CS计算机代考程序代写 algorithm information theory data mining Excel decision tree Tree Learning

Tree Learning
COMP9417 Machine Learning and Data Mining
Term 2, 2020
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Acknowledgements
Material derived from slides for the book “Machine Learning” by T. Mitchell
McGraw-Hill (1997) http://www-2.cs.cmu.edu/~tom/mlbook.html
Material derived from slides by Andrew W. Moore
http:www.cs.cmu.edu/~awm/tutorials
Material derived from slides by Eibe Frank
http://www.cs.waikato.ac.nz/ml/weka
Material derived from slides for the book “Machine Learning” by P. Flach Cambridge University Press (2012) http://cs.bris.ac.uk/~flach/mlbook
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Aims
Aims
This lecture will enable you to describe decision tree learning, the use of entropy and the problem of overfitting. Following it you should be able to:
• define the decision tree representation
• list representation properties of data and models for which decision
trees are appropriate
• reproduce the basic top-down algorithm for decision tree induction
(TDIDT)
• define entropy in the context of learning a Boolean classifier from
examples
• describe the inductive bias of the basic TDIDT algorithm
• define overfitting of a training set by a hypothesis
• describe developments of the basic TDIDT algorithm: pruning, rule
generation, numerical attributes, many-valued attributes, costs,
missing values
• describe regression and model trees
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Introduction
Brief History of Decision Tree Learning Algorithms
• late 1950’s – Bruner et al. in psychology work on modelling concept acquisition
• early 1960s – Hunt et al. in computer science work on Concept Learning Systems (CLS)
• late 1970s – Quinlan’s Iterative Dichotomizer 3 (ID3) based on CLS is efficient at learning on then-large data sets
• 1980s – CART (Classification and Regression Trees) book (Breiman et al. (1984)) and software
• early 1990s – ID3 adds features, develops into C4.5, becomes the “default” machine learning algorithm
• late 1990s – C5.0, commercial version of C4.5 (available from SPSS and www.rulequest.com)
• 2000s – trees become more frequently used in ensembles
• current – still widely available and applied; influential techniques
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Introduction
Why use decision trees?
• Trees in some form are probably still the single most popular data mining tool
• Easy to understand
• Easy to implement
• Easy to use
• Computationally efficient (even on big data) to learn and run
• There are some drawbacks, though — e.g., high variance
• They do classification, i.e., predict a categorical output from
categorical and/or real inputs, or regression
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Introduction
Decision Tree for PlayTennis
Outlook
Sunny Overcast Rain Yes
Humidity
Wind
High Normal No Yes
Strong Weak No Yes
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Tree Learning
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Introduction
A Tree to Predict C-Section Risk
Learned from medical records of 1000 women Negative examples are C-sections
[833+,167-] .83+ .17-
Fetal_Presentation = 1: [822+,116-] .88+ .12-
| Previous_Csection = 0: [767+,81-] .90+ .10-
| | Primiparous = 0: [399+,13-] .97+ .03-
| | Primiparous = 1: [368+,68-] .84+ .16-
| | | Fetal_Distress = 0: [334+,47-] .88+ .12-
| | | | Birth_Weight < 3349: [201+,10.6-] .95+ .05- | | | | Birth_Weight >= 3349: [133+,36.4-] .78+ .22- | | | Fetal_Distress = 1: [34+,21-] .62+ .38-
| Previous_Csection = 1: [55+,35-] .61+ .39- Fetal_Presentation = 2: [3+,29-] .11+ .89- Fetal_Presentation = 3: [8+,22-] .27+ .73-
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Introduction
Decision Tree for Credit Rating
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Introduction
Decision Tree for Fisher’s Iris data
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Introduction
Decision Trees
Decision tree representation:
• Each internal node tests an attribute
• Each branch corresponds to attribute value • Each leaf node assigns a classification
How would we represent the following expressions ? • ∧,∨, XOR
• (A∧B)∨(C∧¬D∧E)
• M of N
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Introduction
Decision Trees
X∧Y
X = t:
| Y = t: true | Y = f: no
X = f: no
X∨Y
X = t: true
X = f:
| Y = t: true | Y = f: no
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Tree Learning
Term 2, 2020
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Introduction
When are Decision Trees the Right Model?
• With Boolean values for the instances X and class Y , the representation adopted by decision-trees allows us to represent Y as a Boolean function of the X
• Given d input Boolean variables, there are 2d possible input values for these variables. Any specific function assigns Y = 1 to some subset of these, and Y = 0 to the rest
• Any Boolean function can be trivially represented by a tree. Each function assigns Y = 1 to some subset of the 2d possible values of X. So, for each combination of values with Y = 1, have a path from root toaleafwithY =1. AllotherleaveshaveY =0
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Introduction
When are Decision Trees the Right Model?
• This is nothing but a re-representation of the truth-table, and will have 2d leaves. More compact trees may be possible, by taking into account what is common between one or more rows with the same Y value
• But, even for Boolean functions, there are some functions for which compact trees may not be possible (the parity and majority functions are examples)1
• In general, although possible in principle to express any Boolean function, our search and prior restrictions may not allow us to find the correct tree in practice
• BUT: If you want readable models that combine logical tests with a probability-based decision, then decision trees are a good start
1Finding the optimal tree is NP-complete (Hyafil and Rivest (1976)).
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Introduction
When to Consider Decision Trees?
• Instances described by a mix of numeric features and discrete attribute–value pairs
• Target function is discrete valued (otherwise use regression trees)
• Disjunctive hypothesis may be required
• Possibly noisy training data
• Interpretability is an advantage
Examples are extremely numerous, including: • Equipment or medical diagnosis
• Credit risk analysis
• Modeling calendar scheduling preferences • etc.
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The Basic Algorithm
Top-Down Induction of Decision Trees (TDIDT)
“ID3” decision tree learning algorithm (Quinlan (1986))
Main loop:
1 A ← the “best” decision attribute for next node
2 Assign A as decision attribute for node
3 For each value of A, create new descendant of node
4 Sort training examples to leaf nodes
5 If training examples perfectly classified, Then STOP, Else iterate over new leaf nodes
Essentially, this is the top-level of the ID3 algorithm — the first efficient symbolic Machine Learning algorithm.
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The Basic Algorithm
Which attribute is best?
[29+,35-] A1=? [29+,35-] A2=? tf tf
[21+,5-] [8+,30-] [18+,33-] [11+,2-]
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Information Gain
Bits
You are watching a set of independent random samples of X You observe that X has four possible values
So you might see: BAACBADCDADDDA…
You transmit data over a binary serial link. You can encode each reading withtwobits(e.g. A=00,B=01,C=10,D=11)
0100001001001110110011111100…
P(X = A) = 1 4
P(X = B) = 1 4
P(X = C) = 1 4
P(X = D) = 1 4
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Information Gain
Fewer Bits
Someone tells you that the probabilities are not equal
It’s possible . . .
. . . to invent a coding for your transmission that only uses 1.75 bits on
average per symbol. How ?
P(X = A) = 1 2
P(X = B) = 1 4
P(X = C) = 1 8
P(X = D) = 1 8
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Information Gain
Fewer Bits
Someone tells you that the probabilities are not equal
It’s possible . . .
. . . to invent a coding for your transmission that only uses 1.75 bits per
symbol on average. How ?
P(X = A) = 1 2
P(X = B) = 1 4
P(X = C) = 1 8
P(X = D) = 1 8
A
0
B
10
C
110
D
111
(This is just one of several ways)
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Information Gain
Fewer Bits
Suppose there are three equally likely values
Here’s a na ̈ıve coding, costing 2 bits per symbol
P (X = A) = 1 3
P(X = B) = 1 3
P(X = C) = 1 3
A
00
B
01
C
10
Can you think of a coding that would need only 1.6 bits per symbol on average ?
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Information Gain
Fewer Bits
Suppose there are three equally likely values
Using the same approach as before, we can get a coding costing 1.6 bits per symbol on average . . .
P (X = A) = 1 3
P(X = B) = 1 3
P(X = C) = 1 3
A
0
B
10
C
11
Thisgivesus,onaverage 1 ×1bitforAand2×1 ×2bitsforBandC, 33
which equals 5 ≈ 1.6 bits. 3
Is this the best we can do ?
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Information Gain
Fewer Bits
Suppose there are three equally likely values
From information theory, the optimal number of bits to encode a symbol withprobabilitypis−log2 p …
Sothebestwecandoforthiscaseis−log 1 bitsforeachofA,Band 23
C, or 1.5849625007211563 bits per symbol
P (X = A) = 1 3
P(X = B) = 1 3
P(X = C) = 1 3
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Term 2, 2020 23 / 100

Information Gain
General Case
Suppose X can have one of m values … V1,V2,…Vm
What’s the smallest possible number of bits, on average, per symbol, needed to transmit a stream of symbols drawn from X’s distribution ? It’s
H(X) = −p1log2p1−p2log2p2−…−pmlog2pm m
= −􏰂pjlog2pj j=1
H(X) is called the entropy of X
P(X = V1) = p1
P(X = V2) = p2
…
P(X = Vm) = pm
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Information Gain
General Case
“High entropy” means X is very uniform and boring “Low entropy” means X is very varied and interesting
Another way to think about this is that entropy is, in some sense, the inverse of information.
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Information Gain
Entropy
1.0
0.5
0.0 0.5 1.0 p+
Consider a 2-class distribution, where:
S is a sample of training examples
p⊕ is the proportion of positive examples in S p⊖ is the proportion of negative examples in S
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Entropy(S)

Information Gain
Entropy
Entropy measures the “impurity” of S
Entropy(S) ≡ −p⊕ log2 p⊕ − p⊖ log2 p⊖
A “pure” sample is one in which all examples are of the same class.
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Information Gain
Entropy
Entropy(S) = expected number of bits needed to encode class (⊕ or ⊖) of randomly drawn member of S (under the optimal, shortest-length code)
Why ?
Information theory: optimal length code assigns − log2 p bits to message
having probability p.
So, expected number of bits to encode ⊕ or ⊖ of random member of S:
p⊕(− log2 p⊕) + p⊖(− log2 p⊖) Entropy(S) ≡ −p⊕ log2 p⊕ − p⊖ log2 p⊖
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Information Gain
Information Gain
• Gain(S, A) = expected reduction in entropy due to sorting on A
Gain(S,A) ≡ Entropy(S) − 􏰂 |Sv|Entropy(Sv)
[29+,35-] A1=? [29+,35-] A2=? tf tf
|S| v∈V alues(A)
[21+,5-] [8+,30-] [18+,33-] [11+,2-]
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Information Gain
Information Gain
Gain(S, A1)
=
= =
= =
􏰀|St| |Sf| 􏰁 Entropy(S) − |S| Entropy(St) + |S| Entropy(Sf )
0.9936 −
((26(−21 log2(21) − 5 log2( 5 ))) +
(38(− 8 log2( 8 ) − 30 log2(30)))) 64 38 38 38 38
0.9936 − ( 0.2869 + 0.4408 ) 0.2658
64 26 26 26 26
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Tree Learning
Term 2, 2020
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Information Gain
Information Gain
Gain(S, A2) = 0.9936 − ( 0.7464 + 0.0828 ) = 0.1643
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Information Gain
Information Gain
So we choose A1, since it gives a larger expected reduction in entropy.
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Information Gain
Training Examples
Day Outlook Temperature Humidity Wind PlayTennis
D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14
Sunny Hot
Sunny Hot Overcast Hot Rain Mild
Rain Cool
Rain Cool Overcast Cool Sunny Mild Sunny Cool Rain Mild Sunny Mild Overcast Mild
Overcast Hot Rain Mild
High Weak No High Strong No High Weak Yes High Weak Yes
Normal Weak Yes Normal Strong No Normal Strong Yes
High Weak No Normal Weak Yes Normal Weak Yes Normal Strong Yes
High Strong Yes Normal Weak Yes High Strong No
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Information Gain
Information gain once more
S: [9+,5-] E =0.940
High
[3+,4-] E =0.985
Gain (S, Humidity )
S: [9+,5-] E =0.940
Which attribute is the best classifier?
Humidity
Wind
= .940 – (7/14).985 – (7/14).592 = .151
= .940 – (8/14).811 – (6/14)1.0 = .048
Normal
[6+,1-] E =0.592
Weak
[6+,2-] E =0.811
Strong
[3+,3-] E =1.00
Gain (S, Wind)
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Tree Learning
Term 2, 2020 34 / 100

Information Gain
Information gain once more
{D1, D2, …, D14} [9+,5−]
Sunny Overcast Rain
{D1,D2,D8,D9,D11} {D3,D7,D12,D13} {D4,D5,D6,D10,D14} [2+,3−] [4+,0−] [3+,2−]
Yes
Which attribute should be tested here?
Ssunny = {D1,D2,D8,D9,D11}
Gain (Ssunny , Humidity) = .970 − (3/5) 0.0 − (2/5) 0.0 = .970
Gain (Ssunny , Temperature) = .970 − (2/5) 0.0 − (2/5) 1.0 − (1/5) 0.0 = .570 Gain (Ssunny , Wind) = .970 − (2/5) 1.0 − (3/5) .918 = .019
Outlook
?
?
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Information Gain
Attribute selection – impurity measures more generally
Estimate class probability of class k at node m of the tree as pˆmk = |Smk|. |Sm |
Classify at node m by predicting the majority class, pˆmk(m). Misclassification error:
Entropy for K class values:
1−pˆmk(m)
K
− 􏰂 pˆmk log pˆmk
k=1
CART (Breiman et al. (1984)) uses the “Gini index”:
K
􏰂 pˆmk (1 − pˆmk ) k=1
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Tree Learning
Term 2, 2020
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Information Gain
Attribute selection – impurity measures more generally
Why not just use accuracy, or misclassification error ? In practice, not found to work as well as others
Entropy and Gini index are more sensitive to changes in the node probabilities than misclassification error.
Entropy and Gini index are differentiable, but misclassification error is not (Hastie et al. (2009)).
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Tree Learning More Generally
Hypothesis Space Search by ID3
+–+
A1
+–+ +
A2
+–+ –
+
…
A2
+–+ –
…
A2
+–+ –
–
… …
A3
A4
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Tree Learning
Term 2, 2020
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Tree Learning More Generally
Hypothesis Space Search by ID3
• This can be viewed as a graph-search problem
• Each vertex in the graph is a decision tree
• Suppose we only consider the two-class case (ω = ω1 or ω2), and all the features xi are Boolean, so each vertex is a binary tree
• A pair of vertices in the graph have an edge if the corresponding trees differ in just the following way: one of the leaf-nodes in one vertex has been replaced by a non-leaf node testing a feature that has not appeared earlier (and 2 leaves)
• This is the full space of all decision trees (is it?). We want to search for a single tree or a small number of trees in this space. How should we do this?
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Tree Learning More Generally
Hypothesis Space Search by ID3
• Usual graph-search technique: greedy or beam search, starting with the vertex corresponding to the “empty tree” (single leaf node)
• Greedy choice: which one to select? The neighbour that results in the greatest increase in P(D|T)
• How?
• Suppose T is changed to T′. Simply use the ratio of P(D|T′)/P(D|T) • Most of the calculation will cancel out: so, we will only need to do the
local computation at the leaf that was converted into a non-leaf node
• RESULT: set of trees with (reasonably) high posterior probabilities given D: we can now use these to answer questions like
P (y′ = ω1| . . .)? or even make a decision or a classification that
y′ = ω1, given input data x
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Tree Learning More Generally
Hypothesis Space Search by ID3
• Hypothesis space is complete! (contains all finite discrete-valued functions w.r.t attributes)
• Target function surely in there…
• Outputs a single hypothesis (which one?)
• Can’t play 20 questions…
• No back tracking
• Local minima…
• Statistically-based search choices
• Robust to noisy data…
• Inductive bias: approx “prefer shortest tree”
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Tree Learning More Generally
Inductive Bias in ID3
Note H is the power set of instances X
→Unbiased?
Not really…
• Preference for short trees, and for those with high information gain attributes near the root
• Bias is a preference for some hypotheses, rather than a restriction of hypothesis space H
• an incomplete search of a complete hypothesis space versus a complete search of an incomplete hypothesis space (as in learning conjunctive concepts)
• Occam’s razor: prefer the shortest hypothesis that fits the data
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Tree Learning More Generally
Occam’s Razor
William of Ockham (c. 1287-1347)
Entities should not be multiplied beyond necessity
Why prefer short hypotheses?
Argument in favour:
• Fewer short hypotheses than long hypotheses
→ a short hyp that fits data unlikely to be coincidence → a long hyp that fits data might be coincidence
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Tree Learning More Generally
Occam’s Razor
Argument opposed:
• There are many ways to define small sets of hypotheses
• e.g., all trees with a prime number of nodes that use attributes beginning with “Z”
• What’s so special about small sets based on size of hypothesis?? Look back to classification lecture to see how to make this work using
Minimum Description Length (MDL)
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Overfitting and How To Avoid It
Why does overfitting occur?
• Greedy search can make mistakes. We know that it can end up in local minima — so a sub-optimal choice earlier might result in a better solution later (i.e. pick a test whose posterior gain (or information gain) is less than the best one
• But there is also another kind of problem. We know that training error is an optimistic estimate of the true error of the model, and that this optimism increases as the training error decreases
• We will see why this is the case later (lectures on Evaluation)
• Suppose we have two models h1 and h2 with training errors e1 and e2
and optimism o1 and o2. Let the true error of each be E1 = e1 + o1
and E2 = e2 + o2
• Ife1 E2,thenwewillsaythath1 hasoverfitthen
training data
• So, a search method based purely on training data estimates may end overfitting the training data
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Overfitting and How To Avoid It
Overfitting in Decision Tree Learning
Consider adding noisy training example #15:
Sunny, Hot, Normal, Strong, PlayTennis = No What effect on earlier tree?
Sunny Overcast Rain Yes
Outlook
Humidity
Wind
High Normal No Yes
Strong Weak No Yes
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Tree Learning
Term 2, 2020
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Overfitting and How To Avoid It
Overfitting in Decision Tree Learning
Outlook
Sunny
{D1,D2,D8,D9,D11} [2+,3−]
{D1, D2, …, D14} [9+,5−]
Overcast
{D3,D7,D12,D13} [4+,0−]
Yes
Rain
{D4,D5,D6,D10,D14} [3+,2−]
?
?
Which attribute should be tested here?
Ssunny = {D1,D2,D8,D9,D11}
Gain (Ssunny , Humidity) = .970 − (3/5) 0.0 − (2/5) 0.0 = .970
Gain (Ssunny , Temperature) = .970 − (2/5) 0.0 − (2/5) 1.0 − (1/5) 0.0 = .570 Gain (Ssunny , Wind) = .970 − (2/5) 1.0 − (3/5) .918 = .019
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Overfitting and How To Avoid It
Overfitting in General
Consider error of hypothesis h over
• training data: errortrain(h)
• entire distribution D of data: errorD(h)
Definition
Hypothesis h ∈ H overfits training data if there is an alternative hypothesis h′ ∈ H such that
and
errortrain(h) < errortrain(h′) errorD(h) > errorD(h′)
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Overfitting and How To Avoid It
Overfitting in Decision Tree Learning
0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5
0 10 20 30 40 50 60 70 80 90 100 Size of tree (number of nodes)
On training data
On test data
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Accuracy

Overfitting and How To Avoid It
Avoiding Overfitting
How can we avoid overfitting? Pruning
• pre-pruning stop growing when data split not statistically significant
• post-pruning grow full tree, then remove sub-trees which are overfitting
Post-pruning avoids problem of “early stopping” How to select “best” tree:
• Measure performance over training data ?
• Measure performance over separate validation data set ?
• MDL: minimize size(tree) + size(misclassif ications(tree)) ?
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Overfitting and How To Avoid It
Avoiding Overfitting
Pre-pruning
• Can be based on statistical significance test
• Stops growing the tree when there is no statistically significant association between any attribute and the class at a particular node
• For example, in ID3: chi-squared test plus information gain
• only statistically significant attributes were allowed to be selected by
information gain procedure
• Problem — as tree grows:
→ typical sample size at nodes get smaller
→ statistical tests become unreliable
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Overfitting and How To Avoid It
Avoiding Overfitting
Pre-pruning
• Simplest approach: stop growing the tree when fewer than some lower-bound on the number of examples at a leaf
• In C4.5, this parameter is the m parameter
• In sklearn, this parameter is min samples leaf
• In sklearn, the parameter min impurity decrease enables stopping when the this falls below a lower-bound
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Overfitting and How To Avoid It
Avoiding Overfitting
Early stopping
• Pre-pruning may suffer from early stopping: may stop the growth of tree prematurely
• Classic example: XOR/Parity-problem
• No individual attribute exhibits a significant association with the class
• Target structure only visible in fully expanded tree • Prepruning won’t expand the root node
• But: XOR-type problems not common in practice
• And: pre-pruning faster than post-pruning
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Overfitting and How To Avoid It
Avoiding Overfitting
Post-pruning
• Builds full tree first and prunes it afterwards
• Attribute interactions are visible in fully-grown tree
• Problem: identification of subtrees and nodes that are due to chance effects
• Two main pruning operations: • Subtree replacement
• Subtree raising
• Possible strategies: error estimation, significance testing, MDL principle
• We examine two methods: Reduced-error Pruning and Error-based Pruning
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Overfitting and How To Avoid It
Reduced-Error Pruning
Split data into training and validation set Do until further pruning is harmful:
• Evaluate impact on validation set of pruning each possible node (plus those below it)
• Greedily remove the one that most improves validation set accuracy
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Overfitting and How To Avoid It
Reduced-Error Pruning
• Good produces smallest version of most accurate subtree • Not so good reduces effective size of training set
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Overfitting and How To Avoid It
Effect of Reduced-Error Pruning
0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5
0 10 20 30 40 50 60 70 80 90 100 Size of tree (number of nodes)
On training data
On test data On test data (during pruning)
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Accuracy

Overfitting and How To Avoid It
Error-based pruning (C4.5 / J48 / C5.0)
Quinlan (1993) describes the successor to ID3 – C4.5 • many extensions – see below
• post-pruning using training set
• includes sub-tree replacement and sub-tree raising • also: pruning by converting tree to rules
• commercial version – C5.0 – is widely used • RuleQuest.com
• now free
• Weka version – J48 – also widely used
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Overfitting and How To Avoid It
Pruning operator: Sub-tree replacement
Bottom-up:
tree is considered for replacement once all its sub-trees have been considered
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Overfitting and How To Avoid It
Error-based pruning: error estimate
Goal is to improve estimate of error on unseen data using all and only data from training set
But how can this work ?
Make the estimate of error pessimistic !
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Overfitting and How To Avoid It
Error-based pruning: error estimate
• Apply pruning operation if this does not increase the estimated error
• C4.5’s method: using upper limit of standard confidence interval derived from the training data
• Standard Bernoulli-process-based method
• Note: statistically motivated, but not statistically valid • However: works well in practice !
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Overfitting and How To Avoid It
Error-based pruning: error estimate
• The error estimate for a tree node is the weighted sum of error estimates for all its subtrees (possibly leaves).
• Upper bound error estimate e for a node (simplified version): 􏰃f · (1 − f)
• f is actual (empirical) error of tree on examples at the tree node
• N is the number of examples at the tree node
• Zc is a constant whose value depends on confidence parameter c
• C4.5’s default value for confidence c = 0.25
• If c = 0.25 then Zc = 0.69 (from standardized normal distribution) COMP9417 ML & DM Tree Learning Term 2, 2020 62 / 100
e = f + Zc · N

Overfitting and How To Avoid It
Error-based pruning: error estimate
• How does this method implement a pessimistic error estimate ?
• What effect will the c parameter have on pruning ?
• Asc↑,z↓
• See example on next slide (note: values not calculated using exactly the above formula)
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Overfitting and How To Avoid It
Error-based pruning: error estimate
• health plan contribution: node measures f = 0.36, e = 0.46
• sub-tree measures:
• none:f=0.33,e=0.47 • half:f=0.5,e=0.72
• full:f=0.33,e=0.47
• sub-trees combined 6 : 2 : 6 gives 0.51
• sub-trees estimated to give greater error so prune away
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Overfitting and How To Avoid It
Rule Post-Pruning
This method was introduced in Quinlan’s C4.5 • Convert tree to equivalent set of rules
• Prune each rule independently of others
• Sort final rules into desired sequence for use
For: simpler classifiers, people prefer rules to trees
Against: does not scale well, slow for large trees & datasets
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Overfitting and How To Avoid It
Converting A Tree to Rules
IF THEN
IF THEN
…
(Outlook = Sunny) ∧ (Humidity = High) P layT ennis = N o
(Outlook = Sunny) ∧ (Humidity = Normal) P layT ennis = Y es
Outlook
Sunny Overcast Rain Yes
Humidity
Wind
High Normal No Yes
Strong Weak
No
Yes
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Overfitting and How To Avoid It
Rules from Trees (Rule Post-Pruning)
Rules can be simpler than trees but just as accurate, e.g., in C4.5Rules: • path from root to leaf in (unpruned) tree forms a rule
• i.e., tree forms a set of rules
• can simplify rules independently by deleting conditions
• i.e., rules can be generalized while maintaining accuracy • greedy rule simplification algorithm
• drop the condition giving lowest estimated error (as for pruning) • continue while estimated error does not increase
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Overfitting and How To Avoid It
Rules from Trees
Select a “good” subset of rules within a class (C4.5Rules):
• goal: remove rules not useful in terms of accuracy
• find a subset of rules which minimises an MDL criterion
• trade-off accuracy and complexity of rule-set
• stochastic search using simulated annealing
Sets of rules can be ordered by class (C4.5Rules):
• order classes by increasing chance of making false positive errors
• set as a default the class with the most training instances not covered by any rule
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Further Issues in Tree Learning
Continuous Valued Attributes
Decision trees originated for discrete attributes only. Now: continuous attributes.
Can create a discrete attribute to test continuous value:
• T emperature = 82.5
• (Temperature>72.3)∈{t,f}
• Usual method: continuous attributes have a binary split • Note:
• discrete attributes – one split exhausts all values
• continuous attributes – can have many splits in a tree
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Further Issues in Tree Learning
Continuous Valued Attributes
• Splits evaluated on all possible split points
• More computation: n − 1 possible splits for n values of an attribute in training set
• Fayyad (1991)
• sort examples on continuous attribute
• find midway boundaries where class changes, e.g. for Temperature (48+60) and (80+90)
22
• Choose best split point by info gain (or evaluation of choice)
• Note: C4.5 uses actual values in data
Temperature: 40 48 60 72 80 90 PlayTennis: No No Yes Yes Yes No
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Further Issues in Tree Learning
Axis-parallel Splitting
Fitting data that is not a good “match” to the possible splits in a tree.
“Pattern Classification” Duda, Hart, and Stork, (2001)
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Further Issues in Tree Learning
Splitting on Linear Combinations of Features
Reduced tree size by allowing splits that are a better “match” to the data.
“Pattern Classification” Duda, Hart, and Stork, (2001)
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Further Issues in Tree Learning
Attributes with Many Values
Problem:
• If attribute has many values, Gain will select it
• Why ? more likely to split instances into “pure” subsets
• Maximised by singleton subsets
• Imagine using Date = June 22, 2020 as attribute • High gain on training set, useless for prediction
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Further Issues in Tree Learning
Attributes with Many Values
One approach: use GainRatio instead GainRatio(S, A) ≡ Gain(S, A)
SplitInformation(S, A)
SplitInformation(S, A) ≡ − 􏰂c |Si| log i=1 |S|
where Si is subset of S for which A has value vi
|Si| 2 |S|
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Further Issues in Tree Learning
Attributes with Many Values
Why does this help ?
• sensitive to how broadly and uniformly attribute splits instances • actually the entropy of S w.r.t. values of A
• i.e., the information of the partition itself
• therefore higher for many-valued attributes, especially if mostly
uniformly distributed across possible values
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Further Issues in Tree Learning
Attributes with Costs
Consider
• medical diagnosis, BloodTest has cost $150 • robotics, Width from 1ft has cost 23 sec.
How to learn a consistent tree with low expected cost?
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Further Issues in Tree Learning
Attributes with Costs
One approach: evaluate information gain relative to cost: • Example
Gain2 (S, A) . C ost(A)
Preference for decision trees using lower-cost attributes.
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Further Issues in Tree Learning
Attributes with Costs
Also: class (misclassification) costs, instance costs, . . .
See5 / C5.0 can use a misclassification cost matrix.
Can give false positives a different cost to false negatives
Forces a different tree structure to be learned to mimimise asymmetric misclassification costs – can help if class distribution is skewed – why ?
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Further Issues in Tree Learning
Unknown Attribute Values
What if some examples missing values of A?
Use training example anyway, sort through tree. Here are 3 possible approaches
• If node n tests A, assign most common value of A among other examples sorted to node n
• assign most common value of A among other examples with same target value
• assign probability pi to each possible value vi of A
• assign fraction pi of example to each descendant in tree
Note: need to classify new (unseen) examples in same fashion
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Further Issues in Tree Learning
Windowing
Early implementations – training sets too large for memory As a solution ID3 implemented windowing:
1. select subset of instances – the window
2. construct decision tree from all instances in the window
3. use tree to classify training instances not in window
4. if all instances correctly classified then halt, else
5. add selected misclassified instances to the window
6. gotostep2
Windowing retained in C4.5 because it can lead to more accurate trees. Related to ensemble learning.
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Learning Non-linear Regression Models with Trees
Non-linear Regression with Trees
Despite some nice properties of Neural Networks, such as generalization to deal sensibly with unseen input patterns and robustness to losing neurons (prediction performance can degrade gracefully), they still have some problems:
• Back-propagation is often difficult to scale – large nets need lots of computing time; may have to be partitioned into separate modules that can be trained independently, e.g. NetTalk, DeepBind
• Neural Networks are not very transparent – hard to understand the representation of what has been learned
Possible solution: exploit success of tree-structured approaches in ML
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Learning Non-linear Regression Models with Trees
Regression trees
• Differences to decision trees:
• Splitting criterion: minimizing intra-subset variation
• Pruning criterion: based on numeric error measure
• Leaf node predicts average class values of training instances reaching
that node
• Can approximate piecewise constant functions • Easy to interpret
• More sophisticated version: model trees
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Learning Non-linear Regression Models with Trees
A Regression Tree and its Prediction Surface
“Elements of Statistical Learning” Hastie, Tibshirani & Friedman (2001)
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Learning Non-linear Regression Models with Trees
Regression Tree on sine dataset
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Learning Non-linear Regression Models with Trees
Regression Tree on CPU dataset
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Learning Non-linear Regression Models with Trees
Tree learning as variance reduction
• The variance of a Boolean (i.e., Bernoulli) variable with success
probability p ̇ is p ̇(1 − p ̇), which is half the Gini index. So we could
interpret the goal of tree learning as minimising the class variance (or
Gini) in the leaves.
• In regression problems we can define the variance in the usual way:
Var(Y)= 1 􏰂(y−y)2 |Y | y∈Y
If a split partitions the set of target values Y into mutually exclusive sets {Y1, . . . , Yl}, the weighted average variance is then
Var({Y1,…,Yl})=􏰂l |Yj|Var(Yj)=…= 1 􏰂y2−􏰂l |Yj|y2j j=1 |Y| |Y|y∈Y j=1 |Y|
The first term is constant for a given set Y and so we want to
maximise the weighted average of squared means in the children.
standard deviation, in case of
√
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Learning Non-linear Regression Models with Trees
Learning a regression tree
Imagine you are a collector of vintage Hammond tonewheel organs. You have been monitoring an online auction site, from which you collected some data about interesting transactions:
# Model Condition Leslie Price
1. B3 excellent no 4513
2. T202 fair yes 625
3. A100 good no 1051
4. T202 good no 270
5. M102 good yes 870
6. A100 excellent no 1770
7. T202 fair no 99
8. A100 good yes 1900
9. E112 fair no 77
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Learning Non-linear Regression Models with Trees
Learning a regression tree
From this data, you want to construct a regression tree that will help you determine a reasonable price for your next purchase.
There are three features, hence three possible splits:
Model = [A100, B3, E112, M102, T202]
[1051, 1770, 1900][4513][77][870][99, 270, 625]
Condition = [excellent, good, fair]
[1770, 4513][270, 870, 1051, 1900][77, 99, 625]
Leslie=[yes,no] [625,870,1900][77,99,270,1051,1770,4513]
The means of the first split are 1574, 4513, 77, 870 and 331, and the weighted average of squared means is 3.21 · 106. The means of the second split are 3142, 1023 and 267, with weighted average of squared means 2.68 · 106; for the third split the means are 1132 and 1297, with weighted average of squared means 1.55 · 106. We therefore branch on Model at the top level. This gives us three single-instance leaves, as well as three A100s and three T202s.
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Learning Non-linear Regression Models with Trees
Learning a regression tree
For the A100s we obtain the following splits:
Condition=[excellent,good,fair] [1770][1051,1900][] Leslie=[yes,no] [1900][1051,1770]
Without going through the calculations we can see that the second split results in less variance (to handle the empty child, it is customary to set its variance equal to that of the parent). For the T202s the splits are as follows:
Condition=[excellent,good,fair] [][270][99,625] Leslie=[yes,no] [625][99,270]
Again we see that splitting on Leslie gives tighter clusters of values. The learned regression tree is depicted on the next slide.
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Learning Non-linear Regression Models with Trees
A regression tree
Model
=A100
=B3
=E122
=M102
=T202
Leslie
=yes =no
Leslie
=yes =no
f̂(x)=4513
f̂(x)=77
A regression tree learned from the Hammond organ dataset.
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f̂(x)=870
f̂(x)=1900
f̂(x)=1411
f̂(x)=625
f̂(x)=185

Learning Non-linear Regression Models with Trees
Model trees
• Like regression trees but with linear regression functions at each node
• Linear regression applied to instances that reach a node after full tree has been built
• Only a subset of the attributes is used for LR
• Attributes occurring in subtree (+maybe attributes occurring in path
to the root)
• Fast: overhead for Linear Regression (LR) not large because usually only a small subset of attributes is used in tree
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Learning Non-linear Regression Models with Trees
Two uses of features (Flach (2012))
Suppose we want to approximate y = cos πx on the interval −1 ≤ x ≤ 1. A linear approximation is not much use here, since the best fit would be
y = 0.
However, if we split the x-axis in two intervals −1 ≤ x < 0 and 0 ≤ x ≤ 1, we could find reasonable linear approximations on each interval. We can achieve this by using x both as a splitting feature and as a regression variable (next slide). COMP9417 ML & DM Tree Learning Term 2, 2020 92 / 100 Learning Non-linear Regression Models with Trees A small model tree COMP9417 ML & DM Tree Learning Term 2, 2020 93 / 100 ŷ = 2x+1 x <0 ≥0 ŷ = −2x+1 1 -1 0 1 -1 Learning Non-linear Regression Models with Trees Model Tree on CPU dataset COMP9417 ML & DM Tree Learning Term 2, 2020 94 / 100 Learning Non-linear Regression Models with Trees Smoothing • Na ̈ıve prediction method – output value of LR model at corresponding leaf node • Improve performance by smoothing predictions with internal LR models • Predicted value is weighted average of LR models along path from root to leaf • Smoothing formula: p′ = np+kq where n+k • p′ • p • q • n • k prediction passed up to next higher node prediction passed to this node from below value predicted by model at this node number of instances that reach node below smoothing constant • Same effect can be achieved by incorporating the internal models into the leaf nodes COMP9417 ML & DM Tree Learning Term 2, 2020 95 / 100 Learning Non-linear Regression Models with Trees Building the tree • Splitting criterion: standard deviation reduction SDR = sd(T) − 􏰂 |Ti| × sd(Ti) i |T| where T1, T2, . . . are the sets from splits of data at node. • Termination criteria (important when building trees for numeric prediction): • Standard deviation becomes smaller than certain fraction of sd for full training set (e.g. 5%) • Too few instances remain (e.g. less than four) COMP9417 ML & DM Tree Learning Term 2, 2020 96 / 100 Learning Non-linear Regression Models with Trees Pruning the tree • Pruning is based on estimated absolute error of LR models • Heuristic estimate: n + v × average absolute error n−v where n is number of training instances that reach the node, and v is the number of parameters in the linear model • LR models are pruned by greedily removing terms to minimize the estimated error • Model trees allow for heavy pruning: often a single LR model can replace a whole subtree • Pruning proceeds bottom up: error for LR model at internal node is compared to error for subtree COMP9417 ML & DM Tree Learning Term 2, 2020 97 / 100 Learning Non-linear Regression Models with Trees Discrete (nominal) attributes • Nominal attributes converted to binary attributes and treated as numeric • Nominal values sorted using average class value for each one • For k-values, k − 1 binary attributes are generated • the ith binary attribute is 0 if an instance’s value is one of the first i in the ordering, 1 otherwise • Best binary split with original attribute provably equivalent to a split on one of the new attributes COMP9417 ML & DM Tree Learning Term 2, 2020 98 / 100 Summary Summary – decision trees • Decision tree learning is a practical method for many classifier learning tasks – still a “Top 10” data mining algorithm – see sklearn.tree.DecisionTreeClassifier • TDIDT family descended from ID3 searches complete hypothesis space - the hypothesis is there, somewhere... • Uses a search or preference bias, search for optimal tree is, in general, not tractable • Overfitting is inevitable with an expressive hypothesis space and noisy data, so pruning is important • Decades of research into extensions and refinements of the general approach, e.g., for numerical prediction, logical trees • Often the “try-first” machine learning method in applications, illustrates many general issues • Performance can be improved with use of “ensemble” methods COMP9417 ML & DM Tree Learning Term 2, 2020 99 / 100 Summary Summary – regression and model trees • Regression trees were introduced in CART – R’s implementation is close to CART, but see sklearn.tree.DecisionTreeRegressor for a basic version • Quinlan proposed the M5 model tree inducer • M5′: slightly improved version that is publicly available (M5Pin Weka is based on this) • Quinlan also investigated combining instance-based learning with M5 • CUBIST: Quinlan’s rule learner for numeric prediction www.rulequest.com • Interesting comparison: Neural nets vs. model trees — both do non-linear regression • other methods also can learn non-linear models COMP9417 ML & DM Tree Learning Term 2, 2020 100 / 100 References Breiman, L., Friedman, J. H., Olshen, R. A., and Stone, C. J. (1984). Classification and Regression Trees. Wadsworth, Belmont. Flach, P. (2012). Machine Learning. Cambridge University Press. Hastie, T., Tibshirani, R., and Friedman, J. (2009). The Elements of Statistical Learning. Springer, 2nd edition. Hyafil, L. and Rivest, R. (1976). Constructing Optimal Binary Decision Trees is NP-Complete. Information Processing Letters, 5(1):15–17. Quinlan, J. R. (1986). Induction of decision trees. Machine Learning, 1(1):81–106. COMP9417 ML & DM Tree Learning Term 2, 2020 100 / 100

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