̂‖ (classify x̂)
4 if ni �= 1 then compute
5 ρj =
{
nj
nj+1
‖x̂−mj‖2 j �= i
nj
nj−1‖x̂−mi‖
2 j = i
6 if ρk ≤ ρj for all j then transfer x̂ to Dk
7 recompute Je,mi,mk
8 until no change in Je in n attempts
9 return m1,m2, . . . ,mc
10 end
A moment’s consideration will show that this procedure is is essentially a sequen-
tial version of the k-means procedure (Algorithm 1) described in Sect. 10.4.3. Where
the k-means procedure waits until all n samples have been reclassified before updat-
ing, the Basic Iterative Minimum-Squared-Error procedure updates after each sample
is reclassified. It has been experimentally observed that this procedure is more suscep-
tible to being trapped in local minima, and it has the further disadvantage of making
the results depend on the order in which the candidates are selected. However, it is at
least a stepwise optimal procedure, and it can be easily modified to apply to problems
in which samples are acquired sequentially and clustering must be done on-line.
One question that plagues all hill-climbing procedures is the choice of the starting
point. Unfortunately, there is no simple, universally good solution to this problem.
One approach is to select c samples randomly for the initial cluster centers, using
them to partition the data on a minimum-distance basis. Repetition with different
random selections can give some indication of the sensitivity of the solution to the
10.9. HIERARCHICAL CLUSTERING 37
starting point. Yet another approach is to find the c-cluster starting point from the
solutions to the (c − a)-cluster problem. The solution for the one-cluster problem is
the total sample mean; the starting point for the c-cluster problem can be the final
means for the (c−a)-cluster problem plus the sample that is farthest from the nearest
cluster center. This approach leads us directly to the so-called hierarchical clustering
procedures, which are simple methods that can provide very good starting points for
iterative optimization.
10.9 Hierarchical Clustering
Up to now, our methods have formed disjoint clusters — in computer science terminol-
ogy, we would say that the data description is “flat.” However, there are many times
when clusters have subclusters, these have sub-subclusters, and so on. In biological
taxonomy, for instance, kingdoms are split into phylums, which are split into subphy-
lums, which are split into orders, and suborders, and families, and subfamilies, and
genus and species, and so on, all the way to a particular individual organism. Thus
we might have kingdom = animal, phylum = Chordata, subphylum = Vertebrata,
class = Osteichthyes, subclass = Actinopterygii, order = Salmoniformes, family =
Salmonidae, genus = Oncorhynchus, species = Oncorhynchus kisutch, and individual
= the particular Coho salmon caught in my net. Organisms that lie in the animal
kingdom — such as a salmon and a moose — share important attributes that are not
present in organisms in the plant kingdom, such as redwood trees. In fact, this kind of
hierarchical clustering permeates classifactory activities in the sciences. Thus we now
turn to clustering methods which will lead to representations that are “hierarchical,”
rather than flat.
10.9.1 Definitions
Let us consider a sequence of partitions of the n samples into c clusters. The first of
these is a partition into n clusters, each cluster containing exactly one sample. The
next is a partition into n− 1 clusters, the next a partition into n− 2, and so on until
the nth, in which all the samples form one cluster. We shall say that we are at level
k in the sequence when c = n− k + 1. Thus, level one corresponds to n clusters and
level n to one cluster. Given any two samples x and x′, at some level they will be
grouped together in the same cluster. If the sequence has the property that whenever
two samples are in the same cluster at level k they remain together at all higher levels,
then the sequence is said to be a hierarchical clustering.
The most natural representation of hierarchical clustering is a corresponding tree,
called a dendrogram, which shows how the samples are grouped. Figure 10.10 shows dendro-
grama dendrogram for a simple problem involving eight samples. Level 1 shows the eight
samples as singleton clusters. At level 2, samples x6 and x7 have been grouped to
form a cluster, and they stay together at all subsequent levels. If it is possible to
measure the similarity between clusters, then the dendrogram is usually drawn to
scale to show the similarity between the clusters that are grouped. In Fig. 10.10, for
example, the similarity between the two groups of samples that are merged at level 5
has a value of roughly 60.
We shall see shortly how such similarity values can be obtained, but first note that
the similarity values can be used to help determine whether groupings are natural or
forced. If the similarity values for the levels are roughly evenly distributed throughout
38 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
the range of possible values, then there is no principled argument that any particular
number of clusters is better or “more natural” than another. Conversely, suppose that
there is a unusually large gap between the similarity values for the levels corresponding
to c = 3 and to c = 4 clusters. In such a case, one can argue that c = 3 is the most
natural number of clusters (Problem 35).
0
10
20
30
40
50
60
70
80
90
100
x1
S
im
il
a
ri
ty
s
ca
le
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Level 7
x2 x3 x4 x5 x6 x7 x8
Level 8
Figure 10.10: A dendrogram can represent the results of hierarchical clustering algo-
rithms. The vertical axis shows a generalized measure of similarity among clusters.
Here, at level 1 all eight points lie in singleton clusters; each point in a cluster is
highly similar to itself, of course. Points x6 and x7 happen to be the most similar,
and are merged at level 2, and so forth.
Another representation for hierarchical clustering is based on sets, in which each
level of cluster may contain sets that are subclusters, as shown in Fig. 10.11. Yet an-
other, textual, representation uses brackets, such as: {{x1, {x2,x3}}, {{{x4,x5}, {x6,x7}},x8}}.
While such representations may reveal the hierarchical structure of the data, they do
not naturally represent the similarities quantitatively. For this reason dendrograms
are generally preferred.
x1
x2
x3
x4
x5
x6
x7
x8
8
7
6
5
3 4
2
Figure 10.11: A set or Venn diagram representation of two-dimensional data (which
was used in the dendrogram of Fig. 10.10) reveals the hierarchical structure but not
the quantitative distances between clusters. The levels are numbered in red.
Because of their conceptual simplicity, hierarchical clustering procedures are among
the best-known of unsupervised methods. The procedures themselves can be divided
according to two distinct approaches — agglomerative and divisive. AgglomerativeAgglomer-
ative (bottom-up, clumping) procedures start with n singleton clusters and form the se-
quence by successively merging clusters. Divisive (top-down, splitting) procedures
divisive start with all of the samples in one cluster and form the sequence by successively
splitting clusters. The computation needed to go from one level to another is usually
10.9. HIERARCHICAL CLUSTERING 39
simpler for the agglomerative procedures. However, when there are many samples
and one is interested in only a small number of clusters, this computation will have
to be repeated many times. For simplicity, we shall concentrate on agglomerative
procedures, and merely touch on some divisive methods in Sect. 10.12.
10.9.2 Agglomerative Hierarchical Clustering
The major steps in agglomerative clustering are contained in the following procedure,
where c is the desired number of final clusters:
Algorithm 4 (Agglomerative hierarchical clustering)
1 begin initialize c, ĉ← n,Di ← {xi}, i = 1, . . . , n
2 do ĉ← ĉ− 1
3 Find nearest clusters, say, Di and Dj
4 Merge Di and Dj
5 until c = ĉ
6 return c clusters
7 end
As described, this procedure terminates when the specified number of clusters has been
obtained and returns the clusters, described as set of points (rather than as mean or
representative vectors). If we continue until c = 1 we can produce a dendrogram like
that in Fig. 10.10. At any level the “distance” between nearest clusters can provide
the dissimilarity value for that level. Note that we have not said how to measure the
distance between two clusters, and hence how to find the “nearest” clusters, required
by line 3 of the Algorithm. The considerations here are much like those involved
in selecting a general clustering criterion function. For simplicity, we shall generally
restrict our attention to the following distance measures:
dmin(Di,Dj) = min
x∈Di
x′∈Dj
‖x− x′‖ (74)
dmax(Di,Dj) = max
x∈Di
x′∈Dj
‖x− x′‖ (75)
davg(Di,Dj) =
1
ninj
∑
x∈Di
∑
x′∈Dj
‖x− x′‖ (76)
dmean(Di,Dj) = ‖mi −mj‖. (77)
All of these measures have a minimum-variance flavor, and they usually yield the same
results if the clusters are compact and well separated. However, if the clusters are
close to one another, or if their shapes are not basically hyperspherical, quite different
results can be obtained. Below we shall illustrate some of the differences.
But first let us consider the computational complexity of a particularly simple
agglomerative clustering algorithm. Suppose we have n patterns in d-dimensional
space, and we seek to form c clusters using dmin(Di,Dj) defined in Eq. 74. We
will, once and for all, need to calculate n(n − 1) inter-point distances — each of
which is an O(d2) calculation — and place the results in an inter-point distance
table. The space complexity is, then, O(n2). Finding the minimum distance pair
(for the first merging) requires that we step through the complete list, keeping the
40 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
index of the smallest distance. Thus for the first agglomerative step, the complexity
is O(n(n − 1)(d2 + 1)) = O(n2d2). For an arbitrary agglomeration step (i.e., from ĉ
to ĉ − 1), we need merely step through the n(n − 1) − ĉ “unused” distances in the
list and find the smallest for which x and x′ lie in different clusters. This is, again,
O(n(n− 1)− ĉ). The full time complexity is thus O(cn2d2), and in typical conditions
n� c.∗
The Nearest-Neighbor Algorithm
When dmin is used to measure the distance between clusters (Eq. 74) the algorithm
is sometimes called the nearest-neighbor cluster algorithm, or minimum algorithmminimum
algorithm Moreover, if it is terminated when the distance between nearest clusters exceeds an
arbitrary threshold, it is called the single-linkage algorithm. Suppose that we think
single-
linkage
algorithm
of the data points as being nodes of a graph, with edges forming a path between the
nodes in the same subset Di. When dmin is used to measure the distance between
subsets, the nearest neighbor nodes determine the nearest subsets. The merging of
Di and Dj corresponds to adding an edge between the nearest pair of nodes in Di
and Dj . Since edges linking clusters always go between distinct clusters, the resulting
graph never has any closed loops or circuits; in the terminology of graph theory, this
procedure generates a tree. If it is allowed to continue until all of the subsets are
linked, the result is a spanning tree — a tree with a path from any node to any otherspanning
tree node. Moreover, it can be shown that the sum of the edge lengths of the resulting
tree will not exceed the sum of the edge lengths for any other spanning tree for that
set of samples (Problem 37). Thus, with the use of dmin as the distance measure, the
agglomerative clustering procedure becomes an algorithm for generating a minimal
spanning tree.
Figure 10.12 shows the results of applying this procedure to Gaussian data. In
both cases the procedure was stopped giving two large clusters (plus three singleton
outliers); a minimal spanning tree can be obtained by adding the shortest possible edge
between the two clusters. In the first case where the clusters are fairly well separated,
the obvious clusters are found. In the second case, the presence of a point located so
as to produce a bridge between the clusters results in a rather unexpected grouping
into one large, elongated cluster, and one small, compact cluster. This behavior is
often called the “chaining effect,” and is sometimes considered to be a defect of this
distance measure. To the extent that the results are very sensitive to noise or to slight
changes in position of the data points, this is certainly a valid criticism.
The Farthest-Neighbor Algorithm
When dmax (Eq. 75) is used to measure the distance between subsets, the algorithm is
sometimes called the farthest-neighbor clustering algorithm, or maximum algorithm.maximum
algorithm If it is terminated when the distance between nearest clusters exceeds an arbitrary
threshold, it is called the complete-linkage algorithm. The farthest-neighbor algorithm
complete-
linkage
algorithm
discourages the growth of elongated clusters. Application of the procedure can be
thought of as producing a graph in which edges connect all of the nodes in a cluster.
In the terminology of graph theory, every cluster constitutes a complete subgraph.
complete
subgraph
The distance between two clusters is determined by the most distant nodes in the two
∗ There are methods for sorting or arranging the entries in the inter-point distance table so as
to easily avoid inspection of points in the same cluster, but these typically do not improve the
complexity results significantly.
10.9. HIERARCHICAL CLUSTERING 41
Figure 10.12: Two Gaussians were used to generate two-dimensional samples, shown
in pink and black. The nearest-neighbor clustering algorithm gives two clusters that
well approximate the generating Gaussians (left). If, however, another particular
sample is generated (red point at the right) and the procedure re-started, the clusters
do not well approximate the Gaussians. This illustrates how the algorithm is sensitive
to the details of the samples.
clusters. When the nearest clusters are merged, the graph is changed by adding edges
between every pair of nodes in the two clusters.
If we define the diameter of a partition as the largest diameter for clusters in
the partition, then each iteration increases the diameter of the partition as little
as possible. As Fig. 10.13 illustrates, this is advantageous when the true clusters
are compact and roughly equal in size. Nevertheless, when this is not the case — as
happens with the two elongated clusters — the resulting groupings can be meaningless.
This is another example of imposing structure on data rather than finding structure
in it.
Compromises
The minimum and maximum measures represent two extremes in measuring the dis-
tance between clusters. Like all procedures that involve minima or maxima, they
tend to be overly sensitive to “outliers” or “wildshots.” The use of averaging is an
obvious way to ameliorate these problems, and davg and dmean (Eqs. 76 & 77) are
natural compromises between dmin and dmax. Computationally, dmean is the simplest
of all of these measures, since the others require computing all ninj pairs of distances
‖x− x′‖. However, a measure such as davg can be used when the distances ‖x− x′‖
are replaced by similarity measures, where the similarity between mean vectors may
be difficult or impossible to define.
10.9.3 Stepwise-Optimal Hierarchical Clustering
We observed earlier that if clusters are grown by merging the nearest pair of clus-
ters, then the results have a minimum variance flavor. However, when the measure
42 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
dmax = large dmax = small
Figure 10.13: The farthest-neighbor clustering algorithm uses the separation between
the most distant points as a criterion for cluster membership. If this distance is set
very large, then all points lie in the same cluster. In the case shown at the left, a
fairly large dmax leads to three clusters; a smaller dmax gives four clusters, as shown
at the right.
of distance between clusters is chosen arbitrarily, one can rarely assert that the re-
sulting partition extremizes any particular criterion function. In effect, hierarchical
clustering defines a cluster as whatever results from applying the clustering procedure.
Nevertheless, with a simple modification it is possible to obtain a stepwise-optimal
procedure for extremizing a criterion function. This is done merely by replacing line 3
of the Basic Iterative Agglomerative Clustering Procedure (Algorithm 4) by a more
general form to get:
Algorithm 5 (Stepwise optimal hierarchical clustering)
1 begin initialize c, ĉ← n,Di ← {xi}, i = 1, . . . , n
2 do ĉ← ĉ− 1
3 Find clusters whose merger changes the criterion the least, say, Di and Dj
4 Merge Di and Dj
5 until c = ĉ
6 return c clusters
7 end
We saw earlier that the use of dmax causes the smallest possible stepwise increase
in the diameter of the partition. Another simple example is provided by the sum-
of-squared-error criterion function Je. By an analysis very similar to that used in
Sect. ??, we find that the pair of clusters whose merger increases Je as little as
possible is the pair for which the “distance”
de(Di,Dj) =
√
ninj
ni + nj
‖mi −mj‖ (78)
is minimum (Problem 34). Thus, in selecting clusters to be merged, this criterion takes
into account the number of samples in each cluster as well as the distance between
clusters. In general, the use of de tends to favor growth by merging singletons or
small clusters with large clusters over merging medium-sized clusters. While the final
partition may not minimize Je, it usually provides a very good starting point for
further iterative optimization.
10.10. *THE PROBLEM OF VALIDITY 43
10.9.4 Hierarchical Clustering and Induced Metrics
Suppose that we are unable to supply a metric for our data, but that we can measure a
dissimilarity value δ(x,x′) for every pair of samples, where δ(x,x′) ≥ 0, with equality dissimil-
arityholding if and only if x = x′. Then agglomerative clustering can still be used, with the
understanding that the nearest pair of clusters is the least dissimilar pair. Interestingly
enough, if we define the dissimilarity between two clusters by
δmin(Di,Dj) = min
x∈Di
x′∈Dj
δ(x,x′) (79)
or
δmax(Di,Dj) = max
x∈Di
x′∈Dj
δ(x,x′) (80)
then the hierarchical clustering procedure will induce a distance function for the given
set of n samples. Furthermore, the ranking of the distances between samples will be
invariant to any monotonic transformation of the dissimilarity values (Problem 18).
We can now define a generalized distance d(x,x′) between x and x′ as the value
of the lowest level clustering for which x and x′ are in the same cluster. To show that
this is a legitimate distance function, or metric, we need to show four things: for all metric
vectors x, x′ and x′′
non-negativity: d(x,x′) ≥ 0
reflexivity: d(x,x′) = 0 if and only if x = x′
symmetry: d(x,x′) = d(x′,x)
triangle inequality: d(x,x′) + d(x′,x′′) ≥ d(x,x′′).
It is easy to see that these requirements are satisfied and hence that dissimilarity can
induce a metric. For our formula for dissimilarity, we have moreover that
d(x,x′′) ≤ max[d(x,x′), d(x′,x′′)] for any x′ (81)
in which case we say that d(·, ·) is an ultrametric (Problem 31). Ultrametric criteria ultra-
metriccan be more immune to local minima problems since stricter ordering of distances
among clusters is maintained.
10.10 *The Problem of Validity
With almost all of the procedures considered thus far we have assumed that the num-
ber of clusters is known. That is a reasonable assumption if we are upgrading a
classifier that has been designed on a small labeled set, or if we are tracking slowly
time-varying patterns. However, it may be an unjustified assumption if we are ex-
ploring a data set whose properties are, at base, unknown. Thus, a recurring problem
in cluster analysis is that of deciding just how many clusters are present.
When clustering is done by extremizing a criterion function, a common approach
is to repeat the clustering procedure for c = 1, c = 2, c = 3, etc., and to see how the
criterion function changes with c. For example, it is clear that the sum-of-squared-
error criterion Je must decrease monotonically with c, since the squared error can
44 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
be reduced each time c is increased merely by transferring a single sample to a new
singleton cluster. If the n samples are really grouped into ĉ compact, well separated
clusters, one would expect to see Je decrease rapidly until ĉ = c, decreasing much
more slowly thereafter until it reaches zero at c = n. Similar arguments have been
advanced for hierarchical clustering procedures and can be apparent in a dendrogram,
the usual assumption being that a large disparity in the levels at which clusters merge
indicates the presence of natural groupings.
A more formal approach to this problem is to devise some measure of goodness of
fit that expresses how well a given c-cluster description matches the data. The chi-
squared and Kolmogorov-Smirnov statistics are the traditional measures of goodness
of fit, but the curse of dimensionality usually demands the use of simpler measures,
some criterion function, which we denote J(c). Since we expect a description in terms
of c+1 clusters to give a better fit than a description in terms of c clusters, we would
like to know what constitutes a statistically significant improvement in J(c).
A formal way to proceed is to advance the null hypothesis that there are exactly
c clusters present, and to compute the sampling distribution for J(c + 1) under this
hypothesis. This distribution tells us what kind of apparent improvement to expect
when a c-cluster description is actually correct. The decision procedure would be
to accept the null hypothesis if the observed value of J(c + 1) falls within limits
corresponding to an acceptable probability of false rejection.
Unfortunately, it is usually very difficult to do anything more than crudely esti-
mate the sampling distribution of J(c + 1). The resulting solutions are not above
suspicion, and the statistical problem of testing cluster validity is still essentially un-
solved. However, under the assumption that a suspicious test is better than none,
we include the following approximate analysis for the simple sum-of-squared-error
criterion which closely parallels our discussion in Chap. ??.
Suppose that we have a set D of n samples and we want to decide whether or not
there is any justification for assuming that they form more than one cluster. Let us
advance the null hypothesis that all n samples come from a normal population with
mean µ and covariance matrix σ2I.∗ If this hypothesis were true, multiple clusters
found would have to have been formed by chance, and any observed decrease in the
sum-of-squared error obtained by clustering would have no significance.
The sum of squared error Je(1) is a random variable, since it depends on the
particular set of samples:
Je(1) =
∑
x∈D
‖x−m‖2, (82)
where m is the sample mean of the full data set. Under the null hypothesis, the
distribution for Je(1) is approximately normal with mean ndσ2 and variance 2ndσ4
(Problem 38). Suppose now that we partition the set of samples into two subsets D1
and D2 so as to minimize Je(2), where
Je(2) =
2∑
i=1
∑
x∈Di
‖x−mi‖2, (83)
mi being the mean of the samples in Di. Under the null hypothesis, this partitioning
is spurious, but it nevertheless results in a value for Je(2) that is smaller than Je(1).
∗ We could of course assume a different cluster form, but in the absence of further information, the
Gaussian can be justified on the grounds we have discussed before.
10.11. COMPETITIVE LEARNING 45
If we knew the sampling distribution for Je(2), we could determine how small Je(2)
would have to be before we were forced to abandon a one-cluster null hypothesis.
Lacking an analytical solution for the optimal partitioning, we cannot derive an exact
solution for the sampling distribution. However, we can obtain a rough estimate by
considering the suboptimal partition provided by a hyperplane through the sample
mean. For large n, it can be shown that the sum of squared error for this partition is
approximately normal with mean n(d− 2/π)σ2 and variance 2n(d− 8/π2)σ4.
This result agrees with out statement that Je(2) is smaller than Je(1), since the
mean of Je(2) for the suboptimal partition — n(d− 2/π)σ2 — is less than the mean
for Je(1) — ndσ2. To be considered significant, the reduction in the sum-of-squared
error must certainly be greater than this. We can obtain an approximate critical value
for Je(2) by assuming that the suboptimal partition is nearly optimal, by using the
normal approximation for the sampling distribution, and by estimating σ2 according
to
σ̂2 =
1
nd
∑
x∈D
‖x−m‖2 = 1
nd
Je(1). (84)
The final result can be stated as follows (Problem 39): Reject the null hypothesis at
the p-percent significance level if
Je(2)
Je(1)
< 1− 2 πd − α √ 2(1− 8/π2d) nd , (85) where α is determined by p = 100 ∞∫ α 1 2π e−u 2/2 du = 100(1− erf(α)), (86) and erf(·) is the standard error function. This provides us with a test for deciding error functionwhether or not the splitting of a cluster is justified. Clearly the c-cluster problem can be treated by applying the same test to all clusters found. 10.11 Competitive Learning A clustering algorithm related to decision-directed versions of k-means (Algorithm 1) is based on neural network learning rules (Chap. ??) and called competitive learning. In both procedures, the number of desired clusters and their centers are initialized, and during clustering each pattern is provisionally classified into one of the clusters. The methods of updating the cluster centers differ, however. In the decision-directed method, each cluster center is calculated as the mean of the current provisional mem- bers. In competitive learning, the adjustment is confined to the single cluster center most similar to the pattern presented. As a result, in competitive learning clus- ters that are “far away” from the current pattern tend not to be altered (but see Sect. 10.11.2) — sometimes considered a desirable property. The drawback is that the solution need not minimize a single global cost or criterion function. We now turn to the specific competitive learning algorithm. For reasons that will become clear, each d-dimensional pattern is augmented (with x0 = 1) and normalized to have length ‖x‖ = 1; thus all patterns lie on the surface of a d-dimensional sphere. 46 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING ω1 ω2 ωc clusters x0 ... ... x2x1 xd normalized input w1 wcw2 - - - - - - ... Figure 10.14: The two-layer network which implements the competitive learning al- gorithm consists of d + 1 input units and c output or cluster units. Each augmented input pattern is normalized to unit length, i.e., ‖x‖ = 1, as is the set of weights at each cluster unit. When a pattern is presented, each of the cluster units computes its net activation netj = wtjx; only the weights at the most active cluster unit are modified. (The suppression of activity in all but the most active cluster units can be implemented by competition among these units, as indicated by the red arrows.) The weights of the most active unit are then modified to be more similar to the pattern presented. The competitive learning algorithm can be understood by its neural network imple- mentation (Fig. 10.14), which resembles a Perceptron network (Chapt. ??, Fig. ??), with input units fully connected to c output or cluster units. Each of the c cluster centers is initialized with a randomly chosen weight vector, also normalized ‖wj‖ = 1, j = 1, . . . c. It is traditional but not required to initialize cluster centers to be c points randomly selected from the data. When a new pattern is presented, each of the cluster units computes its net activation, netj = wtjx. Only the most active neuron (i.e., the closest to the new pattern) is permitted to update its weights. While this selection of the most active unit is algorithmically trivial, it can be implemented in a winner-take-all network, where each cluster unit j inhibits others by an amount proportional to netj , as shown by the red arrows in Fig. 10.14. It is this competition between cluster units, and the rsulting suppression of activity in all but the one with the largest net that gives the algorithm its name. Learning is confined to the weights at the most active unit. The weight vector at this unit is updated to be more like the pattern: w(t + 1) = w(t) + ηx, (87) where η is a learning rate. The weights are then normalized to insure d∑ i=0 w2i = 1. This normalization is needed to keep the classification and clustering based on the position in feature space rather than overall magnitude of w. Without such weight normalization, a single weight, say wj′ , could grow in magnitude and forever give the greatest value netj′ , and through competition thereby prevent other clusters from learning. Figure 10.15 shows the trajectories of three cluster centers in response to a sequence of patterns chosen randomly from the set shown. 10.11. COMPETITIVE LEARNING 47 Algorithm 6 (Competitive learning) 1 begin initialize η, n, c,w1,w2, . . . ,wc 2 xi ← {1,xi} i = 1, . . . n augment all patterns 3 xi ← xi/‖xi‖ i = 1, . . . n normalize all patterns 4 do randomly select a pattern x 5 j ← arg max j′ wtj′x classify x 6 wj ← wj + ηx weight update 7 wj ← wj/‖wj‖ weight normalization 8 until no significant change in w in n attempts 9 return w1,w2, . . . ,wc 10 end x1, w1 x2, w2 x3, w3 Figure 10.15: All of the three-dimensional patterns have been normalized ( 3∑ i=1 x2i = 1), and hence lie on a two-dimensional sphere. Likewise, the weights of the three cluster centers have been normalized. The red curves show the trajectory of the weight vectors; at the end of learning, each lies near the center of a cluster. A drawback of Algorithm 6 is that there is no guarantee that it will terminate, even for a finite, non-pathological data set — the condition in line 8 may never be satisfied and thus the weights may vary forever. A simple heuristic is to decay the learning rate in line 6 , for instance by η(t) = η(0)αt for α < 1 where t is an iteration number. If the initial cluster centers are representative of the full data set, and the rate of decay is set so that the full data set is presented at least several times before the learning is reduced to very small values, then good results can be expected. However if then a novel pattern is added, it cannot be learning, since η is too small. Likewise, such a learning decay scheme is inappropriate if we seek to track gradual changes in the data. In a non-stationary environment, a we may want a clustering algorithm to be stable to prevent ceaseless recoding, and yet plastic, or changeable, in response to a new pattern. (Freezing cluster centers would prevent recoding, but would not per- mit learning of new patterns.) This tradeoff has been called the stability-plasticity dilemma, and we shall see in Sect. 10.11.2 how it can be addressed. First, however, stability- plasticitywe turn to the problem of unknown number of clusters. 48 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING 10.11.1 Unknown number of clusters We have mentioned the problem of unknown number of cluster centers. When the number is unknown, we can proceed in one of two general ways. In the first, we compare some cluster criterion as a function of the number of clusters. If there is a large gap in the criterion values, it suggests a “natural” number of clusters. A second approach is to state a threshold for the creation of a new cluster. This is useful in on-line cases. The drawback is that it depends more strongly on the order of data presentation. Whereas clustering algorithms such as k-means and hierarchical clustering typi- cally have all data present before clustering begins (i.e., are off-line), there are occa- sionally situations in which clustering must be performed on-line as the data streams in, for instance when there is inadequate memory to store all the patterns themselves, or in a time-critical situation where the clusters need to be used even before the full data is present. Our graph theoretic methods can be performed on-line — one merely links the new pattern to an existing cluster based on some similarity measure. In order to make on-line versions of methods such as k-means, we will have to be a bit more careful. Under these conditions, the best approach generally is to represent clusters by their “centers” (e.g., means) and update these centers based solely on its current value and the incoming pattern. Here we shall assume that the number of clusters is known, and return in Sect. ?? to the case where it is not known. Suppose we currently have c cluster centers; they may have been placed initially at random positions, or as the first c patterns presented, or the current state after any number of patterns have been presented. The simplest approach is to alter only the cluster center most similar to a new pattern being presented, and the cluster center is changed to be somewhat more like the pattern (Fig. 10.16). Figure 10.16: In leader-follower clustering, the number of clusters and their centers depend upon the random sequence of data presentations. The three simulations shown employed the same learning rate η, threshold θ, and number of presentations of each point (50), but differ in the random sequence of presentations. Notice that in the simulation on the left, three clusters are generated whereas in the other simulations, only two. If we let wi represent the current center for cluster i, η a learning rate and introduce a threshold θ, a relative of the Basic leader-follower clustering algorithm is then: Algorithm 7 (Basic leader-follower clustering) 1 begin initialize η, θ ← threshold 10.11. COMPETITIVE LEARNING 49 Figure 10.17: Instability can arise when a pattern is assigned different cluster mem- berships at different times. Early in clustering the pattern marked x∗ lies in the black cluster, while later in clustering it lies in the red cluster. Similar pattern presentations can make x∗ alternate arbitrarily between clusters. 2 µ1 ← x 3 do accept new x 4 j ← arg min j′ ‖x− µj′‖ (find nearest cluster) 5 if ‖x− µj‖ < θ 6 then µj ← µj + ηx 7 else add new µ← x 8 µ← µ/‖µ‖ (normalize weight) 9 until no more patterns 10 return µ1,µ2, . . . 11 end Before we analyze some drawbacks of such a leader-follower clustering algorithm, let us consider one popular neural technique for achieving it. 10.11.2 Adaptive Resonance The simplest adaptive resonance networks (or Adaptive Resonance Theory or ART networks) perform a modification of the On-line clustering with cluster creation pro- cedure we have just seen. While the primary motivation for ART was to explain biological learning, we shall not be concerned here with their biological relevance nor with their use in supervised learning (but see Problem 41). The above algorithm, however, can occasionally present a problem, regardless of whether it is implemented via competitive learning. Consider a cluster w1 that originally codes a particular pattern x0, i.e., if x0 is presented, the output node having weights w1 is most activated. Suppose a “hostile” sequence of patterns is presented, i.e., one that sweeps the cluster centers in unusual ways (Fig. 10.17). It is possible that after the cluster centers have been swept, that x0 is coded by w2. Indeed, a particularly devious sequence can lead x0 to be coded by an arbitrary sequence of cluster centers, with any cluster center being active an arbitrary number of times. The network works as follows. First a pattern is presented to the input units. This leads via bottom-up connections wij to activations in the output units. A winner- 50 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING Figure 10.18: Adaptive Resonance network (ART1 for binary patterns). Weights are bidirectional, gain, the orienting system controls the , and hence (indirectly) the number of clusters found. take-all computation leads to only the most activated output unit being active — all other output units are suppressed. Activation is then sent back to the input units via weights wji. This leads, in turn to a modification of the activation of the input units. Very quickly, a stable configuration of output and input units occurs, called a “resonance”(though this has nothing to do with the type of resonance in a driven oscillator). ART networks detect novelty by means of the orienting subsystem. The details need not concern us here, but in broad overview, the orienting subsystem has two inputs: the total number of active input features and the total number of features that are active in the input layer. (Note that these two numbers need not be the same, since the top-down feedback affects the activation of the input units, but not the number of active inputs themselves.) If an input pattern is “too different” from any current cluster centers, then the orienting subsystem sends a reset wave signal that renders the active output unit quiet. This allows a new cluster center to be found, or if all have been explored, then a new cluster center is created. The criterion for “too different” is a single number, set by the user, called the vigilance, ρ(0 ≤ ρ ≤ 1. Denoting the number of active input features as |I| and thevigilance number active in the input layer during a resonance as |R|, then there will be a reset if |R| |I| < ρ, (88) where rho is a user-set number called the vigilance parameter. A low vigilance pa-vigilance parameter rameter means that there can be a poor “match” between the input and the learned cluster and the network will accept it. (Thus vigilance and the ratio of the number of features used by ART, while motivated by proportional considerations, is just one of an infinite number of possible closeness criteria (related to δ). For the same data set, a low vigilance leads to a small number of large coarse clusters being formed, while a high vigilance leads to a large number of fine clusters (Fig. 10.19). We have presented the basic approach and issues with ART1, but these return (though in a more subtle way) in analog versions of ART in the literature. 10.12. *GRAPH THEORETIC METHODS 51 Figure 10.19: The results of ART1 applied to a sequence of binary figures. a) ρ = xx. b) ρ = 0.xx. 10.12 *Graph Theoretic Methods Where the mathematics of normal mixtures and minimum-variance partitions leads us to picture clusters as isolated clumps, the language and concepts of graph theory lead us to consider much more intricate structures. Unfortunately, there is no uniform way of posing clustering problems as problems in graph theory. Thus, the effective use of these ideas is still largely an art, and the reader who wants to explore the possibilities should be prepared to be creative. We begin our brief look into graph-theoretic methods by reconsidering the simple procedures that produce the graphs shown in Fig. 10.6. Here a threshold distance d0 was selected, and two points are placed in the same cluster if the distance between them is less than d0. This procedure can easily be generalized to apply to arbitrary similarity measures. Suppose that we pick a threshold value s0 and say that xi is similar to xj if s(xi,xj) > s0. This defines an n-by-n similarity matrix S = [sij ], with similarity
matrixbinary component
sij =
{
1 if s(xi,xj) > s0
0 otherwise.
(89)
Furthermore, this matrix induces a similarity graph, dual to S, in which nodes corre- similarity
graphspond to points and an edge joins node i and node j if and only if sij = 1.
The clusterings produced by the single-linkage algorithm and by a modified version
of the complete-linkage algorithm are readily described in terms of this graph. With
the single-linkage algorithm, two samples x and x′ are in the same cluster if and only
if there exists a chain x,x1,x2, . . . ,xk,x′ such that x is similar to x1, x1 is similar to
x2, and so on for the whole chain. Thus, this clustering corresponds to the connected
components of the similarity graph. With the complete-linkage algorithm, all samples connected
componentin a given cluster must be similar to one another, and no sample can be in more than
one cluster. If we drop this second requirement, then this clustering corresponds to
the maximal complete subgraphs of the similarity graph — the “largest” subgraphs maximal
complete
subgraph
with edges joining all pairs of nodes. (In general, the clusters of the complete-linkage
algorithm will be found among the maximal complete subgraphs, but they cannot be
determined without knowing the unquantized similarity values.)
52 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
In the preceding section we noted that the nearest-neighbor algorithm could be
viewed as an algorithm for finding a minimal spanning tree. Conversely, given a
minimal spanning tree we can find the clusterings produced by the nearest-neighbor
algorithm. Removal of the longest edge produces the two-cluster grouping, removal of
the next longest edge produces the three-cluster grouping, and so on. This amounts
to a divisive hierarchical procedure, and suggests other ways of dividing the graph
into subgraphs. For example, in selecting an edge to remove, we can compare its
length to the lengths of other edges incident upon its nodes. Let us say that an edge
is inconsistent if its length l is significantly larger than l̄, the average length of allinconsis-
tent edge other edges incident on its nodes. Figure 10.20 shows a minimal spanning tree for a
two-dimensional point set and the clusters obtained by systematically removing all
edges for which l > 2l̄ in this way. This criterion is sensitive to local conditions gives
results that are quite different from merely removing the two longest edges.
Figure 10.20: The removal of inconsistent edges — ones with length significantly larger
than the average incident upon a node — may yield natural clusters. The original
data is shown at the left and its minimal spanning tree is shown in the middle. At
virtually every node, incident edges are of nearly the same length. Each of the two
nodes shown in red are exceptions: their incident edges are of very different lengths.
When the two such inconsistent edges are removed, three clusters are produced, as
shown at the right.
When the data points are strung out into long chains, a minimal spanning tree
forms a natural skeleton for the chain. If we define the diameter path as the longestdiameter
path path through the tree, then a chain will be characterized by the shallow depth of
the branching off the diameter path. In contrast, for a large, uniform cloud of data
points, the tree will usually not have an obvious diameter path, but rather several
distinct, near-diameter paths. For any of these, an appreciable number of nodes will
be off the path. While slight changes in the locations of the data points can cause
major rerouting of a minimal spanning tree, they typically have little effect on such
statistics.
One of the useful statistics that can be obtained from a minimal spanning tree
is the edge length distribution. Figure 10.21 shows a situation in which two dense
clusters are embedded in a sparse set of points; the lengths of the edges of the min-
imal spanning tree exhibit two distinct clusters which would easily be detected by a
minimum-variance procedure. By deleting all edges longer than some intermediate
value, we can extract the dense cluster as the largest connected component of the
remaining graph. While more complicated configurations can not be disposed of this
easily, the flexibility of the graph-theoretic approach suggests that it is applicable to
a wide variety of clustering problems.
10.13. COMPONENT ANALYSIS 53
1
2
3
4
5
6
7
8
9
length
n
u
m
b
er
1
2
3
4
5
6
7
8
9
length
n
u
m
b
er
Figure 10.21: A minimal spanning tree is shown at the left; its bimodal edge length
distribution is evident in the histogram below. If all links of intermediate or high
length are removed (red), the two natural clusters are revealed (right).
10.13 Component analysis
Component analysis is an unsupervised approach to finding the “right” features from
the data. We shall discuss two leading methods, each having a somewhat different
goal. In principal component analysis (PCA), we seek to represent the d-dimensional
data in a lower-dimensional space. This will reduce the degrees of freedom, reduce
the space and time complexities. The goal is to represent data in a space that best
describes the variation in a sum-squared error sense, as we shall see. In independent
component analysis (ICA) we seek those directions that show the independence of
signals. This method is particularly helpful for segmenting signals from multiple
sources. As with standard clustering methods, it helps greatly if we know how many
independent components exist ahead of time.
10.13.1 Principal component analysis (PCA)
The basic approach in principal componements or Karhunen-Loéve transform is con- Karhunen-
Loéve
transform
ceptually quite simple. First, the d-dimensional mean vector µ and d× d covariance
matrix Σ are computed for the full data set. Next, the eigenvectors and eigenvalues
are computed (cf. Appendix ??), and sorted according to decreasing eigenvalue. Call
these eigenvectors e1 with eigenvalue λ1, e2 with eigenvalue λ2, and so on. Next, the
largest k such eigenvectors are chosen. In practice, this is done by looking at a spec-
trum of eigenvectors. Often there will be xxx implying an inherent dimensionality of
the subspace governing the “signal.” The other dimensions are noise. Form a k × k
matrix A whose columns consist of the k eigenvectors.
Preprocess data according to:
x′ = At(x− µ). (90)
It can be shown that this representation minimizes a squared error criterion (Prob-
lem 42).
54 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
10.13.2 Non-linear component analysis
We have just seen how to find a k-dimensional linear subspace of feature space that
best represents the full data according to a minimum-square-error sense. If the data
set is not well described by a sample mean and covariance matrix, but instead in-
volves complicated interactions of features, then the linear subspace may be a poor
representation. In such a case a non-linear component may be needed.
A neural network approach to such non-linear component analysis employs a net-
work with five layers of units, as shown in Fig. 10.22. The middle layer consists of
k < d linear units, and it is here that the non-linear components will be revealed.
It is important that the two other internal layers have nonlinear units (Problem 44).
The entire network is trained using the techniques of Chapt. ?? as an auto-encoderauto-
encoder or auto-associator. That is, each d-dimensional pattern is presented as input and as
the target or desired output. When trained on a sum-squared error criterion, such a
network readily learns the auto-encoder problem.
The top two layers of the trained network are discarded, and the rest used for
non-linear components. For each input pattern x, the output of the k units of the
three-layer network correspond to the non-linear components.
output
nonlinear
input
nonlinear
x
1
x
2
x
d
x
1
x
2
x
d
x
1
x
2
x
d
Γ(F
2
)
x
F
2
F1
F1
1 k
Figure 10.22: A five-layer neural network with two layers of non-linear units (e.g.,
sigmoidal), trained to be an auto-encoder, develops an internal representation that
corresponds to the non-linear principal components of the full data set. (Bias units
are not shown.) The process can be viewed in feature space (at the right). The
transformation F1 is a non-linear projection onto a k-dimensional non-linear subspace
denoted Γ(F2). Points in Γ(F2) are mapped via F2 back to the the d-dimensional
space of the original data.
We can understand the function of the full five-layer network in terms of two succes-
sive mappings, F1 is a projection from the d-dimensional input onto a k-dimensional
nonlinear subspace, followed by F2, a mapping from that subspace back to the full
d-dimensional space, as shown in the right of the figure.
Learning in the original network is highly nonlinear, and during training care
must be taken so as to avoid a poor local minimum (Chap. ??). Naturally, one
must take care to set an appropriate number k of units. Recall that in (linear)
principal component analysis, the number of components k could be chosen based
on the spectrum of eigenvectors. If the eigenvalues are ordered by magnitude, any
significant drop between successive values indicates a “natural” number dimension
to the subspace. Likewise, suppose five-layer networks are trained, with different
numbers k of units in the middle layer. Assuming poor local minima have been
10.13. COMPONENT ANALYSIS 55
avoided, the training error will surely decrease for successively larger values of k. If
the improvement k + 1 over k is small, this may indicate that k is the “natural”
dimension of the nonlinear subspace.
We should not conclude that principal component analysis is always beneficial
for classification. If the noise is large compared to the difference between categories,
then component analysis will find the directions of the noise, rather than the signal,
as illustrated in Fig. 10.23. In such cases, we seek to ignore the noise, and instead
extract the directions that are indicative of the categories — a technique we consider
next.
z1
z2
ω2
ω1
x1
x2
Figure 10.23: Features from two classes are as shown, along with nonlinear compo-
nents of the full data set. Apparently, these classes are well separated along the y2
direction, but the large noise gives the largest nonlinear component to be y1. Prepro-
cessing by keeping merely the largest nonlinear component would retain the “noise”
and discard the “signal,” giving poor recognition. The same defect arises in linear
principal components, where the compoenents are linear and everywhere perpendic-
ular.
10.13.3 *Independent component analysis (ICA)
Suppose there are c independent scalar source signals xi(t) for i = 1, . . . c where we
can consider t to be a time index 1 ≤ t ≤ T . For notational convenience we group the
c values at an instant into a vector x(t) and assume, further, that the vector has zero
mean. Because of our independence assumption, and an assumption of no noise, we
the multivariate density function can be written as
p(x(t)) =
c∏
i=1
p(xi(t)). (91)
Suppose that a d-dimensional data (or sensor) vector is observed at each moment,
y(t) = Ax(t), (92)
where A is a c× d scalar matrix, and below we shall require d ≥ c.
The method is perhaps best illustrated in its most typical use. Suppose there are
c sound sources being sensed by d microphones, all in a room. Each microphone gets
a mixture of the sources, with amplitudes depending upon the distances (Fig. 10.24).
(We shall ignore any effects of delays.)
56 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
2 4 6 8 10
- 1
- 0.5
0.5
1
2 4 6 8 10
- 1
- 0.5
0.5
1
2 4 6 8 10
- 1
- 0.5
0.5
1
2 4 6 8 10
- 1
- 0.5
0.5
1
2 4 6 8 10
-1
-0.5
0.5
1
x
1
(t)
x
2
(t)
t
t
2 4 6 8 10
- 1
- 0.5
0.5
1
2 4 6 8 10
-1
-0.5
0.5
1
t
t
t
t
t
s
1
(t) = a
11
x
1
(t) + a
12
x
2
(t)
s
2
(t) = a
21
x
1
(t) + a
22
x
2
(t)
s
3
(t) = a
31
x
1
(t) + a
32
x
2
(t)
A
W
x
1
(t) = w
11
s
1
(t) + w
12
s
2
(t)+ w
13
s
3
(t)^
x
2
(t) = w
21
s
1
(t) + w
22
s
2
(t)+ w
23
s
3
(t)^
Figure 10.24: Independent component analysis (ICA) is an unsupervised method that
can be applied to the problem of blind source separation. In such problems, two or
more source signals (assumed independent) x1(t), x2(t), . . . , xd(t) are combined to
yield a sum signal, s1(t)+s2(t)+ . . .+sc(t) where c ≥ d. (This figure illustrates a case
with only two components.) Given merely the linear signals, and the assumption of
the number of components, d, the task of ICA is to recover the source signals. This
is equivalent to finding a matrix W that is the inverse of A. In general appalications
of ICA, one seeks to extraction independent components from the sensed signals,
whether or not they arose from a linear mixture of initial sources.
The task of independent component analysis is to recover the source signals from
the sensed signals. More specifically, we seek a real matrix W such that
z(t) = Wy(t) = WAx(t), (93)
where z is an estimate of the sources x(t). Of couse we seek W = A−1, but neither
A nor its inverse are known.
We approach the determination of A by maximum-likelihood techniques. We use
an estimate of the density, parameterized by a p̂(y; a) and seek the parameter vector
a that minimizes the diffrerence between the source distribution and the estimate.
That is, a is the basis vectors of A and thus p̂(y; a) is an estimate of the p(y).
This difference can be quantified by the Kullback-Liebler divergence:
D(p(y), p̂(y; a)) = D(p(y)||p̂(y; a))
=
∫
p(y)log
p(y)
p̂(y; a)
dy
= H(y)−
∫
p(y)logp̂(y; a)dy (94)
The log-likelihood is
l(a) =
1
n
n∑
i=1
logp̂(xi; a). (95)
10.13. COMPONENT ANALYSIS 57
and using the law of large numbers, the Kullback-Liebler divergence can be written
as
l(a) = −
∫
p(y)logp(y)dy −
∫
p(y)log
p(y)
p̂(y; a)
dy
= H(y)︸ ︷︷ ︸
indep. of W
−D(p(y)||p̂(y; a)), (96)
where the entropy H(y) is independent of W. Thus we maximize the log-likelihood
by minimizing the Kullback-Liebler divergence with respect to the estimated density
p̂(y; a):
∂l(a)
∂W
= − ∂
∂W
D(p(y)||p̂(y; a)). (97)
Because A is an invertible matrix, and because the Kullback-Liebler divergence is
invariant under invertible transformation (Problem 47), we have
∂l(a)
∂W
= − ∂
∂W
D(p(x)||p̂(z)). (98)
∂H(yyy)
∂WWW
=
∂
∂WWW
log[|WWW|] + ∂
∂WWW
log
[
n∏
i=1
∣∣∣∂xxi
∂yyi
∣∣∣
]
= [WWW−1]t − φ(xxx)zzzt, (99)
where φ(xxx) is the score function, the gradient fector of the log likelihood: score
function
φ(z) = −∂p(z)/∂z
p(z)
= −
∂p(z1)/∂z1
p(z1)
...
∂p(zq)/∂zq
p(zq)
(100)
Thus the learning rule is
∂H(xxx)
∂xxx
= [xxxt]−1 − φ(xx)yyt. (101)
A simpler form comes if we merely scale, following the natural gradient
∆xxx ∝ ∂H(xxx)
∂xxx
WWtWW = [I− φ(xx)xxt]WWW. (102)
This, then is the learning algorithm.
An assumption is that at most one of the sources is Gaussian distributed (Prob-
lem 46). Indeed this method is most successful if the distributions are highly skewed
or otherwise deviate markedly from Gaussian.
We can understand the difference between PCA and ICA in the following way.
Imagine that there were two sources that are correlated and large correlated signals
in a particular direction. PCA would find that direction, and indeed would reduce the
sum-squared error. Such components are not independent, and would not be useful
for separating the sources. As such, they would not be found by ICA. Instead, ICA
58 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
would find those directions that are best for separating the sources — even if those
directions have small eigenvectors.
Generally speaking, when used as preprocessing for classification, independent
component analysis has several characteristics that make it more desirable than linear
or non-linear principal component analysis. As we saw in Fig. 10.23, such principal
components need not be effective in separating classes. Recall that the sensed input
consists of a signal (due to the true categories) plus noise. If the noise is large much
larger than the signal, principal components will depend more upon the noise than on
the signal. Since the different categories are, we assume, independent, independent
component analysis is likely to extract those features that are useful in distinguishing
the classes.
10.14 Low-Dimensional Representations and Multi-
dimensional Scaling (MDS)
Part of the problem of deciding whether or not a given clustering means anything
stems from our inability to visualize the structure of multidimensional data. This
problem is further aggravated when similarity or dissimilarity measures are used that
lack the familiar properties of distance. One way to attack this problem is to try to
represent the data points as points in some lower-dimensional space in such a way
that the distances between points in the that space correspond to the dissimilarities
between points in the original space. If acceptably accurate representations can be
found in two or perhaps three dimensions, this can be an extremely valuable way to
gain insight into the structure of the data. The general process of finding a configura-
tion of points whose interpoint distances correspond to similarities or dissimilarities
is often called multidimensional scaling.
Let us begin with the simpler case where it is meaningful to talk about the dis-
tances between the n samples x1, . . . ,xn. Let yi be the lower-dimensional image of
xi, δij be the distance between xi and xj , and dij be the distance between yi and
yj (Fig. 10.25). Then we are looking for a configuration of image points y1, . . . ,yn
for which the n(n − 1)/2 distances dij between image points are as close as possi-
ble to the corresponding original distances δij . Since it will usually not be possible
to find a configuration for which dij = δij for all i and j, we need some criterion
for deciding whether or not one configuration is better than another. The following
sum-of-squared-error functions are all reasonable candidates:
Jee =
∑
i
(c) Summarize and interpret your answer in words.
12. Consider the univariate normal mixture
p(x|µ1, . . . , µc) =
c∑
j=1
P (ωj)√
2πσ
exp
[
−1
2
(
x− µj
σ
)2]
in which all of the c components have the same, known, variance σ2. Suppose that
the means are so far apart compared to σ that for any observed x all but one of the
terms in this sum are negligible. Use a heuristic argument to show that the value of
max
µ1,…,µc
{ 1
n
ln p(x1, . . . , xn|µ1, . . . , µc)
}
ought to be approximately
c∑
j=1
P (ωj)ln P (ωj)−
1
2
ln [2πσe]
when the number n of independently drawn samples is large. (Here e is the base of
the natural logarithms.)
13. Let x1, . . . ,xn be n d-dimensional samples and Σ be any non-singular d-by-d
matrix. Show that the vector x that minimizes
m∑
k=1
(xk − x)tΣ−1(xk − x)
is the sample mean, x̄ = 1/n
n∑
k=1
xk.
14. Perform the differentiation in Eq. 26 to derive Eqs. 27 & 28.
15. Show that the computational complexity of Algorithm 1 is O(ndcT ), where n,
is the number of d-dimensional patterns, c the assumed number of clusters and T the
number of iterations.
16. Fill in the steps of the derivation of Eqs. 19 – 21.⊕
Section 10.5
17. Consider the combinatorics of exhaustive inspection of clusters of n samples into
c clusters.
(a) Show that there are exactly
1
c!
c∑
i=1
(
c
i
)
(−1)c−iin
such distinct clusterings.
(b) How many clusters are there for n = 100 and c = 5?
10.14. PROBLEMS 73
(c) Find an approximation for your answer to (a) for the case n � c. Use your
answer to estimate the number of clusterings of 1000 points into 10 clusters.
⊕
Section 10.6
18. Prove that the ranking of distances between samples discussed in Sect. ?? is
invariant to any monotonic transformation of the dissimilarity values. Do this as
follows:
(a) Define the value vk for the clustering at level k, and for level 1 let v1 = 0. For all
higher levels, vk is the minimum dissimilarity between pairs of distinct clusters
at level k − 1. Explain why with both δmin and δmax the value vk either stays
the same or increases as k increases.
(b) Assume that no two of the n samples are identical, so that v2 > 0. Use this to
prove monotonicity, i.e., that 0 = v1 ≤ v2 ≤ v3 ≤ · · · ≤ vn.
⊕
Section 10.7
19. Derive Eq. 50 from Eq. 49 using the definition given in Eq. 51.
20. If a set of n samples D is partitioned into c disjoint subsets D1, . . . ,Dc, the
sample mean mi for samples in Di is undefined if Di is empty. In such a case, the
sum of squared errors involves only the non-empty subsets:
Je =
∑
Di �=∅
∑
x∈Di
‖x−mi‖2.
Assuming that n ≥ c, show there are no empty subsets in a partition that minimizes
Je. Explain your answer in words.
21. Consider a set of n = 2k+ 1 samples, k of which coincide at x = −2, k at x = 0,
and one at x = a > 0.
(a) Show that the two-cluster partitioning that minimizes Je groups the k samples
at x = 0 with the one at x = a if a2 < 2(k + 1).
(b) What is the optimal grouping if a2 > 2(k + 1)?
22. Let x1 =
(
4
5
)
, x2 =
(
1
4
)
, x3 =
(
0
1
)
, and x4 =
(
5
0
)
, and consider the following three
partitions:
1. D1 = {x1,x2},D2 = {x3,x4}
2. D1 = {x1,x4},D2 = {x2,x3}
3. D1 = {x1,x2,x3},D2 = {x4}
Show that by the sum-of-square error Je criterion (Eq. ??), the third partition is
favored, whereas by the invariant Jd (Eq. 63) criterion the first two partitions are
favored.
23. Let x1 =
(
xx
xx
)
, x2 =
(
xx
xx
)
, x3 =
(
xx
xx
)
, and x4 =
(
xx
xx
)
, and consider the following
three partitions:
1. D1 = {x1,x2},D2 = {x3,x4}
74 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
2. D1 = {x1,x4},D2 = {x2,x3}
3. D1 = {x1,x2,x3},D2 = {x4}
(a) Find the clustering that minimizes the sum-of-squared error criterion, Je (Eq. ??).
(b) Find the clustering that minimizes the trace criterion, Jd (Eq. 63).
24. Consider the problem of invariance to transformation of the feature space.
(a) Show the eigenvalues λ1, . . . , λd of S
−1
W SB are invariant to nonsingular linear
transformations of the data.
(b) Show that the eigenvalues ν1, . . . , νd of S
−1
T SW are related to those of S
−1
W SB
by νi = 1/(1 + λi).
(c) Use your above results to show that Jd = |SW |/|ST | is invariant to nonsingular
linear transformations of the data.
25. Recall the definitions of the within-cluster and the between-cluster scatter ma-
trices (Eqs. 57 & 58). Define the total scatter matrix to be ST = SW + SB . Show
that the following measures (Eqs. 65 & 66) are invariant to linear transformations of
the data.
(a) trS−1T SW =
d∑
i=1
1
1+λi
(b) |SW |/|ST | =
d∏
i=1
1
1+λi
(c) |S−1W SB | =
d∏
i=1
λi
(d) What is the typical value of the criterion in (c)? Why, therefore, is that criterion
not very useful?
26. Show that the clustering criterion Jd in Eq. 63 is invariant to linear transforma-
tions of the space as follows. Let T be a nonsingular matrix and consider the change
of variables x′ = Tx.
(a) Write the new mean vectors m′i and scatter matrices S
′
i in terms of the old
values and T.
(b) Calculate J ′d in terms of the (old) Jd and show that they differ solely by an
overall scalar factor.
(c) Since this factor is the same for all partitions, argue that Jd and J ′d rank the
partitions in the same way, and hence that the optimal clustering based on Jd
is invariant to nonsingular linear transformations of the data.
27. Consider the problems that might arise when using the determinant criterion for
clustering.
(a) Show that the rank of the within-cluster scatter matrix Si can not exceed ni−1,
and thus the rank of SW can not exceed
∑
(ni − 1) = n− c.
10.14. PROBLEMS 75
(b) Use your answer to explain why the between cluster scatter matrix SB may
become singular. (Of course, if the samples are confined to a lower dimensional
subspace it is possible to have SW be singular even though n− c ≥ d.)
⊕
Section 10.8
28. One way to generalize the basic-minimum-squared-error procedure is to define
the criterion function
JT =
c∑
i=1
∑
x∈Di
(x−mi)tS−1T (x−mi),
where mi is the mean of the ni samples in Di and ST is the total scatter matrix.
(a) Show that JT is invariant to nonsingular linear transformations of the data.
(b) Show that the transfer of a sample x̂ from Di to Dj causes JT to change to
J∗T = JT +
[ nj
nj + 1
(x̂−mj)tS−1T (x̂−mj)−
ni
ni − 1
(x̂−mi)tS−1T (x̂−mi)
]
.
(c) Using this result, write pseudocode for an iterative procedure for minimizing JT
(cf. Computer Exercise 20).
29. Consider how the transfer of a single point from one cluster to another affects
the mean and sum-squared error, and thereby derive Eqs. 71 & 72.⊕
Section 10.9
30. Let a similarity measure be defined as s(x,x′) = xtx′/(‖x‖ · ‖x′‖).
(a) Interpret this similarity measure if the d features have binary values, where
xi = 1 if x possesses the ith feature and xi = −1 if it does not.
(b) Show that for this case the squared Euclidean distance satisfies
‖x− x′‖2 = 2d(1− s(x,x′)).
31. Let d be the dimensionality of the space, q a scalar parameter (q > 1). For each
of the measures shown, state whether it represents a metric (or not), and whether it
represents an ultrametric (or not).
(a) s(x,x′) = ‖x− x′‖2 (squared Euclidean)
(b) s(x,x′) = ‖x− x′‖ (Euclidean)
(c) s(x,x′) =
(
d∑
k=1
|xk − x′k|q
)1/q
(Minkowski)
(d) s(x,x′) = xtx′/‖x‖‖x′‖ (cosine)
(e) s(x,x′) = xtx′ (dot product)
76 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
(f) s(x,x′) = minα ‖x + αT(x)− x′‖2 (one-sided tangent distance)
where T is a linear transform and α a vector of coefficients (cf. Chap. ??, Sect.
??).
32. Let cluster Di contain ni samples, and let dij be some measure of the distance
between two clusters Di and Dj . In general, one might expect that if Di and Dj are
merged to form a new cluster Dk, then the distance from Dk to some other cluster
Dh is not simply related to dhi and dhj . However, consider the equation
dhk = αdhi + αidhj + βdij + γ|dhi − dhj |.
Show that the following choices for the coefficients αi, αj , β, and γ lead to the distance
functions indicated.
(a) dmin : αi = αj = 0.5, β = 0, γ = −0.5.
(b) dmax : αi = αj = 0.5, β = 0, γ = +0.5.
(c) davg : αi =
ni
ni+nj
, αj =
nj
ni+nj
, β = γ = 0.
(d) d2mean : αi =
ni
ni+nj
, αj =
nj
ni+nj
, β = −αiαj , γ = 0.
33. Consider a hierarchical clustering procedure in which clusters are merged so as
to produce the smallest increase in the sum-of-squared error at each step. If the ith
cluster contains ni samples with sample mean mi, show that the smallest increase
results from merging the pair of clusters for which
ninj
ni + nj
‖mi −mj‖2
is minimum.
34. Assume we are clustering using the sum-of-squared error criterion Je (Eq. ??).
Show that a “distance” measure between clusters can be derived, Eq. 78, such that
merging the “closest” such clusters increases Je as little as possible.
35. Create by hand a dendrogram for the following eight points in one dimension:
{−5.5,−4.1,−3.0,−2.6, 10.1, 11.9, 12.3, 13.6}. Define the similarity between to clus-
ters to be 20 − dmin(Di,Dj), where dmin(Di,Dj) is given in Eq. 74. Based on your
dendrogram, argue that two is the natural number of clusters.
36. Create by hand a dendrogram for the following 10 points in one dimension:
{−2.2,−2.0,−0.3, 0.1, 0.2, 0.4, 1.6, 1.7, 1.9, 2.0}. Define the similarity between to clus-
ters to be 20 − dmin(Di,Dj), where dmin(Di,Dj) is given in Eq. 74. Based on your
dendrogram, argue that three is the natural number of clusters.
37. Assume that the nearest-neighbor cluster algorithm has been allowed to continue
fully, thereby giving a tree with a path from any node to any other node. Show that
the sum of the edge lengths of this resulting tree will not exceed the sum of the edge
lengths for any other spanning tree for that set of samples.⊕
Section 10.10
38. Assume that a large number n of d-dimensional samples has been chosen from
a multidimensional Gaussian, i.e., p(x) ∼ N(m,Σ), where Σ is an arbitrary positive-
definite covariance matrix.
10.14. PROBLEMS 77
(a) Prove that the distribution of the criterion function Je(1) given in Eq. 82 is
normal with mean ndσ2. Express σ in terms of Σ.
(b) Prove that the variance of this distribution is 2ndσ4.
(c) Consider a suboptimal partition of the Gaussian by a hyperplane through the
sample mean. Show that for large n, the sum of squared error for this partiction
is approximately normal with mean n(d− 2/π)σ2 and variance 2n(d− 8/π2)σ4,
where σ is given in part (a).
39. Derive Eqs. 85 & 86.⊕
Section 10.12
40. Consider a simple greedy algorithm for creating a spanning tree.
(a) Write pseudocode for creating a minimal spanning tree linking n points in d
dimension.
(b) Let k denote the average linkage per node. What is the average space complexity
of your algorithm?
(c) What is the average time complexity?
⊕
Section 10.11
41. Consider the adaptive resonance clustering algorithm.
(a) Show that the standard ART algorithm cannot learn the XOR problem.
(b) Explain how the number of clusters generated by the adaptive resonance algo-
rithm depends upon the order of presentation of the samples.
(c) Discuss the benefits and drawbacks of adaptive resonance in stationary and in
non-stationary environments.
⊕
Section 10.13
42. Show that minimizing a mean-squared error criterion for d-dimensional data
leads to the k-dimensional representation (k < d) of the Karhunen-Loéve transform
(Eq. 90) as follows. For simplicity, assume that the data set has zero mean. (If the
mean is not zero, we can always subtract off the mean from each vector to define a
new vectors.)
(a) The (scalar) projection of a vector x onto a unit vector e, a(e) = xte, is, of
course, a random variable. Define the variance of a to be σ2 = Ex[a2]. Show
that σ2 = etΣe, where Σ = Ex[xxt] is the correlation matrix.
(b) A vector e that yields an extremal or stationary value of this variance must obey
σ2(e + δe) = σ2(e), where δe is a small perturbation. Show that this condition
implies (δe)tΣe = 0 at such a stationary point.
(c) Consider small variations δe that do not change the length of the vector, i.e.,
ones in which δe is perpendicular to e. Use this condition and your above results
to show that (δe)tΣλ(δe)te = 0, where λ is a scalar. Show that the necessary
and sufficient solution is Σe = λe — that is, the eigenvector equation of Eq. 99.
78 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
(d) Define a sum-squared-error criterion for a set of points in d-dimensional space
and their projections onto a k-dimensional linear subspace (k < d). Use your
results above to show that in order to minimize your criterion, the subspace
shoud be spanned by the k largest eigenvectors of the correlation matrix.
43. Show that a neural net auto-association network consisting of d − k − d input,
hidden and output layer (with k < d)
(a) Show that a neural net auto-association network consisting of d− k − d input,
hidden and output layer (with k < d) and linear hidden units performs principal
component analysis by considering the minimization it solves. Trained on sum
squared error.
(b) Show that a neural net auto-association network consisting of d− k − d input,
hidden and output layer (with k < d)
(c) Show that the five layer neural net auto-association network of Fig. ?? consisting
of d− k− r− k− d where both layers having k units are nonlinear will perform
nonlinear dimensionality reduction.
44. Consider the use of neural networks for nonlinear principal componet analysis.
(a) Prove that if all units in the five-layer network of Fig. 10.22 are linear, and the
network trained to serve as an auto-encoder, then the representation learned at
the middle layer corresponds to the linear principal component of the data.
(b) State briefly why this also implies that a three-layer network (input, hidden,
output) cannot be used for non-linear principal component analysis, even if the
middle layer consists of non-linear units.
45. The derivation of the Independent component analysis algorithm, summarized
in Eq. 99, assumed that the sources and sum signals were all scalars, that there was
no noise, and that the number of observations, T , is equal to the number of points
generated by each source.
(a) Relax all of these conditions to generalize the method to vectors, x1(t) + . . . +
xc(t). Assume, moreover, that the sum signal is corrupted by additive Gaussian
noise of zero mean, but unknown covariance: p(y) ∼ N(0,Σ).
(b) Suppose the noise is sufficiently small (|Σ| � 1), and that the dimensionality
of the vectors is set to d = 1. Show that your learning rule reduces to that of
Eq. 99.
46. Use the fact that the sum samples from two Gaussians is again a Gaussian to
show why independent component analysis can not isolate sources perfectly if more
than one has a Gaussian distribution.
47. It is a fact that the Kullback-Liebler divergence is invariant under general
invertible transforms. Prove this for the special case of linear transforms, as used in
Sect. 10.13.⊕
Section 10.14
48. Consider the use of multidimensional scaling for representing the points x1 =
(1, 0)t,x2 = (0, 0)t and x3 = (0, 1)t in one dimensions. To obtain a unique solution,
assume that the image points satisfy 0 = y1 < y2 < y3.
10.14. COMPUTER EXERCISES 79
(a) Show that the criterion function Jee is minimized by the configuration with
y2 = (1 +
√
2)/3 and y3 = 2y2.
(b) Show that the criterion function Jff is minimized by the configuration with
y2 = (2 +
√
2)/4 and y3 = 2y2.
Computer exercises
Several exercises make use of the data in the following table.
sample x1 x2 x3 sample x1 x2 x3
1 -7.82 -4.58 -3.97 11 6.18 2.81 5.82
2 -6.68 3.16 2.71 12 6.72 -0.93 -4.04
3 4.36 -2.19 2.09 13 -6.25 -0.26 0.56
4 6.72 0.88 2.80 14 -6.94 -1.22 1.13
5 -8.64 3.06 3.50 15 8.09 0.20 2.25
6 -6.87 0.57 -5.45 16 6.81 0.17 -4.15
7 4.47 -2.62 5.76 17 -5.19 4.24 4.04
8 6.73 -2.01 4.18 18 -6.38 -1.74 1.43
9 -7.71 2.34 -6.33 19 4.08 1.30 5.33
10 -6.91 -0.49 -5.68 20 6.27 0.93 -2.78⊕
Section 10.4
1. Consider the univariate normal mixture
p(x|θ) = P (ω1)√
2πσ1
exp
[
−1
2
(
x− µ1
σ1
)2]
+
1− P (ω1)√
2πσ2
exp
[
−1
2
(
x− µ2
σ2
)2]
.
Write a general program for computing the maximum likelihood values of the pa-
rameters, and apply it to the 20 x1 points in the table above under the following
assumptions of what is known and what is unknown:
(a) Known: P (ω1) = 0.5, σ1 = σ2 = 1; Unknown: µ1 and µ2.
(b) Known: P (ω1) = 0.5; Unknown: σ1 = σ2 = σ, µ1 and µ2.
(c) Known: P (ω1) = 0.5; Unknown: σ1, σ2, µ1 and µ2.
(d) Unknown: P (ω1), σ1, σ2, µ1 and µ2.
2. Write a program to implement k-means clustering (Algorithm 1), and apply it to
the three-dimensional data in the table for the following assumed numbers of clusters,
and starting points.
(a) Let c = 2, m1(0) = (1, 1, 1)t and m2(0) = (−1, 1,−1)t.
(b) Let c = 2, m1(0) = (0, 0, 0)t and m2(0) = (1, 1,−1)t. Compare your final
solution with that from part (a), and explain any differences, including the
number of iterations for convergence.
(c) Let c = 3, m1(0) = (0, 0, 0)t, m2(0) = (1, 1, 1)t and m3(0) = (−1, 0, 2)t.
80 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
(d) Let c = 3, m1(0) = (−0.1, 0, 0.1)t, m2(0) = (0,−0.1, 0.1)t and m3(0) =
(−0.1,−0.1, .1)t. Compare your final solution with that from part (c), and
explain any differences, including the number of iterations for convergence.
3. Repeat Computer exercise 2, but use instead a fuzzy k-means algorithm (Algo-
rithm 1) with the “blending” be set by b = 2 (Eqs. 27 & 28).
4. Explore the problems that can come with mis-specifying the number of clusters in
the fuzzy k-means algorithm (Algorithm 2) using the following one-dimensional data:
D = {−5.0,−4.5,−4.1,−3.9, 2.5, 2.8, 3.1, 3.9, 4.5}.
(a) Use your program in the four conditions defined by c = 2 and c = 3, and b = 1
and b = 4. In each cases initialize the cluster centers to distinct values, but ones
near x = 0.
(b) Compare your solutions to the c = 3, b = 4 case to the c = 3, b = 1 case, and
discuss any sources of the differences.
5. Show how a few labeled samples in a k-means algorithm can improve clustering
of unlabeled samples in the following, somewhat extreme case.
(a) Generate 50 two-dimensional samples for each of four spherical Gaussians, p(x|ωi) ∼
N(µi, I), where µ1 =
(−2
−2
)
, µ2 =
(−2
2
)
, µ3 =
(
2
2
)
, and µ4 =
(
2
−2
)
.
(b) Choose c = 4 initial positions for the cluster means randomly from the full 200
samples. What is the probability that your random selection yields exactly one
cluster center for each component density? (Make the simplifying assumption
that the component densities do not overlap significantly.)
(c) Using the four samples selected in part (b), run a k-means clusterer on the full
200 points. (If the four points in fact come from different components, re-select
samples to insure that at least two come from the same component density
before using your clusterer.)
(d) Now assume you have some label information, in particular four samples known
to come from distinct component densities. Using these as your initial cluster
centers, run a k-means clusterer on the full 200 points.
(e) Discuss the value of a few labeled samples for clustering in light of the final
clusters given in (c & d).
⊕
Section 10.5
6. Explore unsupervised Bayesian learning of the mean of a Gaussian distribution
following way.
(a) Generate a data set D of 30 points, uniformly distributed in the interval −10 ≤
x ≤ +10.
(b) Assume the data comes from a normal distribution with known variance, but
unknown mean, i.e., p(x) ∼ N(µ, 2) — that is, the unknown parameter θ in
Eq. 37 is simply the scalar µ. Assume a wide prior for the parameter: p(µ)
is uniform in the range −10 ≤ µ ≤ +10. Plot posterior probabilities for k =
0, 1, 2, 3, 4, 5, 10, 15, 20, 25, 30 points from D.
10.14. COMPUTER EXERCISES 81
(c) Now assume instead a narrow prior, i.e., p(µ) uniform in the range−1 ≤ µ ≤ +1,
and repeat part (b) using the same order of data presentation.
(d) Are your curves for part (b) and part (c) the same for small number of points?
For large number of points? Explain.
7. Write a decision-directed clusterer related to k-means in the following way.
(a) First, generate a set D of n = 1000 three-dimensional points in the unit square,
0 ≤ xi ≤ 1, i = 1, 2.
(b) Randomly choose c = 4 of these points as the initial cluster centers mj , j =
1, 2, 3, 4.
(c) The core of the algorithm operates as follows: First, each sample xi, is classified
by the nearest cluster center mj . Next, each mean mj is calculated to be
the mean of the samples in ωj . If there is no change in the centers after n
presentations, halt.
(d) Use your algorithm to plot four trajectories of the position of the cluster centers.
(e) What is the space and the time complexities of this algorithm? State any
assumptions you invoke.
⊕
Section 10.6
8. Explore the role of metrics, similarity measures and thresholds on cluster formation
in the following way.
(a) First, generate a two-dimensional data set consisting of two parts: D1 contains
100 points whose distance from the origin is chosen uniformly in the range
3 ≤ r ≤ 5, and angular position uniform in the range 0 ≤ φ < 2π; likewise, D2
consists of 50 points of distance 0 ≤ r ≤ 2 and angle 0 ≤ φ < 2π. The full data
set used below is D = D1 ∪ D2.
(b) Write a simple clustering algorithm that links any two points x and x′ if
d(x,x′) < θ, where θ is a threshold selected by the user, and distance is calcu-
lated by means of a general Minkowski metric (Eq. 44),
d(x,x′) =
(
d∑
k=1
|xk − x′k|q
)1/q
.
Let q = 2 (Euclidean distance) and apply your algorithm to the data D for
the following thresholds: θ = 0.01, 0.05, 0.1, 0.5, 1, 5. In each case, plot all 150
points and differentiate the clusters by color or other plotting convention.
(c) Repeat part (b) with q = 1 (city block distance).
(d) Repeat part (b) with q = 4.
(e) Discuss how the metric affects the “natural” number of clusters implied by your
results.
82 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
⊕
Section 10.7
9. Explore different clustering criteria by exhaustive search in the following way. Let
D be the first seven three-dimensional points in the table above.
(a) If we assume that any cluster must have at least one point, how many cluster
configurations are possible for the seven points?
(b) Write a program to search through each of the cluster configurations, and for
each compute the following criteria: Je (Eq. 49), Jd (Eq. 63),
d∑
i=1
λi (Eq. 64),
Jf = trS
−1
T SW (Eq. 65) and |S|/|ST | (Eq. 66). show the optimal clusters for
each of your four criteria.
(c) Perform a whitening transformation on your points are repeat part (b).
(d) In light of your results, discuss which of the criteria are invariant to the whitening
transformation.
⊕
Section 10.8
10. Show that the Basic Iterative Least-Squares clustering algorithm gives solutions
and final criterion values that depend upon starting conditions in the following way.
Implement Algorithm 3 for c = 3 clusters and apply it to the data in the table above.
For each simulations, list the final clusters as sets of points (identified by their index
in the table), along with the corresponding value of the criterion function.
(a) m1(0) = (1, 1, 1)t, m2(0) = (−1,−1,−1)t and m3(0) = (0, 0, 0)t.
(b) m1(0) = (0.1, 0.1, 0.1)t, m2(0) = (−0.1,−0.1,−0.1)t and m3(0) = (0, 0, 0)t.
(c) m1(0) = (2, 0, 2)t, m2(0) = (−2, 0,−2)t and m3(0) = (1, 1, 1)t.
(d) m1(0) = (0.5, 1, 0.2)t, m2(0) = (0.2,−1, 0.5)t and m3(0) = (0.2, 0.4, 0.6)t.
(e) Explain why your final answers differ.
⊕
Section 10.9
11. Implement the basic hierarchical agglomerative clustering algorithm (Algo-
rithm 4), as well as a method for drawing dendrograms based on its results. Apply
your algorithm and draw dendrograms to the date in the table above using the dis-
tance measure indicated below. Define the similarity between two clusters to be linear
in distance, with similarity = 100 for singleton clusters (c = 20) and similarity = 0
for the single cluster (c = 1).
(a) dmin (Eq. 74)
(b) dmax (Eq. 75)
(c) davg (Eq. 76)
(d) dmean (Eq. 77)
10.14. COMPUTER EXERCISES 83
12. Explore the use of cluster dendrograms for selecting the “most natural” number
of clusters.
(a) Write a program to perform hierarchical clustering and display a dendrogram,
using measure of distance to be selected from the Eqs. 74 – 77.
(b) Write a program to generate n/c points from each of c one-dimensional Gaus-
sians, p(x|ωi) ∼ N(µi, σ2i ), i = 1, . . . , c. Use your program to generate n = 50
points, 25 in each of two clusters, with µ1 = 0, µ2 = 1, and σ21 = σ
2
2 = 1.
Repeate with µ2 = 4.
(c) Use your program from (a) to generate dendrograms for each of the two data
sets generated in (??).
(d) The difference in similarity values for successive levels is a random variable,
which we can model as a normal distribution with mean and variance. Suppose
we define the “most natural” number of clusters according to the largest gap in
similarity values, and that this largest gap is significant if it differs “significantly”
from the distribution. State your criterion analytically, and show that one of
the cases in (??) indeed has two clusters.
⊕
Section 10.10
⊕
Section 10.12
13. xxx⊕
Section 10.11
14. Implement a basic competitive learning clustering algorithm (Algorithm 6) and
apply it to the three-dimensional data in the table above as follows.
(a) First, preprocess the data by augmenting each vector with x0 = 1 and normal-
izing to unit length. In this way, each point lies on the surface of a hypersphere.
(b) Set c = 2, and let the inital (normalized) weght vectors correspond to patterns
1 & 2. Let the learning rate be η = 0.1. Present the patterns in cyclic order,
1, 2, . . . , 20, 1, 2, . . . , 20, 1, 2, . . ..
(c) Modify your program so as to reduce the learning rate by multiplying by the
constant factor α < 1 after each pattern presentation, so the learning rate
approaches zero exponentially. Repeat your simulation of part (b) with such
decay, where α = 0.99. Compare your final clusterings with those from using
α = 0.5.
(d) Repeat part (c) but with the patterns chosen in a random order, i.e., with the
probability of presenting any given pattern being 1/20 per trial. Discuss the
role of random versus sequenced pattern presentation on the final clusterings.
⊕
Section 10.13
15. PCA exercise
16. Explore the use of independent component analysis for blind source separation
in the following example.
84 CHAPTER 10. UNSUPERVISED LEARNING AND CLUSTERING
(a) Generate 100 points for t = 1, . . . , 100 for x1(t) = xxx and x2(t) = xxx. Gen-
erate 100 points each for three sensors according to:
x1(t) = xxx
x2(t) = xxx
and three sensors:
s1(t) = xxx
s2(t) = xxx
s3(t) = xxx
(Of course, in this blind source separation task, neither the source signals nor
the mixing parameters are known.)
(b) xxx
17. Repeat Computer exercise 16, but for three sources:
x1(t) = xxx
x2(t) = xxx
x3(t) = xxx
and four sensors:
s1(t) = xxx
s2(t) = xxx
s3(t) = xxx
s4(t) = xxx
⊕
Section 10.14
18. Write a computer program that uses the general maximum-likelihood equation of
Sect. ?? iteratively to estimate the unknown means, variances, and prior probabilities.
Use this program to find maximum-likelihood estimates of these parameters for the
data in Table ??.
19. hill climbing for clustering. Start at BAD and at GOOD starting places. Note
that do not get same answer.
20. Write a program to perform the minimization of in Problem 28.
21. what if you have the wrong number of clusters?. . . .xxx
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Index
agglomerative clustering, see cluster-
ing, agglomerative
Agglomerative hierarchical clustering
Algorithm, 39
ART, see clustering, adaptive resonance
auto-associator, 54
auto-encoder, 54
auto-encoder network, 67
Basic minimum-squared-error cluster-
ing
Algorithm, 36
Bayes
formula, 18
blind source separation, 56
classification
food, 3
classifier
Bayes, 17
unsupervised, 17
Cluster
similarity measure, 37
cluster
chaining, 40
criterion
chi-squared, 44
determininant, 32
invariant, 32
Kolmogorov-Smirnov, 44
local minimum, 34
dendrogram, 37
diameter
path, 52
level, 37
mean, 35
singleton, 36
tree
minimal spanning, 52
validity, 38
cluster criterion
trace, 32
cluster membership
fuzzy, 14
cluster validity, 67
clustering
Adaptive Resonance, 67
adaptive resonance, 49–50
agglomerative, 38–42
Bayesian, 67
bottom-up, see clustering, agglom-
erative
chaining, 67
complete-linkage, 40, 51
criterion
squared error, 67
sum-of-squared error, 29
trace, 67
criterion function, 29
decision directed, 23
divisive, 38
farthest-neighbor, 40–41
fuzzy k-means, 14–16, 67
gradient ascent, 35
graph theoretic, 51–52
graph-theoretic, 67
hierarchical, 37, 67
divisive, 67
stepwise-optimal, 41–42
hypothesis
null, 44
iterative algorithms, 67
k-means, 13–14, 67
leader-follower, 48–50, 67
maximum algorithm, 40
maximum-likelihood, 67
minimum algorithm, 40
motivations, 3
nearest-neighbor, 40
nonparametric method, 25
optimization
iterative, 35
88
INDEX 89
single-linkage, 40, 51
small sample, 19
solution
unique, 5
splitting, see clustering, divisive
starting point, 36
clusteringt
hierarchical
agglomerative, 67
Competitive learning
Algorithm, 47
competitive learning, 45–47, 67
component analysis, 53–58, 67
connected component, 51
criterion
scattering, 31
sum squared error, 29
curse of dimensionality, 44
data description
flat, 37
hierarchical, 37
data matrix, 65
data mining, 3
dendrogram, see cluster, dendrogram,
67
density
component, 4
joint, 6
mixture, 4, 20
discrimination versus representation, 66
dissimilarity, 43, 60
clustering, 39
distance
Kullback-Liebler, 56
distance function
as dissimiliarity measure, 25
divisive clustering, see clustering, divi-
sive
entropy, 57
for independent component anal-
ysis, 67
error
sum-of-squared, 58
error function (erf), 45
estimate
maximum-likelihood
clustering, 6
factor analysis, 65
Factorization Theorem, 20
family (taxonomic), 37
feature space
isotropic, 25
rescaling, 25
flat data description, 37
frequency ratios, 12
function
Dirac delta, 19
fuzzy k-means clustering
Algorithm, 15
genus, 37
graph
similarity, 51
hierarchical data description, 37
Hierarchical dimensionality reduction
Algorithm, 66
hyperellipsoid, 24
hypersphere, 27
hypothesis testing
and clustering, 67
identifiability, 5, 19
discrete distribution, 5
inconsistent edge, 52
inner product, 27
invariance
dilation, 28
rotation, 28
translation, 25
k-means clustering, see clustering, k-
means
Algorithm, 13
kingdom, 37
Kohonen map, see self-organizing fea-
ture map
Kronecker delta, 12
Kullback-Leibler divergence, see distance,
Kullback-Liebler
law of large numbers, 57
Leader-follower clustering
Algorithm, 48
learning
Bayesian vs. maximum-likelihood,
22
competitive, see competitive learn-
ing
90 INDEX
supervised vs. unsupervised, 19
unsupervised
batch protocol, 23
Bayesian, 18
computational complexity, 19
decision-directed, 23
learning rate
decay, 47
likelihood
gradient ascent solution, 8
LVQ, see learning vector quantization
matrix
covariance, 12
diagonal, 13
data, see data matrix
scatter, 31
total, 31
similarity, 51
maximimum-likelihood
solution
non-uniqueness, 9
maximum-likelihood
solution
singular, 11
unsupervised, 6
non-uniqueness, 8
MDS, see multidimensional scaling
mean
sample, 24
metric
clustering, 25
dissimilarity, 43
Euclidean, 25
induced, 43
Mahalanobis, 27
Minkowski, 26
non-negativity, 43
properties, 43
reflexivity, 43
symmetry, 43
Tanimoto, 28
triangle inequality, 43
minimal spanning tree, see tree, mini-
mal spanning
mixing parameter, see parameter, mix-
ing
mixture
discrete distribution, 5
monotonicity constraint, 60
multidimensional scaling, 58–61, 67
optimization
iterative, 13, 37
order (taxonomic), 37
orienting subsystem, 50
outlier, 41
outlier pattern, 29
parameter
mixing, 4
partition
minimum variance, 29
PCA, see principal component analysis
phoneme, 3
phylum, 37
preprocessing, 3
principal component, 26
principal component analysis, 53, 67
principal components, 32
nonlinear, 67
probability
posterior, 6
representation, 66
saddle point, 10
sample independence, 18
scatter matrix
eigenvector, 32
invariant, 32
score function, 57
search
bias, 10
segmentation, 67
self-organizing feature map, 61–65, 67
sensor vector, 55
set diagram, see Venn diagram
similarity function, 27
similarity graph, see graph, similarity
similarity measure, 25
skeleton, see tree, spanning, minimal
SOM, see self-organizing feature map
source separation
blind, 67
species, 37
stability-plasticity, 47
standardized data, 26
Stepwise optimal hierarchical cluster-
ing
Algorithm, 42
INDEX 91
subcluster, 37, 67
subfamily (taxonomic), 37
subgraph
complete, 40
maximal complete, 51
suborder (taxonomic), 37
subphylum, 37
sufficient statistic, 20, 21
sufficient statistics
in unsupervised learning, 24
taxonomy, 37
topologically ordered map, see self-organizing
feature map
trace criterion, see cluster criterion, trace
tree
minimal spanning, 40
spanning
minimal, 52
tree (graph), 40
triangle inequality, see metric, triangle
inequality
two-joint arm
self-organizing map example, 61
ultrametric, 43
unidentifiable
complete, 5
unsupervised learning
convergence rate, 22
vector quantization, 68
Venn diagram, 38
vigilance parameter, 50
Voronoi tesselation, 14
VQ, see vector quantization
weight normalization, 46
whitening transform, 26
wild shot pattern, see outlier pattern
Contents
A Mathematical foundations 3
A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
A.2 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
A.2.1 Notation and preliminaries . . . . . . . . . . . . . . . . . . . . 6
A.2.2 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
A.2.3 Outer product . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
A.2.4 Derivatives of matrices . . . . . . . . . . . . . . . . . . . . . . . 8
A.2.5 Determinant and trace . . . . . . . . . . . . . . . . . . . . . . . 9
A.2.6 Matrix inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 10
A.2.7 Eigenvectors and eigenvalues . . . . . . . . . . . . . . . . . . . 11
A.3 Lagrange optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
A.4 Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
A.4.1 Discrete random variables . . . . . . . . . . . . . . . . . . . . . 12
A.4.2 Expected values . . . . . . . . . . . . . . . . . . . . . . . . . . 13
A.4.3 Pairs of discrete random variables . . . . . . . . . . . . . . . . 14
A.4.4 Statistical independence . . . . . . . . . . . . . . . . . . . . . . 15
A.4.5 Expected values of functions of two variables . . . . . . . . . . 15
A.4.6 Conditional probability . . . . . . . . . . . . . . . . . . . . . . 16
A.4.7 The Law of Total Probability and Bayes’ rule . . . . . . . . . . 17
A.4.8 Vector random variables . . . . . . . . . . . . . . . . . . . . . . 18
A.4.9 Expectations, mean vectors and covariance matrices . . . . . . 19
A.4.10 Continuous random variables . . . . . . . . . . . . . . . . . . . 20
A.4.11 Distributions of sums of independent random variables . . . . . 22
A.4.12 Univariate normal density . . . . . . . . . . . . . . . . . . . . . 23
A.5 Gaussian derivatives and integrals . . . . . . . . . . . . . . . . . . . . 24
A.5.1 Multivariate normal densities . . . . . . . . . . . . . . . . . . . 26
A.5.2 Bivariate normal densities . . . . . . . . . . . . . . . . . . . . . 28
A.6 Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
A.6.1 Chi-squared test . . . . . . . . . . . . . . . . . . . . . . . . . . 31
A.7 Information theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
A.7.1 Entropy and information . . . . . . . . . . . . . . . . . . . . . 32
A.7.2 Relative entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 33
A.7.3 Mutual information . . . . . . . . . . . . . . . . . . . . . . . . 34
A.8 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . 34
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1
2 CONTENTS
Appendix A
Mathematical foundations
Our goal here is to present the basic results and definitions from linear algebra,probability theory, information theory and computational complexity that serve
as the mathematical foundations for pattern recognition. We will try to give intuitive
insight whenever appropriate, but do not attempt to prove these results; systematic
expositions can be found in the references.
A.1 Notation
Here are the terms and notation used throughout the book. In addition, there are
numerous specialized variables and functions whose definitions and usage should be
clear from the text.
variables, symbols and operations
� approximately equal to
≡ equivalent to (or defined to be)
∝ proportional to
∞ infinity
x→ a x approaches a
t← t + 1 in an algorithm: assign to variable t the new value t + 1
lim
x→a
f(x) the value of f(x) in the limit as x approaches a
arg max
x
f(x) the value of x that leads to the maximum value of f(x)
arg min
x
f(x) the value of x that leads to the minimum value of f(x)
�x� ceiling of x, i.e., the least integer not smaller than x (e.g., �3.5� = 4)
x
floor of x, i.e., the greatest integer not larger than x (e.g., 3.5
= 3)
m mod n m modulo n, the remainder when m is divided by n (e.g., 7 mod 5
= 2)
ln(x) logarithm base e, or natural logarithm of x
log(x) logarithm base 10 of x
log2(x) logarithm base 2 of x
3
4 APPENDIX A. MATHEMATICAL FOUNDATIONS
exp[x] or ex exponential of x, i.e., e raised to the power of x
∂f(x)/∂x partial derivative of f with respect to x
b∫
a
f(x)dx the integral of f(x) between a and b. If no limits are written, the
full space is assumed
F (x; θ) function of x, with implied dependence upon θ
Q.E.D., quod erat demonstrandum (“which was to be proved ”) —
used to signal the end of a proof
mathematical operations
x̄ mean or average value of x
E [f(x)] the expected value of function f(x) where x is a random variable
Ey[f(x, y)] the expected value of function over several variables, f(x, y), taken
over a subset y of them
Varf [·] the variance, i.e., Ef [(x− Ef [x])2]
n∑
i=1
ai the sum from i = 1 to n: a1 + a2 + … + an
n∏
i=1
ai the product from i = 1 to n: a1 × a2 × …× an
f(x) � g(x) convolution of f(x) with g(x)
vectors and matrices
Rd d-dimensional Euclidean space
x,A, … boldface is used for (column) vectors and matrices
f(x) vector-valued function (note the boldface) of a scalar
f(x) vector-valued function (note the boldface) of a vector
I identity matrix, square matrix having 1s on the diagonal and 0
everywhere else
1i vector of length i consisting solely of 1’s
diag(a1, a2, …, ad) matrix whose diagonal elements are a1, a2, …, ad, and off-diagonal
elements 0
xt transpose of vector x
‖x‖ Euclidean norm of vector x
Σ covariance matrix
tr[A] the trace of A, i.e., the sum of its diagonal components: tr[A] =
d∑
i=1
aii
A−1 the inverse of matrix A
A† pseudoinverse of matrix A
|A| or Det[A] determinant of A
λ eigenvalue
e eigenvector
ui unit vector in the ith direction in Euclidean space
A.1. NOTATION 5
sets
A,B, C,D, … “Calligraphic” font generally denotes sets or lists, e.g., data set
D = {x1, …,xn}
x ∈ D x is an element of set D
x /∈ D x is not an element of set D
A ∪ B union of two sets, i.e., the set containing all elements of A and B
|D| the cardinality of set D, i.e., the number of (possibly non-distinct)
elements in it; occassionally written card|D|
max
x
[D] the maximum x value in set D
probability, distributions and complexity
ω state of nature
P (·) probability
p(·) probability density
P (a, b) the joint probability , i.e., the probability of having both a and b
p(a, b) the joint probability density, i.e., the probability density of having
both a and b
Pr{·} the probability of a condition being met, e.g., Pr{x < x0} means
the probability that x is less than x0
p(x|θ) the conditional probability density of x given θ
w weight vector
λ(·, ·) loss function
∇ =
d
dx1
d
dx2
...
d
dxd
gradient operator in Rd, sometimes written grad
∇θ =
d
dθ1
d
dθ2
...
d
dθd
gradient operator in θ coordinates, sometimes written gradθ
θ̂ maximum likelihood value of θ
∼ “has the distribution,” e.g., p(x) ∼ N(µ, σ2) means that the density
of x is normal, with mean µ and variance σ2
N(µ, σ2) normal or Gaussian distribution with mean µ and variance σ2
N(µ,Σ) multidimensional normal or Gaussian distribution with mean µ
and covariance matrix Σ
U(xl, xu) a one-dimensional uniform distribution between xl and xu
U(xl,xu) a d-dimensional uniform density, i.e., uniform density within the
smallest axes-aligned bounding box that contains both xl and xu,
and zero elsewhere
T (µ, δ) triangle distribution, having center µ and full half-width δ
δ(x) Dirac delta function
Γ(·) Gamma function
n! n factorial = n× (n− 1)× (n− 2)× ...× 1(
n
k
)
= n!
k!(n−k)! binomial coefficient, n choose k for n and k integers
6 APPENDIX A. MATHEMATICAL FOUNDATIONS
O(h(x)) big oh order of h(x)
Θ(h(x)) big theta order of h(x)
Ω(h(x)) big omega order of h(x)
sup
x
f(x) the supremum value of f(x) — the global maximum of f(x) over
all values of x
A.2 Linear algebra
A.2.1 Notation and preliminaries
A d-dimensional column vector x and its transpose xt can be written as
x =
x1
x2
...
xd
and x
t = (x1 x2 . . . xd), (1)
where all components can take on real values. We denote an n × d (rectangular)
matrix M and its d× n transpose Mt as
M =
m11 m12 m13 . . . m1d
m21 m22 m23 . . . m2d
...
...
...
. . .
...
mn1 mn2 mn3 . . . mnd
and (2)
Mt =
m11 m21 . . . mn1
m12 m22 . . . mn2
m13 m23 . . . mn3
...
...
. . .
...
m1d m2d . . . mnd
. (3)
In other words, the jith entry of Mt is the ijth entry of M.
A square (d × d) matrix is called symmetric if its entries obey mij = mji; it is
called skew-symmetric (or anti-symmetric) if mij = −mji. A general matrix is called
non-negative if mij ≥ 0 for all i and j. A particularly important matrix is the identity
matrix, I — a d×d (square) matrix whose diagonal entries are 1’s, and all other entriesidentity
matrix 0. The Kronecker delta function or Kronecker symbol, defined as
Kronecker
delta δij =
{
1 if i = j
0 otherwise,
(4)
can serve to define the entries of an identity matrix. A general diagonal matrix (i.e.,
one having 0 for all off diagonal entries) is denoted diag(m11,m22, ...,mdd), the entries
being the successive elements m11,m22, . . . ,mdd. Addition of vectors and of matrices
is component by component.
We can multiply a vector by a matrix, Mx = y, i.e.,
A.2. LINEAR ALGEBRA 7
m11 m12 . . . m1d
m21 m22 . . . m2d
...
...
. . .
...
mn1 mn2 . . . mnd
x1
x2
...
xd
=
y1
y2
...
...
yn
, (5)
where
yj =
d∑
i=1
mjixi. (6)
Note that the number of columns of M must equal the number of rows of x. Also, if
M is not square, the dimensionality of y differs from that of x.
A.2.2 Inner product
The inner product of two vectors having the same dimensionality will be denoted here inner
productas xty and yields a scalar:
xty =
d∑
i=1
xiyi = y
tx. (7)
It is sometimes also called the scalar product or dot product and denoted x • y, or
more rarely (x, y). The Euclidean norm or length of the vector is Euclidean
norm
‖x‖ =
√
xtx. (8)
we call a vector “normalized” if ‖x‖ = 1. The angle between two d-dimensional
vectors obeys
cos θ =
xty
||x|| ||y||
, (9)
and thus the inner product is a measure of the colinearity of two vectors — a natural
indication of their similarity. In particular, if xty = 0, then the vectors are orthogonal,
and if ||xty|| = ||x|| ||y||, the vectors are colinear. From Eq. 9, we have immediately
the Cauchy-Schwarz inequality, which states
‖xty‖ ≤ ||x|| ||y||. (10)
We say a set of vectors {x1,x2, . . . ,xn} is linearly independent if no vector in the linear
independ-
ence
set can be written as a linear combination of any of the others. Informally, a set of d
linearly independent vectors spans an d-dimensional vector space, i.e., any vector in
that space can be written as a linear combination of such spanning vectors.
A.2.3 Outer product
The outer product (sometimes called matrix product or dyadic product) of two vectors matrix
productyields a matrix
8 APPENDIX A. MATHEMATICAL FOUNDATIONS
M = xyt =
x1
x2
...
xd
(y1 y2 . . . yn) =
x1y1 x1y2 . . . x1yn
x2y1 x2y2 . . . x2yn
...
...
. . .
...
xdy1 xdy2 . . . xdyn
, (11)
that is, the components of M are mij = xiyj . Of course, if the dimensions of x and
y are not the same, then M is not square.
A.2.4 Derivatives of matrices
Suppose f(x) is a scalar-valued function of d variables xi, i = 1, 2, ...d, which we
represent as the vector x. Then the derivative or gradient of f with respect to this
vector is computed component by component, i.e.,
∇f(x) = gradf(x) = ∂f(x)
∂x
=
∂f(x)
∂x1
∂f(x)
∂x2
...
∂f(x)
∂xd
. (12)
If we have an n-dimensional vector-valued function f (note the use of boldface),
of a d-dimensional vector x, we calculate the derivatives and represent them as the
Jacobian matrixJacobian
matrix
J(x) =
∂f(x)
∂x
=
∂f1(x)
∂x1
. . .
∂f1(x)
∂xd
...
. . .
...
∂fn(x)
∂x1
. . .
∂fn(x)
∂xd
. (13)
If this matrix is square, its determinant (Sect. A.2.5) is called simply the Jacobian or
occassionally the Jacobian determinant.
If the entries of M depend upon a scalar parameter θ, we can take the derivative
of M component by component, to get another matrix, as
∂M
∂θ
=
∂m11
∂θ
∂m12
∂θ
. . . ∂m1d
∂θ
∂m21
∂θ
∂m22
∂θ
. . . ∂m2d
∂θ
...
...
. . .
...
∂mn1
∂θ
∂mn2
∂θ
. . . ∂mnd
∂θ
. (14)
In Sect. A.2.6 we shall discuss matrix inversion, but for convenience we give here the
derivative of the inverse of a matrix, M−1:
∂
∂θ
M−1 = −M−1 ∂M
∂θ
M−1. (15)
A.2. LINEAR ALGEBRA 9
Consider a matrix M that is independent of x. The following vector derivative
identities can be verified by writing out the components:
∂
∂x
[Mx] = M (16)
∂
∂x
[ytx] =
∂
∂x
[xty] = y (17)
∂
∂x
[xtMx] = [M + Mt]x. (18)
In the case where M is symmetric (as for instance a covariance matrix, cf. Sect. A.4.10),
then Eq. 18 simplifies to
∂
∂x
[xtMx] = 2Mx. (19)
We first recall the use of second derivatives of a scalar function of a scalar x in
writing a Taylor series (or Taylor expansion) about a point:
f(x) = f(x0) +
df(x)
dx
∣∣∣∣∣
x=x0
(x− x0) +
1
2!
d2f(x)
dx2
∣∣∣∣∣
x=x0
(x− x0)2 + O((x− x0)3). (20)
Analogously, if our scalar-valued f is a instead function of a vector x, we can expand
f(x) in a Taylor series around a point x0:
f(x) = f(x0) +
[
∂f
∂x︸︷︷︸
J
]t
x=x0
(x− x0) +
1
2!
(x− x0)t
[
∂2f
∂x2︸︷︷︸
H
]t
x=x0
(x− x0) + O(||x− x0||3), (21)
where H is the Hessian matrix, the matrix of second-order derivatives of f(·), here Hessian
matrixevaluated at x0. (We shall return in Sect. A.8 to consider the O(·) notation and the
order of a function used in Eq. 21 and below.)
A.2.5 Determinant and trace
The determinant of a d × d (square) matrix is a scalar, denoted |M|, and reveals
properties of the matrix. For instance, if we consider the columns of M as vectors, if
these vectors are not linearly independent, then the determinant vanishes. In pattern
recognition, we have particular interest in the covariance matrix Σ, which contains
the second moments of a sample of data. In this case the absolute value of the
determinant of a covariance matrix is a measure of the d-dimensional hypervolume
of the data that yielded Σ. (It can be shown that the determinant is equal to the
product of the eigenvalues of a matrix, as mentioned in Sec. A.2.7.) If the data
lies in a subspace of the full d-dimensional space, then the columns of Σ are not
linearly independent, and the determinant vanishes. Further, the determinant must
be non-zero for the inverse of a matrix to exist (Sec. A.2.6).
The calculation of the determinant is simple in low dimensions, and a bit more
involved in high dimensions. If M is itself a scalar (i.e., a 1 × 1 matrix M), then
|M | = M . If M is 2×2, then |M| = m11m22−m21m12. The determinant of a general
square matrix can be computed by a method called expansion by minors, and this expansion
by minors
10 APPENDIX A. MATHEMATICAL FOUNDATIONS
leads to a recursive definition. If M is our d × d matrix, we define Mi|j to be the
(d− 1)× (d− 1) matrix obtained by deleting the ith row and the jth column of M:
j
i
m11 m12 · · ·
⊗
· · · · · · m1d
m21 m22 · · ·
⊗
· · · · · · m2d
...
...
. . .
⊗
· · · · · ·
...
...
... · · ·
⊗
· · · · · ·
...⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗
...
... · · ·
⊗
· · ·
. . .
...
md1 md2 · · ·
⊗
· · · · · · mdd
= Mi|j . (22)
Given the determinants |Mx|1|, we can now compute the determinant of M the ex-
pansion by minors on the first column giving
|M| = m11|M1|1| −m21|M2|1|+ m31|M3|1| − · · · ±md1|Md|1|, (23)
where the signs alternate. This process can be applied recursively to the successive
(smaller) matrixes in Eq. 23.
Only for a 3×3 matrix, this determinant calculation can be represented by “sweep-
ing” the matrix, i.e., taking the sum of the products of matrix terms along a diagonal,
where products from upper-left to lower-right are added with a positive sign, and those
from the lower-left to upper-right with a minus sign. That is,
|M| =
∣∣∣∣∣∣
m11 m12 m13
m21 m22 m23
m31 m32 m33
∣∣∣∣∣∣ (24)
= m11m22m33 + m13m21m32 + m12m23m31
−m13m22m31 −m11m23m32 −m12m21m33.
Again, this “sweeping” mnemonic does not work for matrices larger than 3× 3.
For any matrix we have |M| = |Mt|. Furthermore, for two square matrices of
equal size M and N, we have |MN| = |M| |N|.
The trace of a d × d (square) matrix, denoted tr[M], is the sum of its diagonal
elements:
tr[M] =
d∑
i=1
mii. (25)
Both the determinant and trace of a matrix are invariant with respect to rotations of
the coordinate system.
A.2.6 Matrix inversion
So long as its determinant does not vanish, the inverse of a d× d matrix M, denoted
M−1, is the d× d matrix such that
MM−1 = I. (26)
We call the scalar Cij = (−1)i+j |Mi|j | the i, j cofactor or equivalently the cofactor ofcofactor
A.2. LINEAR ALGEBRA 11
the i, j entry of M. As defined in Eq. 22, Mi|j is the (d− 1)× (d− 1) matrix formed
by deleting the ith row and jth column of M. The adjoint of M, written Adj[M], is adjoint
the matrix whose i, j entry is the j, i cofactor of M. Given these definitions, we can
write the inverse of a matrix as
M−1 =
Adj[M]
|M|
. (27)
If M is not square (or if M−1 in Eq. 27 does not exist because the columns of M are
not linearly independent) we typically use instead the pseudoinverse M†, defined as pseudo-
inverse
M† = [MtM]−1Mt. (28)
The pseudoinverse is useful because it insures M†M = I.
A.2.7 Eigenvectors and eigenvalues
Given a d× d matrix M a very important class of linear equations is of the form
Mx = λx (29)
for scalar λ, which can be rewritten
(M− λI)x = 0, (30)
where I the identity matrix, and 0 the zero vector. The solution vector x = ei and
corresponding scalar λ = λi to Eq. 29 are called the eigenvector and associated eigen-
value. There are d (possibly non-distinct) solution vectors {e1, e2, . . . , ed} each with
an associated eigenvalue {λ1, λ2, . . . , λd}. Under multiplication by M the eigenvectors
are changed only in magnitude — not direction:
Mej = λjej . (31)
If M is diagonal, then the eigenvectors are parallel to the coordinate axes.
One method of finding the eigenvectors and eigenvalues is to solve the character-
istic equation (or secular equation), character-
istic
equation
secular
equation
|M− λI| = λd + a1λd−1 + . . . + ad−1λ + ad = 0, (32)
for each of its d (possibly non-distinct) roots λj . For each such root, we then solve a
set of linear equations to find its associated eigenvector ej .
Finally, it can be shown that the determinant of a matrix is just the product of
its eigenvalues:
|M| =
d∏
i=1
λi. (33)
12 APPENDIX A. MATHEMATICAL FOUNDATIONS
A.3 Lagrange optimization
Suppose we seek the position x0 of an extremum of a scalar-valued function f(x),
subject to some constraint. If a constraint can be expressed in the form g(x) = 0,
then we can find the extremum of f(x) as follows. First we form the Lagrangian
function
L(x, λ) = f(x) + λg(x)︸ ︷︷ ︸
=0
, (34)
where λ is a scalar called the Lagrange undetermined multiplier. We convert this con-undeter-
mined
multiplier
strained optimization problem into an unconstrained problem by taking the derivative,
∂L(x, λ)
∂x
=
∂f(x)
∂x
+ λ
∂g(x)
∂x
= 0, (35)
and using standard methods from calculus to solve the resulting equations for λ and
the extremizing value of x. (Note that the last term on the left hand side does not
vanish, in general.) The solution gives the x position of the extremum, and it is a
simple matter of substitution to find the extreme value of f(·) under the constraints.
A.4 Probability Theory
A.4.1 Discrete random variables
Let x be a discrete random variable that can assume any of the finite number m of
different values in the set X = {v1, v2, . . . , vm}. We denote by pi the probability that
x assumes the value vi:
pi = Pr{x = vi}, i = 1, . . . ,m. (36)
Then the probabilities pi must satisfy the following two conditions:
pi ≥ 0 and
m∑
i=1
pi = 1. (37)
Sometimes it is more convenient to express the set of probabilities {p1, p2, . . . , pm}
in terms of the probability mass function P (x), which must satisfy the following twoprobability
mass
function
conditions:
P (x) ≥ 0 and∑
x∈X
P (x) = 1and (38)
∑
x/∈X
P (x) = 0. (39)
A.4. PROBABILITY THEORY 13
A.4.2 Expected values
The expected value, mean or average of the random variable x is defined bymean
E [x] = µ =
∑
x∈X
xP (x) =
m∑
i=1
vipi. (40)
If one thinks of the probability mass function as defining a set of point masses, with
pi being the mass concentrated at x = vi, then the expected value µ is just the center
of mass. Alternatively, we can interpret µ as the arithmetic average of the values in a
large random sample. More generally, if f(x) is any function of x, the expected value
of f is defined by
E [f(x)] =
∑
x∈X
f(x)P (x). (41)
Note that the process of forming an expected value is linear, in that if α1 and α2 are
arbitrary constants,
E [α1f1(x) + α2f2(x)] = α1E [f1(x)] + α2E [f2(x)]. (42)
It is sometimes convenient to think of E as an operator — the (linear) expectation
operator. Two important special-case expectations are the second moment and the expectation
operator
second
moment
variance:
variance
E [x2] =
∑
x∈X
x2P (x) (43)
Var[x] ≡ σ2 = E [(x− µ)2] =
∑
x∈X
(x− µ)2P (x), (44)
where σ is the standard deviation of x. The variance can be viewed as the moment of standard
deviationinertia of the probability mass function. The variance is never negative, and is zero
if and only if all of the probability mass is concentrated at one point.
The standard deviation is a simple but valuable measure of how far values of x
are likely to depart from the mean. Its very name suggests that it is the standard
or typical amount one should expect a randomly drawn value for x to deviate or
differ from µ. Chebyshev’s inequality (or Bienaymé-Chebyshev inequality) provides a Chebyshev’s
inequalitymathematical relation between the standard deviation and |x− µ|:
Pr{|x− µ| > nσ} ≤ 1
n2
. (45)
This inequality is not a tight bound (and it is useless for n < 1); a more practical rule
of thumb, which strictly speaking is true only for the normal distribution, is that 68%
of the values will lie within one, 95% within two, and 99.7% within three standard
deviations of the mean (Fig. A.1). Nevertheless, Chebyshev’s inequality shows the
strong link between the standard deviation and the spread of a distribution. In
14 APPENDIX A. MATHEMATICAL FOUNDATIONS
addition, it suggests that |x−µ|/σ is a meaningful normalized measure of the distance
from x to the mean (cf. Sect. A.4.12).
By expanding the quadratic in Eq. 44, it is easy to prove the useful formula
Var[x] = E [x2]− (E [x])2. (46)
Note that, unlike the mean, the variance is not linear. In particular, if y = αx, where
α is a constant, then Var[y] = α2Var[x]. Moreover, the variance of the sum of two
random variables is usually not the sum of their variances. However, as we shall see
below, variances do add when the variables involved are statistically independent.
In the simple but important special case in which x is binary valued (say, v1 = 0
and v2 = 1), we can obtain simple formulas for µ and σ. If we let p = Pr{x = 1},
then it is easy to show that
µ = p and
σ =
√
p(1− p). (47)
A.4.3 Pairs of discrete random variables
Let x and y be random variables which can take on values in X = {v1, v2, . . . , vm},
and Y = {w1, w2, . . . , wn}, respectively. We can think of (x, y) as a vector or a point
in the product space of x and y. For each possible pair of values (vi, wj) we have aproduct
space joint probability pij = Pr{x = vi, y = wj}. These mn joint probabilities pij are non-
negative and sum to 1. Alternatively, we can define a joint probability mass function
P (x, y) for which
P (x, y) ≥ 0 and∑
x∈X
∑
y∈Y
P (x, y) = 1. (48)
The joint probability mass function is a complete characterization of the pair of ran-
dom variables (x, y); that is, everything we can compute about x and y, individually
or together, can be computed from P (x, y). In particular, we can obtain the separate
marginal distributions for x and y by summing over the unwanted variable:marginal
distribu-
tion
Px(x) =
∑
y∈Y
P (x, y)
Py(y) =
∑
x∈X
P (x, y). (49)
We will occassionally use subscripts, as in Eq. 49, to emphasize the fact that
Px(x) has a different functional form than Py(y). It is common to omit them and
write simply P (x) and P (y) whenever the context makes it clear that these are in
fact two different functions — rather than the same function merely evaluated with
different variables.
A.4. PROBABILITY THEORY 15
A.4.4 Statistical independence
Variables x and y are said to be statistically independent if and only if
P (x, y) = Px(x)Py(y). (50)
We can understand such independence as follows. Suppose that pi = Pr{x = vi} is
the fraction of the time that x = vi, and qj = Pr{y = wj} is the fraction of the time
that y = wj . Consider those situations where x = vi. If it is still true that the fraction
of those situations in which y = wj is the same value qj , it follows that knowing the
value of x did not give us any additional knowledge about the possible values of y;
in that sense y is independent of x. Finally, if x and y are statistically independent,
it is clear that the fraction of the time that the specific pair of values (vi, wj) occurs
must be the product of the fractions piqj = P (vi)P (wj).
A.4.5 Expected values of functions of two variables
In the natural extension of Sect. A.4.2, we define the expected value of a function
f(x, y) of two random variables x and y by
E [f(x, y)] =
∑
x∈X
∑
y∈Y
f(x, y)P (x, y), (51)
and as before the expectation operator E is linear:
E [α1f1(x, y) + α2f2(x, y)] = α1E [f1(x, y)] + α2E [f2(x, y)]. (52)
The means (first moments) and variances (second moments) are:
µx = E [x] =
∑
x∈X
∑
y∈Y
xP (x, y)
µy = E [y] =
∑
x∈X
∑
y∈Y
yP (x, y)
σ2x = V [x] = E [(x− µx)2] =
∑
x∈X
∑
y∈Y
(x− µx)2P (x, y)
σ2y = V [y] = E [(y − µy)2] =
∑
x∈X
∑
y∈Y
(y − µy)2P (x, y). (53)
An important new “cross-moment” can now be defined, the covariance of x and covar-
iancey:
σxy = E [(x− µx)(y − µy)] =
∑
x∈X
∑
y∈Y
(x− µx)(y − µy)P (x, y). (54)
We can summarize Eqs. 53 & 54 using vector notation as:
µ = E [x] =
∑
x∈{XY}
xP (x) (55)
Σ = E [(x− µ)(x− µ)t], (56)
16 APPENDIX A. MATHEMATICAL FOUNDATIONS
where {XY} respresents the space of all possible values for all components of x and
Σ is the covariance matrix (cf., Sect. A.4.9).
The covariance is one measure of the degree of statistical dependence between x
and y. If x and y are statistically independent, then σxy = 0. If σxy = 0, the variables
x and y are said to be uncorrelated. It does not follow that uncorrelated variables mustuncorre-
lated be statistically independent — covariance is just one measure of dependence. However,
it is a fact that uncorrelated variables are statistically independent if they have a
multivariate normal distribution, and in practice statisticians often treat uncorrelated
variables as if they were statistically independent. If α is a constant and y = αx, which
is a case of strong statistical dependence, it is also easy to show that σxy = ασ2x. Thus,
the covariance is positive if x and y both increase or decrease together, and is negative
if y decreases when x increases.
There is an important Cauchy-Schwarz inequality for the variances σx and σy andCauchy-
Schwarz
inequality
the covariance σxy. It can be derived by observing that the variance of a random
variable is never negative, and thus the variance of λx + y must be non-negative no
matter what the value of the scalar λ. This leads to the famous inequality
σ2xy ≤ σ2xσ2y, (57)
which is analogous to the vector inequality (xty)2 ≤ ‖x‖2 ‖y‖2 given in Eq. 8.
The correlation coefficient, defined ascorrelation
coefficient
ρ =
σxy
σxσy
, (58)
is a normalized covariance, and must always be between −1 and +1. If ρ = +1,
then x and y are maximally positively correlated, while if ρ = −1, they are maxi-
mally negatively correlated. If ρ = 0, the variables are uncorrelated. It is common for
statisticians to consider variables to be uncorrelated for practical purposes if the mag-
nitude of their correlation coefficient is below some threshold, such as 0.05, although
the threshold that makes sense does depend on the actual situation.
If x and y are statistically independent, then for any two functions f and g
E [f(x)g(y)] = E [f(x)]E [g(y)], (59)
a result which follows from the definition of statistical independence and expectation.
Note that if f(x) = x − µx and g(y) = y − µy, this theorem again shows that
σxy = E [(x− µx)(y − µy)] is zero if x and y are statistically independent.
A.4.6 Conditional probability
When two variables are statistically dependent, knowing the value of one of them
lets us get a better estimate of the value of the other one. This is expressed by the
following definition of the conditional probability of x given y:
Pr{x = vi|y = wj} =
Pr{x = vi, y = wj}
Pr{y = wj}
, (60)
or, in terms of mass functions,
P (x|y) = P (x, y)
P (y)
. (61)
A.4. PROBABILITY THEORY 17
Note that if x and y are statistically independent, this gives P (x|y) = P (x). That
is, when x and y are independent, knowing the value of y gives you no information
about x that you didn’t already know from its marginal distribution P (x).
Consider a simple illustration of a two-variable binary case where both x and y
are either 0 or 1. Suppose that a large number n of pairs of xy-values are randomly
produced. Let nij be the number of pairs in which we find x = i and y = j, i.e., we
see the (0, 0) pair n00 times, the (0, 1) pair n01 times, and so on, where n00 + n01 +
n10 + n11 = n. Suppose we pull out those pairs where y = 1, i.e., the (0, 1) pairs and
the (1, 1) pairs. Clearly, the fraction of those cases in which x is also 1 is
n11
n01 + n11
=
n11/n
(n01 + n11)/n
. (62)
Intuitively, this is what we would like to get for P (x|y) when y = 1 and n is large.
And, indeed, this is what we do get, because n11/n is approximately P (x, y) and
n11/n
(n01+n11)/n
is approximately P (y) for large n.
A.4.7 The Law of Total Probability and Bayes’ rule
The Law of Total Probability states that if an event A can occur in m different ways
A1, A2, . . . , Am, and if these m subevents are mutually exclusive — that is, cannot
occur at the same time — then the probability of A occurring is the sum of the
probabilities of the subevents Ai. In particular, the random variable y can assume
the value y in m different ways — with x = v1, with x = v2, . . ., and x = vm. Because
these possibilities are mutually exclusive, it follows from the Law of Total Probability
that P (y) is the sum of the joint probability P (x, y) over all possible values for x.
Formally we have
P (y) =
∑
x∈X
P (x, y). (63)
But from the definition of the conditional probability P (y|x) we have
P (x, y) = P (y|x)P (x), (64)
and after rewriting Eq. 64 with x and y exchanged and a trivial math, we obtain
P (x|y) = P (y|x)P (x)∑
x∈X
P (y|x)P (x)
, (65)
or in words,
posterior =
likelihood× prior
evidence
,
where these terms are discussed more fully in Chapt. ??.
Equation 65 is called Bayes’ rule. Note that the denominator, which is just P (y), is
obtained by summing the numerator over all x values. By writing the denominator in
this form we emphasize the fact that everything on the right-hand side of the equation
is conditioned on x. If we think of x as the important variable, then we can say that
the shape of the distribution P (x|y) depends only on the numerator P (y|x)P (x); the
18 APPENDIX A. MATHEMATICAL FOUNDATIONS
denominator is just a normalizing factor, sometimes called the evidence, needed to evidence
insure that the P (x|y) sum to one.
The standard interpretation of Bayes’ rule is that it “inverts” statistical connec-
tions, turning P (y|x) into P (x|y). Suppose that we think of x as a “cause” and y as
an “effect” of that cause. That is, we assume that if the cause x is present, it is easy
to determine the probability of the effect y being observed; the conditional probability
function P (y|x) — the likelihood — specifies this probability explicitly. If we observelikelihood
the effect y, it might not be so easy to determine the cause x, because there might
be several different causes, each of which could produce the same observed effect.
However, Bayes’ rule makes it easy to determine P (x|y), provided that we know both
P (y|x) and the so-called prior probability P (x), the probability of x before we makeprior
any observations about y. Said slightly differently, Bayes’ rule shows how the prob-
ability distribution for x changes from the prior distribution P (x) before anything is
observed about y to the posterior P (x|y) once we have observed the value of y.posterior
A.4.8 Vector random variables
To extend these results from two variables x and y to d variables x1, x2, . . . , xd, it is
convenient to employ vector notation. As given by Eq. 48, the joint probability mass
function P (x) satisfies P (x) ≥ 0 and
∑
P (x) = 1, where the sum extends over all
possible values for the vector x. Note that P (x) is a function of d variables, and can
be a very complicated, multi-dimensional function. However, if the random variables
xi are statistically independent, it reduces to the product
P (x) = Px1(x1)Px2(x2) · · ·Pxd(xd)
=
d∏
i=1
Pxi(xi). (66)
where we have used the subscripts just to emphasize the fact that the marginal distri-
butions will generally have a different form. Here the separate marginal distributions
Pxi(xi) can be obtained by summing the joint distribution over the other variables.
In addition to these univariate marginals, other marginal distributions can be ob-
tained by this use of the Law of Total Probability. For example, suppose that we have
P (x1, x2, x3, x4, x5) and we want P (x1, x4), we merely calculate
P (x1, x4) =
∑
x2
∑
x3
∑
x5
P (x1, x2, x3, x4, x5). (67)
One can define many different conditional distributions, such as P (x1, x2|x3) or
P (x2|x1, x4, x5). For example,
P (x1, x2|x3) =
P (x1, x2, x3)
P (x3)
, (68)
where all of the joint distributions can be obtained from P (x) by summing out the un-
wanted variables. If instead of scalars we have vector variables, then these conditional
distributions can also be written as
P (x1|x2) =
P (x1,x2)
P (x2)
, (69)
A.4. PROBABILITY THEORY 19
and likewise, in vector form, Bayes’ rule becomes
P (x1|x2) =
P (x2|x1)P (x1)∑
x1
P (x2|x1)P (x1)
. (70)
A.4.9 Expectations, mean vectors and covariance matrices
The expected value of a vector is defined to be the vector whose components are
the expected values of the original components. Thus, if f(x) is an n-dimensional,
vector-valued function of the d-dimensional random vector x,
f(x) =
f1(x)
f2(x)
...
fn(x)
, (71)
then the expected value of f is defined by
E [f ] =
E [f1(x)]
E [f2(x)]
...
E [fn(x)]
=
∑
x
f(x)P (x). (72)
In particular, the d-dimensional mean vector µ is defined by mean
vector
µ = E [x] =
E [x1]
E [x2]
...
E [xd]
=
µ1
µ2
...
µd
=
∑
x
xP (x). (73)
Similarly, the covariance matrix Σ is defined as the (square) matrix whose ijth element covariance
matrixσij is the covariance of xi and xj :
σij = σji = E [(xi − µi)(xj − µj)] i, j = 1 . . . d, (74)
as we saw in the two-variable case of Eq. 54. Therefore, in expanded form we have
Σ =
E [(x1 − µ1)(x1 − µ1)] E [(x1 − µ1)(x2 − µ2)] . . . E [(x1 − µ1)(xd − µd)]
E [(x2 − µ2)(x1 − µ1)] E [(x2 − µ2)(x2 − µ2)] . . . E [(x2 − µ2)(xd − µd)]
...
...
. . .
...
E [(xd − µd)(x1 − µ1)] E [(xd − µd)(x2 − µ2)] . . . E [(xd − µd)(xd − µd)]
=
σ11 σ12 . . . σ1d
σ21 σ22 . . . σ2d
...
...
. . .
...
σd1 σd2 . . . σdd
=
σ21 σ12 . . . σ1d
σ21 σ
2
2 . . . σ2d
...
...
. . .
...
σd1 σd2 . . . σ
2
d
. (75)
We can use the vector product (x− µ)(x− µ)t, to write the covariance matrix as
Σ = E [(x− µ)(x− µ)t]. (76)
20 APPENDIX A. MATHEMATICAL FOUNDATIONS
Thus, Σ is symmetric, and its diagonal elements are just the variances of the in-
dividual elements of x, which can never be negative; the off-diagonal elements are
the covariances, which can be positive or negative. If the variables are statistically
independent, the covariances are zero, and the covariance matrix is diagonal. The
analog to the Cauchy-Schwarz inequality comes from recognizing that if w is any d-
dimensional vector, then the variance of wtx can never be negative. This leads to the
requirement that the quadratic form wtΣw never be negative. Matrices for which
this is true are said to be positive semi-definite; thus, the covariance matrix Σ must
be positive semi-definite. It can be shown that this is equivalent to the requirement
that none of the eigenvalues of Σ can be negative.
A.4.10 Continuous random variables
When the random variable x can take values in the continuum, it no longer makes
sense to talk about the probability that x has a particular value, such as 2.5136,
because the probability of any particular exact value will almost always be zero.
Rather, we talk about the probability that x falls in some interval (a, b); instead of
having a probability mass function P (x) we have a probability mass density functionmass
density p(x). The mass density has the property that
Pr{x ∈ (a, b)} =
b∫
a
p(x) dx. (77)
The name density comes by analogy with material density. If we consider a small
interval (a, a + ∆x) over which p(x) is essentially constant, having value p(a), we see
that p(a) = Pr{x ∈ (a, a + ∆x)}/∆x. That is, the probability mass density at x = a
is the probability mass Pr{x ∈ (a, a + ∆x)} per unit distance. It follows that the
probability density function must satisfy
p(x) ≥ 0 and
∞∫
−∞
p(x) dx = 1. (78)
In general, most of the definitions and formulas for discrete random variables carry
over to continuous random variables with sums replaced by integrals. In particular,
the expected value, mean and variance for a continuous random variable are defined
by
E [f(x)] =
∞∫
−∞
f(x)p(x) dx
µ = E [x] =
∞∫
−∞
xp(x) dx (79)
Var[x] = σ2 = E [(x− µ)2] =
∞∫
−∞
(x− µ)2p(x) dx,
A.4. PROBABILITY THEORY 21
and, as in Eq. 46, the variance obeys σ2 = E [x2]− (E [x])2.
The multivariate situation is similarly handled with continuous random vectors x.
The probability density function p(x) must satisfy
p(x) ≥ 0 and
∞∫
−∞
p(x) dx = 1, (80)
where the integral is understood to be a d-fold, multiple integral, and where dx is the
element of d-dimensional volume dx = dx1dx2 · · · dxd. The corresponding moments
for a general n-dimensional vector-valued function are
E [f(x)] =
∞∫
−∞
∞∫
−∞
· · ·
∞∫
−∞
f(x)p(x) dx1dx2 . . . dxd =
∞∫
−∞
f(x)p(x) dx (81)
and for the particular d-dimensional functions as above, we have
µ = E [x] =
∞∫
−∞
xp(x) dx (82)
Σ = E [(x− µ)(x− µ)t] =
∞∫
−∞
(x− µ)(x− µ)tp(x) dx.
If the components of x are statistically independent, then the joint probability density
function factors as
p(x) =
d∏
i=1
p(xi) (83)
and the covariance matrix is diagonal.
Conditional probability density functions are defined just as conditional mass func-
tions. Thus, for example, the density for x given y is given by
p(x|y) = p(x, y)
p(y)
(84)
and Bayes’ rule for density functions is
p(x|y) = p(y|x)p(x)∞∫
−∞
p(y|x)p(x) dx
, (85)
and likewise for the vector case.
Occassionally we will need to take the expectation with respect to a subset of the
variables, and in that case we must show this as a subscript, for instance
Ex1 [f(x1, x2)] =
∞∫
−∞
f(x1, x2)p(x1) dx1. (86)
22 APPENDIX A. MATHEMATICAL FOUNDATIONS
A.4.11 Distributions of sums of independent random variables
It frequently happens that we know the densities for two independent random variables
x and y, and we need to know the density of their sum z = x+ y. It is easy to obtain
the mean and the variance of this sum:
µz = E [z] = E [x + y] = E [x] + E [y] = µx + µy,
σ2z = E [(z − µz)2] = E [(x + y − (µx + µy))2] = E [((x− µx) + (y − µy))2]
= E [(x− µx)2] + 2 E [(x− µx)(y − µy)]︸ ︷︷ ︸
=0
+E [(y − µy)2] (87)
= σ2x + σ
2
y,
where we have used the fact that the cross-term factors into E [x− µx]E [y− µy] when
x and y are independent; in this case the product is manifestly zero, since each of
the component expectations vanishes. Thus, in words, the mean of the sum of two
independent random variables is the sum of their means, and the variance of their
sum is the sum of their variances. If the variables are random yet not independent —
for instance y = −x, where x is randomly distributed — then the variance is not the
sum of the component variances.
It is only slightly more difficult to work out the exact probability density function
for z = x+y from the separate density functions for x and y. The probability that z is
between ζ and ζ+∆z can be found by integrating the joint density p(x, y) = p(x)p(y)
over the thin strip in the xy-plane between the lines x + y = ζ and x + y = ζ + ∆z.
It follows that, for small ∆z,
Pr{ζ < z < ζ + ∆z} =
{ ∞∫
−∞
p(x)p(ζ − x) dx
}
∆z, (88)
and hence that the probability density function for the sum is the convolution of theconvolution
probability density functions for the components:
p(z) = p(x) � p(y) =
∞∫
−∞
p(x)p(z − x) dx. (89)
As one would expect, these results generalize. It is not hard to show that:
• The mean of the sum of d independent random variables x1, x2, . . . , xd is the
sum of their means. (In fact the variables need not be independent for this to
hold.)
• The variance of the sum is the sum of their variances.
• The probability density function for the sum is the convolution of the separate
density functions:
p(z) = p(x1) � p(x2) � . . . � p(xd). (90)
A.4. PROBABILITY THEORY 23
A.4.12 Univariate normal density
One of the most important results of probability theory is the Central Limit Theorem,Central
Limit
Theorem
which states that, under various conditions, the distribution for the sum of d inde-
pendent random variables approaches a particular limiting form known as the normal
distribution. As such, the normal or Gaussian probability density function is very Gaussian
important, both for theoretical and practical reasons. In one dimension, it is defined
by
p(x) =
1√
2πσ
e−1/2((x−µ)/σ)
2
. (91)
The normal density is traditionally described as a “bell-shaped curve”; it is com-
pletely determined by the numerical values for two parameters, the mean µ and the
variance σ2. This is often emphasized by writing p(x) ∼ N(µ, σ2), which is read as
“x is distributed normally with mean µ and variance σ2.” The distribution is sym-
metrical about the mean, the peak occurring at x = µ and the width of the “bell”
is proportional to the standard deviation σ. The parameters of a normal density in
Eq. 91 satisfy the following equations:
E [1] =
∞∫
−∞
p(x) dx = 1
E [x] =
∞∫
−∞
x p(x) dx = µ (92)
E [(x− µ)2] =
∞∫
−∞
(x− µ)2p(x) dx = σ2.
Normally distributed data points tend to cluster about the mean. Numerically, the
probabilities obey
Pr{|x− µ| ≤ σ} � 0.68
Pr{|x− µ| ≤ 2σ} � 0.95 (93)
Pr{|x− µ| ≤ 3σ} � 0.997,
as shown in Fig. A.1.
A natural measure of the distance from x to the mean µ is the distance |x − µ|
measured in units of standard deviations:
r =
|x− µ|
σ
, (94)
the Mahalanobis distance from x to µ. (In the one-dimensional case, this is sometimes Mahalanobis
distancecalled the z-score.) Thus for instance the probability is 0.95 that the Mahalanobis
distance from x to µ will be less than 2. If a random variable x is modified by
(a) subtracting its mean and (b) dividing by its standard deviation, it is said to be
standardized. Clearly, a standardized normal random variable u = (x−µ)/σ has zero standardized
mean and unit standard deviation, that is,
24 APPENDIX A. MATHEMATICAL FOUNDATIONS
-4 -3 -2 -1 0 1 2 3 4
p(u)
68%
95%
99.7%
u
Figure A.1: A one-dimensional Gaussian distribution, p(u) ∼ N(0, 1), has 68% of its
probability mass in the range |u| ≤ 1, 95% in the range |u| ≤ 2, and 99.7% in the
range |u| ≤ 3.
p(u) =
1√
2π
e−u
2/2, (95)
which can be written as p(u) ∼ N(0, 1). Table A.1 shows the probability that a value,
chosen at random according to p(u) ∼ N(0, 1), differs from the mean value by less
than a criterion z.
Table A.1: The probability a sample drawn from a standardized Gaussian has absolute
value less than a criterion, i.e., Pr[|u| ≤ z]
z Pr[|u| ≤ z] z Pr[|u| ≤ z] z Pr[|u| ≤ z]
0.0 0.0 1.0 0.682 2.0 0.954
0.1 0.080 1.1 0.728 2.1 0.963
0.2 0.158 1.2 0.770 2.326 0.980
0.3 0.236 1.3 0.806 2.5 0.988
0.4 0.310 1.4 0.838 2.576 0.990
0.5 0.382 1.5 0.866 3.0 0.9974
0.6 0.452 1.6 0.890 3.090 0.9980
0.7 0.516 1.7 0.910 3.291 0.999
0.8 0.576 1.8 0.928 3.5 0.9996
0.9 0.632 1.9 0.942 4.0 0.99994
A.5 Gaussian derivatives and integrals
Because of the prevalence of Gaussian functions throughout statistical pattern recog-
nition, we often have occassion to integrate and differentiate them. The first three
A.5. GAUSSIAN DERIVATIVES AND INTEGRALS 25
derivatives of a one-dimensional (standardized) Gaussian are
∂
∂x
[
1√
2πσ
e−x
2/(2σ2)
]
= −x√
2πσ3
e−x
2/(2σ2) =
−x
σ2
p(x)
∂2
∂x2
[
1√
2πσ
e−x
2/(2σ2)
]
= 1√
2πσ5
(
−σ2 + x2
)
e−x
2/(2σ2) =
−σ2 + x2
σ4
p(x) (96)
∂3
∂x3
[
1√
2πσ
e−x
2/(2σ2)
]
= 1√
2πσ7
(
3xσ2 − x3
)
e−x
2/(2σ2) =
−3xσ2 − x3
σ6
p(x),
and are shown in Fig. A.2.
-4 -2 2 4
x
f '''
f
f '
f ''
Figure A.2: A one-dimensional Gaussian distribution and its first three derivatives,
shown for f(x) ∼ N(0, 1).
An important finite integral of the Gaussian is the so-called error function, defined error
functionas
erf(u) =
√
2
π
u∫
0
e−x
2/2dx. (97)
As can be seen from Fig. A.1, erf(0) = 0, erf(1) = 0.68 and lim
x→∞
erf(x) = 1. There
is no closed analytic form for the error function, and thus we typically use tables,
approximations or numerical integration for its evaluation (Fig. A.3).
In calculating moments of Gaussians, we need the general integral of powers of x
weighted by a Gaussian. Recall first the definition of a gamma function gamma
function
Γ(n + 1) =
∞∫
0
xne−xdx, (98)
where the gamma function obeys
Γ(n) = nΓ(n− 1) (99)
and Γ(1/2) =
√
π. For n an integer we have Γ(n+1) = n ×(n−1)×(n−2) . . . 1 = n!,
read “n factorial.” factorial
26 APPENDIX A. MATHEMATICAL FOUNDATIONS
1 2 3 4
0.2
0.4
0.6
0.8
1
erf(u)1-erf(u)
1/u2
u
Figure A.3: The error function corresponds to the area under a standardized Gaussian
(Eq. 97) between −u and u, i.e., it describes the probability that a sample drawn
from a standardized Gaussian obeys |x| ≤ u. Thus, the complementary probability,
1 − erf(u) is the probability that a sample is chosen with |x| > u. Chebyshev’s
inequality states that for an arbitrary distribution having standard deviation = 1,
this latter probability is bounded by 1/u2. As shown, this bound is quite loose for a
Gaussian.
Changing variables in Eq. 98, we find the moments of a (normalized) Gaussian
distribution as
2
∞∫
0
xn
e−x
2/(2σ2)
√
2πσ
dx =
2n/2σn√
π
Γ
(
n + 1
2
)
, (100)
where again we have used a pre-factor of 2 and lower integration limit of 0 in order
give non-trivial (i.e., non-vanishing) results for odd n.
A.5.1 Multivariate normal densities
Normal random variables have many desirable theoretical properties. For example, it
turns out that the convolution of two Gaussian functions is again a Gaussian function,
and thus the distribution for the sum of two independent normal random variables is
again normal. In fact, sums of dependent normal random variables also have normal
distributions. Suppose that each of the d random variables xi is normally distributed,
each with its own mean and variance: p(xi) ∼ N(µi, σ2i ). If these variables are
independent, their joint density has the form
p(x) =
d∏
i=1
p(xi) =
d∏
i=1
1√
2πσi
e−1/2((xi−µi)/σi)
2
=
1
(2π)d/2
d∏
i=1
σi
exp
[
−1
2
d∑
i=1
(
xi − µi
σi
)2]
. (101)
This can be written in a compact matrix form if we observe that for this case the
covariance matrix is diagonal, i.e.,
A.5. GAUSSIAN DERIVATIVES AND INTEGRALS 27
Σ =
σ21 0 . . . 0
0 σ22 . . . 0
…
…
. . .
…
0 0 . . . σ2d
, (102)
and hence the inverse of the covariance matrix is easily written as
Σ−1 =
1/σ21 0 . . . 0
0 1/σ22 . . . 0
…
…
. . .
…
0 0 . . . 1/σ2d
. (103)
Thus, the exponent in Eq. 101 can be rewritten using
d∑
i=1
(
xi − µi
σi
)2
= (x− µ)tΣ−1(x− µ). (104)
Finally, by noting that the determinant of Σ is just the product of the variances, we
can write the joint density compactly in terms of the quadratic form
p(x) =
1
(2π)d/2|Σ|1/2
e
−1
2
(x− µ)tΣ−1(x− µ)
. (105)
This is the general form of a multivariate normal density function, where the covari-
ance matrix Σ is no longer required to be diagonal. With a little linear algebra, it
can be shown that if x obeys this density function, then
µ = E [x] =
∞∫
−∞
x p(x) dx
Σ = E [(x− µ)(x− µ)t] =
∞∫
−∞
(x− µ)(x− µ)tp(x) dx, (106)
just as one would expect. Multivariate normal data tend to cluster about the mean
vector, µ, falling in an ellipsoidally-shaped cloud whose principal axes are the eigen-
vectors of the covariance matrix. The natural measure of the distance from x to the
mean µ is provided by the quantity
r2 = (x− µ)tΣ−1(x− µ), (107)
which is the square of the Mahalanobis distance from x to µ. It is not as easy
to standardize a vector random variable (reduce it to zero mean and unit covariance
matrix) as it is in the univariate case. The expression analogous to u = (x−µ)/σ is u =
Σ−1/2(x−µ), which involves the “square root” of the inverse of the covariance matrix.
The process of obtaining Σ−1/2 requires finding the eigenvalues and eigenvectors of
Σ, and is just a bit beyond the scope of this Appendix.
28 APPENDIX A. MATHEMATICAL FOUNDATIONS
A.5.2 Bivariate normal densities
It is illuminating to look at the bivariate normal density, that is, the case of two
normally distributed random variables x1 and x2. It is convenient to define σ21 =
σ11, σ
2
2 = σ22, and to introduce the correlation coefficient ρ defined by
ρ =
σ12
σ1σ2
. (108)
With this notation, the covariance matrix becomes
Σ =
[
σ11 σ12
σ21 σ22
]
=
[
σ21 ρσ1σ2
ρσ1σ2 σ
2
2
]
, (109)
and its determinant simplifies to
|Σ| = σ21σ22(1− ρ2). (110)
Thus, the inverse covariance matrix is given by
Σ−1 =
1
σ21σ
2
2(1− ρ2)
[
σ22 −ρσ1σ2
−ρσ1σ2 σ21
]
=
1
1− ρ2
[
1/σ21 −ρ/(σ1σ2)
−ρ/(σ1σ2 1/σ22
]
. (111)
Next we explicitly expand the quadratic form in the normal density:
(x− µ)tΣ−1(x− µ)
= [(x1 − µ1) (x2 − µ2)]
1
1− ρ2
[
1/σ21 −ρ/(σ1σ2)
−ρ/(σ1σ2) 1/σ22
] [
x1 − µ1
x2 − µ2
]
=
1
1− ρ2
[(
x1 − µ1
σ1
)2
− 2ρ
(
x1 − µ1
σ1
) (
x2 − µ2
σ2
)
+
(
x2 − µ2
σ2
)2]
. (112)
Thus, the general bivariate normal density has the form
px1x2(x1, x2) =
1
2πσ1σ2
√
1− ρ2
× (113)
exp
[
− 1
2(1− ρ2)
[(x1 − µ1
σ1
)2
− 2ρ
(x1 − µ1
σ1
)(x2 − µ2
σ2
)
+
(x2 − µ2
σ2
)2]]
.
As we can see from Fig. A.4, p(x1, x2) is a hill-shaped surface over the x1x2 plane.
The peak of the hill occurs at the point (x1, x2) = (µ1, µ2), i.e., at the mean vector µ.
The shape of the hump depends on the two variances σ21 and σ
2
2 , and the correlation
coefficient ρ. If we slice the surface with horizontal planes parallel to the x1x2 plane,
we obtain the so-called level curves, defined by the locus of points where the quadratic
form
(x1 − µ1
σ1
)2
− 2ρ
(x1 − µ1
σ1
)(x2 − µ2
σ2
)
+
(x2 − µ2
σ2
)2
(114)
A.5. GAUSSIAN DERIVATIVES AND INTEGRALS 29
is constant. It is not hard to show that |ρ| ≤ 1, and that this implies that the level
curves are ellipses. The x and y extent of these ellipses are determined by the variances
σ21 and σ
2
2 , and their eccentricity is determined by ρ. More specifically, the principal
axes of the ellipse are in the direction of the eigenvectors ei of Σ, and the different principal
axeswidths in these directions
√
λi. For instance, if ρ = 0, the principal axes of the ellipses
are parallel to the coordinate axes, and the variables are statistically independent. In
the special cases where ρ = 1 or ρ = −1, the ellipses collapse to straight lines. Indeed,
the joint density becomes singular in this situation, because there is really only one
independent variable. We shall avoid this degeneracy by assuming that |ρ| < 1.
p(x)
x1
µ
µ2|1
x2
x1ˆ
Figure A.4: A two-dimensional Gaussian having mean µ and non-diagonal covariance
Σ. If the value on one variable is known, for instance x1 = x̂1, the distribution over
the other variable is Gaussian with mean µ2|1.
One of the important properties of the multivariate normal density is that all
conditional and marginal probabilities are also normal. To find such a density explic-
itly, which we denote px2|x1(x2|x1), we substitute our formulas for px1x2(x1, x2) and
px1(x1) in the defining equation
px2|x1(x2|x1) =
px1x2(x1, x2)
px1(x1)
=
[
1
2πσ1σ2
√
1− ρ2
e
− 1
2(1−ρ2)
[(
x1−µ1
σ1
)2
−2ρ
(
x1−µ1
σ1
)
+
(
x2−µ2
σ2
)2]]
×
[√
2πσ1e
1
2
(
x1−µ1
σ1
)2]
(115)
=
1
√
2πσ2
√
1− ρ2
exp
[
− 1
2(1− ρ2)
[
x2 − µ2
σ2
− ρx1 − µ1
σ1
]2]
=
1
√
2πσ2
√
1− ρ2
exp
−1
2
(
x2 − [µ2 + ρσ2σ1 (x1 − µ1)]
σ2
√
1− ρ2
)2
.
30 APPENDIX A. MATHEMATICAL FOUNDATIONS
Thus, we have verified that the conditional density px1|x2(x1|x2) is a normal distri-
bution. Moreover, we have explicit formulas for the conditional mean µ2|1 and theconditional
mean conditional variance σ2
2|1:
µ2|1 = µ2 + ρ
σ2
σ1
(x1 − µ1) and
σ22|1 = σ
2
2(1− ρ2), (116)
as illustrated in Fig. A.4.
These formulas provide some insight into the question of how knowledge of the
value of x1 helps us to estimate x2. Suppose that we know the value of x1. Then
a natural estimate for x2 is the conditional mean, µ2|1. In general, µ2|1 is a linear
function of x1; if the correlation coefficient ρ is positive, the larger the value of x1,
the larger the value of µ2|1. If it happens that x1 is the mean value µ1, then the best
we can do is to guess that x2 is equal to µ2. Also, if there is no correlation between
x1 and x2, we ignore the value of x1, whatever it is, and we always estimate x2 by
µ2. Note that in that case the variance of x2, given that we know x1, is the same
as the variance for the marginal distribution, i.e., σ2
2|1 = σ
2
2 . If there is correlation,
knowledge of the value of x1, whatever the value is, reduces the variance. Indeed,
with 100% correlation there is no variance left in x2 when the value of x1 is known.
A.6 Hypothesis testing
Suppose samples are drawn either from distribution D0 or they are not. In pattern
classification, we seek to determine which distribution was the source of any sample,
and if it is indeed D0, we would classify the point accordingly, into ω1, say. Hypothesis
testing addresses a somewhat different but related problem. We assume initially that
distribution D0 is the source of the patterns; this is called the null hypothesis, and
often denoted H0. Based on the value of any observed sample we ask whether we can
reject the null hypothesis, that is, state with some degree of confidence (expressed as
a probability) that the sample did not come from D0.
For instance, D0 might be a standardized Gaussian, p(x) ∼ N(0, 1), and our null
hypothesis is that a sample comes from a Gaussian with mean µ = 0. If the value of
a particular sample is small (e.g., x = 0.3), it is likely that it came from the D0; after
all, 68% of the samples drawn from that distribution have absolute value less than
x = 1.0 (cf. Fig. A.1). If a sample’s value is large (e.g., x = 5), then we would be
more confident that it did not come from D0. At such a situation we merely conclude
that (with some probability) the sample was drawn from a distribution with µ "= 0.
Viewed another way, for any confidence — expressed as a probability — there
exists a criterion value such that if the sampled value differs from µ = 0 by more
than that criterion, we reject the null hypothesis. (It is traditional to use confidences
of .01 or .05.) We then say that the difference of the sample from 0 is statistically
significant. For instance, if our null hypothesis is a standardized Gaussian, then ifstatistical
signifi-
cance
our sample differs from the value x = 0 by more than 2.576, we could reject the null
hypothesis “at the .01 confidence level,” as can be deduced from Table A.1. A more
sophisticated analysis could be applied if several samples are all drawn from D0 or
if the null hypothesis involved a distribution other than a Gaussian. Of course, this
usage of “significance” applies only to the statistical properties of the problem — it
implies nothing about whether the results are “important.” Hypothesis testing is of
A.6. HYPOTHESIS TESTING 31
great generality, and useful when we seek to know whether something other than the
assumed case (the null hypothesis) is the case.
A.6.1 Chi-squared test
Hypothesis testing can be applied to discrete problems too. Suppose we have n
patterns — n1 of which are known to be in ω1, and n2 in ω2 — and we are interested
in determining whether a particular decision rule is useful or informative. In this case,
the null hypothesis is a random decision rule — one that selects a pattern and with
some probability P places it in a category which we will call the “left” category, and
otherwise in the “right” category. We say that a candidate rule is informative if it
differs signficantly from such a random decision.
What we need is a clear mathematical definition of statistical significance under
these conditions. The random rule (the null hypothesis) would place Pn1 patterns
from ω1 and Pn2 from ω2 independently in the left category and the remainder in
the right category. Our candidate decision rule would differ significantly from the
random rule if the proportions differed significantly from those given by the random
rule. Formally, we let niL denote the number of patterns from category ωi placed in
the left category by our candidate rule. The so-called chi-squared statistic for this
case is
χ2 =
2∑
k=1
(niL − nie)2
nie
. (117)
where according to the null hypothesis, the number of patterns in category ωi that we
expect to be placed in the left category is nie = Pni. Clearly χ2 is non-negative, and
is zero if and only if all the observed match the expected numbers. The higher the χ2,
the less likely it is that the null hypothesis is true. Thus, for a sufficiently high χ2, the
difference between the expected and observed distributions is statistically significant,
we can reject the null hypothesis, and can consider our candidate decision rule is
“informative.” For any desired level of significance — such as .01 or .05 — a table
gives the critical values of χ2 that allow us to reject the null hypothesis (Table A.2).
There is one detail that must be addressed: the number of degrees of freedom.
In the situation described above, once the probability P is known, there is only one
free variable needed to describe a candidate rule. For instance, once the number of
patterns from ω1 placed in the left category are known, all other values are determined
uniquely. Hence in this case the number of degrees of freedom is 1. If there were more
categories, or if the candidate decision rule had more possible outcomes, then df would
be greater than 1. The higher the number of degrees of freedom, the higher must be
the computed χ2 to meet a disired level of significance.
We denote the critical values as, for instance, χ2
.01(1)
= 6.64, where the subscript
denotes the significance, here .01, and the integer in parentheses is the degrees of
freedom. (In the Table, we conform to the usage in statistics, where this positive
integer is denoted df , despite the possible confusion in calculus where it denotes an
infinitessimal real number.) Thus if we have one degree of freedom, and the observed
χ2 is greater than 6.64, then we can reject the null hypothesis, and say that, at the
.01 confidence level our results did not come from a (weighted) random decision.
32 APPENDIX A. MATHEMATICAL FOUNDATIONS
Table A.2: Critical values of chi-square (at two confidence levels) for different degrees
of freedom (df)
df .05 .01 df .05 .01 df .05 .01
1 3.84 6.64 11 19.68 24.72 21 32.67 38.93
2 5.99 9.21 12 21.03 26.22 22 33.92 40.29
3 7.82 11.34 13 22.36 27.69 23 35.17 41.64
4 9.49 13.28 14 23.68 29.14 24 36.42 42.98
5 11.07 15.09 15 25.00 30.58 25 37.65 44.31
6 12.59 16.81 16 26.30 32.00 26 38.88 45.64
7 14.07 18.48 17 27.59 33.41 27 40.11 46.96
8 15.51 20.09 18 28.87 34.80 28 41.34 48.28
9 16.92 21.67 19 30.14 37.57 29 42.56 49.59
10 18.31 23.21 20 31.41 37.57 30 43.77 50.89
A.7 Information theory
A.7.1 Entropy and information
Assume we have a discrete set of symbols {v1 v2 . . . vm} with associated probabilities
Pi. The entropy of the discrete distribution — a measure of the randomness or
unpredictability of a sequence of symbols drawn from it — is
H = −
m∑
i=1
Pi log2 Pi, (118)
where since we use the logarithm base 2 entropy is measured in bits. In case anybit
of the probabilities vanish, we use the relation 0 log 0 = 0. One bit corresponds
to the uncertainty that can be resolved by the answer to a single yes/no question.
(For continuous distributions, we often use logarithm base e, denoted ln, in which
case the unit is nat.) The expectation operator (cf. Eq. 41) can be used to write
H = E [log 1/P ], where we think of P as being a random variable whose possible
values are P1, P2, . . . , Pm. The term log21/P is sometimes called the surprise — ifsurprise
Pi = 0 except for one i, then there is no surprise when the corresponding symbol
occurs.
Note that the entropy does not depend on the symbols themselves, just on their
probabilities. For a given number of symbols m, the uniform distribution in which
each symbol is equally likely, is the maximum entropy distribution (and H = log2 m
bits) — we have the maximum uncertainty about the identity of each symbol that
will be chosen. Clearly if x is equally likely to take on integer values 0, 1, ..., 7, we
need 3 bits to describe the outcome and H = log22
3 = 3. Conversely, if all the pi
are 0 except one, we have the minimum entropy distribution (H = 0 bits) — we are
certain as to the symbol that will appear.
For a continuous distribution, the entropy is
H = −
∞∫
−∞
p(x) ln p(x)dx, (119)
A.7. INFORMATION THEORY 33
and again H = E [ln 1/p]. It is worth mentioning that among all continuous density
functions having a given mean µ and variance σ2, it is the Gaussian that has the
maximum entropy (H = .5 + log2 (
√
2πσ) bits). We can let σ approach zero to find
that a probability density in the form of a Dirac delta function, i.e., Dirac
delta
δ(x− a) =
{
0 if x "= a
∞ if x = a, with
∞∫
−∞
δ(x)dx = 1, (120)
has the minimum entropy (H = −∞ bits). For a Dirac function, we are sure that the
value a will be selected each time.
Our use of entropy in continuous functions, such as in Eq. 119, belies some sub-
tle issues which are worth pointing out. If x had units, such as meters, then the
probability density p(x) would have to have units of 1/x. There would be something
fundamentally wrong in taking the logarithm of p(x) — the argument of the loga-
rithm function should be dimensionless. What we should really be dealing with is a
dimensionless quantity, say p(x)/p0(x), where p0(x) is some reference density function
(cf., Sect. A.7.2).
For discrete variable x and arbitrary function f(·), we have H(f(x)) ≤ H(x), i.e.,
processing decreases entropy. For instance, if f(x) = const, the entropy will vanish.
Another key property of the entropy of a discrete distribution is that it is invariant to
“shuffling” the event labels. The related question with continuous variables concerns
what happens when one makes a change of variables. In general, if we make a change of
variables, such as y = x3 or even y = 10x, we will get a different value for the integral
of
∫
q(y)log q(y) dy, where q is the induced density for y. If entropy is supposed
to measure the intrinsic disorganization, it doesn’t make sense that y would have a
different amount of intrinsic disorganization than x, since one is always derivable from
the other; only if there were some randomness (e.g., shuffling) incorporated into the
mapping could we say that one is more disorganized than the other.
Fortunately, in practice these concerns do not present important stumbling blocks
since relative entropy and differences in entropy are more fundamental than H taken
by itself. Nevertheless, questions of the foundations of entropy measures for continu-
ous variables are addressed in books listed in Bibliographical Remarks.
A.7.2 Relative entropy
Suppose we have two discrete distributions over the same variable x, p(x) and q(x).
The relative entropy or Kullback-Leibler distance (which is closely related to cross Kullback-
Leibler
distance
entropy, information divergence and information for discrimination) is a measure of
the “distance” between these distributions:
DKL(p(x), q(x)) =
∑
x
q(x)ln
q(x)
p(x)
. (121)
The continuous version is
DKL(p(x), q(x)) =
∞∫
−∞
q(x)ln
q(x)
p(x)
dx. (122)
34 APPENDIX A. MATHEMATICAL FOUNDATIONS
Although DKL(p(·), q(·)) ≥ 0 and DKL(p(·), q(·)) = 0 if and only if p(·) = q(·), the
relative entropy is not a true metric, since DKL is not necessarily symmetric in the
interchange p↔ q and furthermore the triangle inequality need not be satisfied.
A.7.3 Mutual information
Now suppose we have two distributions over possibly different variables, e.g., p(x) and
q(y). The mutual information is the reduction in uncertainty about one variable due
to the knowledge of the other variable
I(p; q) = H(p)−H(p|q) =
∑
x,y
r(x, y)log
r(x, y)
p(x)q(y)
, (123)
where r(x, y) is the joint distribution of finding value x and y. Mutual information
is simply the relative entropy between the joint distribution r(x, y) and the product
distribution p(x)q(y) and as such it measures how much the distributions of the vari-
ables differ from statistical independence. Mutual information does not obey all the
properties of a metric. In particular, the metric requirement that if p(x) = q(y) then
I(x; y) = 0 need not hold, in general. As an example, suppose we have two binary
random variables with r(0, 0) = r(1, 1) = 1/2, so r(0, 1) = r(1, 0) = 0. According to
Eq. 123, the mutual information between p(x) and q(y) is log 2 = 1.
The relationships among the entropy, relative entropy and mutual information are
summarized in Fig. A.5. The figure shows, for instance, that the joint entropy H(p, q)
is always larger than individual entropies H(p) and H(q); that H(p) = H(p|q) +
I(p; q), and so on.
H(p,q)
H(q|p)I(p;q)
H(p)
H(q)
H(p|q)
Figure A.5: The mathematical relationships among the entropy of distributions p and
q, mutual information I(p, q), and conditional entropies H(p|q) and H(q|p). From this
figure one can quickly see relationships among the information functions. For instance
we can see immediately that I(p; p) = H(p); that if I(p; q) = 0 then H(q|p) = H(q);
that H(p, q) = H(p|q) + H(q), and so forth.
A.8 Computational complexity
In order to analyze and describe the difficulty of problems and the algorithms de-
signed to solve such problems, we turn now to the technical notion of computational
complexity. For instance, calculating the covariance matrix for a samples is somehow
“harder” than calculating the mean. Furthermore, some algorithms for computing
some function may be faster or take less memory, than another algorithm. We seek
A.8. COMPUTATIONAL COMPLEXITY 35
to specify such differences, independent of the current computer hardware (which is
always changing anyway).
To this end we use the concept of the order of a function and the asymptotic
notations “big oh,” “big omega,” and “big theta.” The three asymptotic bounds
most often used are:
Asymptotic upper bound O(g(x)) = {f(x): there exist positive constants c and
x0 such that 0 ≤ f(x) ≤ cg(x) for all x ≥ x0}
Asymptotic lower bound Ω(g(x)) = {f(x): there exist positive constants c and
x0 such that 0 ≤ cg(x) ≤ f(x) for all x ≥ x0}
Asymptotically tight bound Θ(g(x)) = {f(x): there exist positive constants c1, c2,
and x0 such that 0 ≤ c1g(x) ≤ f(x) ≤ c2g(x) for all x ≥ x0}
x x x
x
0
x
0 x0
f(x)
c g(x)
c g(x)
c2 g(x)
c1 g(x)
f(x) = Ω(g(x))f(x) = O(g(x)) f(x) = Θ(g(x))
a) b) c)
f(x)
f(x)
Figure A.6: Three types of asymptotic bounds: a) f(x) = O(g(x)). b) f(x) =
Ω(g(x)). c) f(x) = Θ(g(x)).
Consider the asymptotic upper bound. We say that f(x) is “of order big oh of g(x)” big oh
(written f(x) = O(g(x)) if there exist constants c0 and x0 such that f(x) ≤ c0g(x)
for all x > x0. We shall assume that all our functions are positive and dispense
with taking absolute values. This means simply that for sufficiently large x, an upper
bound on f(x) grows no worse than g(x). For instance, if f(x) = a + bx + cx2 then
f(x) = O(x2) because for sufficiently large x, the constant, linear and quadratic terms
can be “overcome” by proper choice of c0 and x0. The generalization to functions
of two or more variables is straightforward. It should be clear that by the definition
above, the (big oh) order of a function is not unique. For instance, we can describe
our particular f(x) as being O(x2), O(x3), O(x4), O(x2 ln x), and so forth. We use
big omega notation, Ω(·), for lower bounds, and little omega, ω(·), for the tightest
lower bound. Of these, the big oh notation has proven to be most useful since we
generally want an upper bound on the resources when solving a problem.
The lower bound on the complexity of the problem is denoted Ω(g(x)), and is there-
fore the lower bound on any algorithm algorithm that solves that problem. Similarly,
if the complexity of an algorithm is O(g(x)), it is an upper bound on the complexity
of the problem it solves. The complexity of some problems — such as computing the
mean of a discrete set — is known, and thus once we have found an algorithm having
equal complexity, the only possible improvement could be on lowering the constants
of proportionality. The complexity of other problems — such as inverting a matrix
36 APPENDIX A. MATHEMATICAL FOUNDATIONS
— is not yet known, and if fundamental analysis cannot derive it, we must rely on
algorithm developers who find algorithms whose complexity
Approximately.
Such a rough analysis does not tell us the constants c and x0. For a finite size
problem it is possible that a particular O(x3) algorithm is simpler than a particular
O(x2) algorithm, and it is occasionally necessary for us to determine these constants
to find which of several implemementations is the simplest. Nevertheless, for our
purposes the big oh notation as just described is generally the best way to describe
the computational complexity of an algorithm.
Suppose we have a set of n vectors, each of which is d-dimensional and we want to
calculate the mean vector. Clearly, this requires O(nd) multiplications. Sometimes we
stress space and time complexities, which are particularly relevant when contemplat-
ing parallel hardware implementations. For instance, the d-dimensional sample mean
could be calculated with d separate processors, each adding n sample values. Thus
we can describe this implementation as O(d) in space (i.e., the amount of memoryspace
complexity or possibly the number of processors) and O(n) in time (i.e., number of sequential
steps). Of course for any particular algorithm there may be a number of time-space
time
complexity
tradeoffs.
Bibliographical Remarks
There are several good books on linear systems, such as [14], and matrix computations
[8]. Lagrange optimization and related techniques are covered in the definitive book
[2]. While [13] and [3] are of foundational and historic interest, readers seeking clear
presentations of the central ideas in probability should consult [10, 7, 6, 21]. A
handy reference to terms in probability and statistics is [20]. A number of hypothesis
testing and statistical significance, elementary, such as [24], and more advanced [18,
25]. Shannon’s foundational paper [22] should be read by all students of pattern
recognition. It, and many other historically important papers on information theory
can be found in [23]. An excellent textbook at the level of this one is [5] and readers
seeking a more abstract and formal treatment should consult [9]. The study of time
complexity of algorithms began with [12], and space complexity [11, 19]. The multi-
volume [15, 16, 17] contains a description of computational complexity, the big oh
and other asymptotic notations. Somewhat more accessible treatments can be found
in [4] and [1].
Bibliography
[1] Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. The Design and Anal-
ysis of Computer Algorithms. Addison-Wesley, Reading, MA, 1974.
[2] Dimitri P. Bertsekas. Constrained Optimization and Lagrange Multiplier Meth-
ods. Athena Scientific, Belmont, MA, 1996.
[3] Patrick Billingsley. Probability and Measure. Wiley, New York, NY, second
edition, 1986.
[4] Thomas H. Cormen, Charles E. Leiserson, and Ronald L. Rivest. Introduction
to Algorithms. MIT Press, Cambridge, MA, 1990.
[5] Thomas M. Cover and Joy A. Thomas. Elements of Information Theory. Wiley
Interscience, New York, NY, 1991.
[6] Alvin W. Drake. Fundamentals of Applied Probability Theory. McGraw-Hill,
New York, NY, 1967.
[7] William Feller. An Introduction to Probability Theory and Its Applications, vol-
ume 1. Wiley, New York, NY, 1968.
[8] Gene H. Golub and Charles F. Van Loan. Matrix Computations. Johns Hopkins
University Press, Baltimore, MD, third edition, 1996.
[9] Robert M. Gray. Entropy and Information Theory. Springer-Verlag, New York,
NY, 1990.
[10] Richard W. Hamming. The Art of Probability for Scientists and Engineers.
Addison-Wesley, New York, NY, 1991.
[11] Juris Hartmanis, Philip M. Lewis II, and Richard E. Stearns. Hierarchies of
memory limited computations. Proceedings of the Sixth Annual IEEE Symposium
on Switching Circuit Theory and Logical Design, pages 179–190, 1965.
[12] Juris Hartmanis and Richard E. Stearns. On the computational complexity of
algorithms. Transactions of the American Mathematical Society, 117:285–306,
1965.
[13] Harold Jeffreys. Theory of Probability. Oxford University Press, Oxford, UK,
1961 reprint edition, 1939.
[14] Thomas Kailath. Linear Systems. Prentice-Hall, Englewood Cliffs, NJ, 1980.
37
38 BIBLIOGRAPHY
[15] Donald E. Knuth. The Art of Computer Programming, volume 1. Addison-
Wesley, Reading, MA, 1 edition, 1973.
[16] Donald E. Knuth. The Art of Computer Programming, volume 3. Addison-
Wesley, Reading, MA, 1 edition, 1973.
[17] Donald E. Knuth. The Art of Computer Programming, volume 2. Addison-
Wesley, Reading, MA, 1 edition, 1981.
[18] Erich L. Lehmann. Testing Statistical Hypotheses. Springer, New York, NY,
1997.
[19] Philip M. Lewis II, Richard E. Stearns, and Juris Hartmanis. Memory bounds
for recognition of context-free and context-sensitive languages. Proceedings of the
Sixth Annual IEEE Symposium on Switching Circuit Theory and Logical Design,
pages 191–202, 1965.
[20] Francis H. C. Marriott. A Dictionary of Statistical Terms. Longman Scientific
& Technical, Essex, UK, fifth edition, 1990.
[21] Yuri A. Rozanov. Probability Theory: A Concise Course. Dover, New York, NY,
1969.
[22] Claude E. Shannon. A mathematical theory of communication. Bell Systems
Technical Journal, 27:379–423, 623–656, 1948.
[23] David Slepian, editor. Key Papers in the Development of Information Theory.
IEEE Press, New York, NY, 1974.
[24] Richard C. Sprinthall. Basic Statistical Analysis. Allyn & Bacon, Needham
Heights, MA, fifth edition, 1996.
[25] Rand R. Wilcox. Introduction to Robust Estimation and Hypotheses Testing.
Academic Press, New York, NY, 1997.
Index
†, see matrix, pseudoinverse
ρ, see correlation, coefficient
E [·], see expectation
adjoint, see matrix, adjoint
asymptotic lower bound, see lower bound,
asymptotic
asymptotic notation, 35
asymptotic tight bound, see tight bound,
asymptotic
asymptotic upper bound, see upper bound,
asymptotic
average, see expected value
Bayes’ rule, 17, 21
vector, 19
Bienaymé-Chebychev, see Chebyshev’s
inequality
big oh, 35
big omega, 35
big theta, 35
Cauchy-Schwarz inequality, 7, 16
vector analog, 20
Central Limit Theorem, 23
Chebyshev’s inequality, 13
chi-square, 31
table, 32
chi-squared statistic, 31
cofactor
matrix, see matrix, cofactor
complexity
space, 36
time, 36
computational complexity, 34–36
conditional probability, see probabil-
ity, conditional
confidence level, 31
convolution, 22
correlation
coefficient, 16, 28, 30
covariance, 15, 20
matrix, see matrix, covariance
normalized, 16
cross entropy, see distance, Kullback-
Leibler
cross moment, see covariance
density
Gaussian
bivariate, 28
conditional mean, 30
marginal, 29
mean, 23
univariate, 23
variance, 23
joint
singular, 29
distance
Euclidean, 7
Kullback-Leibler, 33
Mahalanobis, 23, 27
distribution
Gaussian, 23
area, 13
covariance, 28
eigenvector, 27
moment, 26
multivariate, 26
principal axes, 27
univariate, 23
joint, 18
marginal, 18
maximum entropy, 32
prior, 18
dot product, see inner product
dyadic product, see matrix product
eigenvalue, 11
eigenvector, 11
entropy, 32
continuous distribution, 32
discrete, 33
39
40 INDEX
relative, 33
surprise, 32
error function, 25
Euclidean norm, see distance, Euclidean
events
mutually exclusive, 17
evidence, 18
expectation
continuous, 20
entropy, 32
linearity, 13, 15
vector, 19
expected value, 13
two variables, 15
factorial, 25
function
Dirac delta, 33
gamma, 25
Kronecker, 6
vector valued, 21
gamma function, see function, gamma
Gaussian
table, 24
unidimensional, 23
Gaussian derivative, 24–25
gradient, 8
Hessian matrix, see matrix, Hessian
hypothesis
null, see null hypothesis
hypothesis testing, 30
identity matrix, see matrix, identity
independence
statistical, 15
independent variables
sum, 22
information
bit, see bit
divergence, see distance, Kullback-
Leibler
for discrimination, see distance, Kullback-
Leibler
mutual, 34
information theory, 32–34
inner product, 7
Jacobian, 8
Jacobian matrix, see matrix, Jacobian
Kronecker delta, see function, Kronecker
Kullback-Leibler, see distance, Kullback-
Leibler
Lagrange optimization, see optimiza-
tion, Lagrange
Lagrange undetermined multiplier, 12
Law of Total Probability, 17
level curves, 28
likelihood, 18
linear independence, 7
matrix columns, 11
little omega, 35
lower bound
asymptotic, 35
Mahalanobis distance, see distance, Ma-
halanobis
marginal, 14
distribution, 14
mass function
probability, see probability, mass
function
matrix
addition, 6
adjoint, 11
anti-symmetric, 6
covariance, 9
determinant, 27, 28
diagonal, 20, 21, 26
eigenvalues, 20
inverse, 27, 28
derivative, 8–9
determinant, 9–10
hypervolume, 9
Hessian, 9
identity (I), 6
inverse
derivative, 8
inversion, 10–12
Jacobian, 8
multiplication, 6
non-negative, 6
positive semi-definite, 20
product, see outer product
pseudoinverse, 11
skew-symmetric, 6
square, 6
symmetric, 6, 9
trace, 10
INDEX 41
maximum entropy, 32
mean, see expected value
calculation
computational complexity, 34
two variables, 15
mean vector, see vector, mean
moment
cross, see covariance
second, 13
multiple integral, 21
mutual information, see information,
mutual
normal, see distribution, Gaussian
null hypothesis, 30
optimization
Lagrange, 12
outer product, 7, 19
principal axes, see axes, principal
prior, 18
prior distribution, see distribution, prior
probability
conditional, 16–17
density, 20
joint, 21
joint, 14, 17
mass, 16, 20
joint, 14
mass function, 12
total
law, see Bayes’ rule
probability theory, 12–24
product space, 14
random variable
discrete, 12
vector, 18–20
scalar product, see inner product
second moment, see moment, second
significance
level, see confidence level
statistical, 30
space-time tradeoff, 36
standard deviation, 13, 23
statistic
chi-squared, see chi-squared statis-
tic
statistical
independence
expectation, 16
statistical dependence, 16
statistical independence, see indepen-
dence, statistical, 16, 20
Gaussian, 29
vector, 18
statistical significance, see significance,
statistical
surprise, 32
Taylor series, 9
tight bound
asymptotic (Θ(·)), 35
trace, see matrix, trace
transpose, 6
unpredictability, see entropy
upper bound
asymptotic, 35
variable
random
continuous, 20–21
discrete, 14
standardized, 27
standardized, 23
variables
uncorrelated, 16
variance, 13
nonlinearity, 14
two variables, 15
vector, 6
addition, 6
colinearity, 7
linearly independent, 7
mean, 19
orthogonal, 7
space, 7
span, 7
vector product, see outer product
z score, 23