ACS6124 Multisensor and Decision Systems: I Multisensor Systems
Tutorial Questions
1. Describe the characteristics of the following change types:
(a) abrupt change
(b) incipient change
(c) intermittent change
(d) If the measurement of a system output is represented by the measurement model y(t) = x(t)+v(t), where x(t) is the true system parameter value, y(t) is the sensor output and v(t) is zero mean white noise signal, give a mathematical representation of how the following changes are modelled:
i. Additive change
ii. Multiplicative change
2. An object tracking system measures the position of the object in clutter and so the model used to alert if the object changes its position. The position of the object is typically θ0 distance from a reference point on a straight road.
(a) GiveamathematicalexpressionfortheBayesoptimaldecisionrulebasedonasinglemea- surement y(t) for the hypothesis that the position has not changed against the null hypoth- esis that the position has changed.
(b) By assuming that the position sensor noise is zero mean Gaussian distributed with vari- ance σ2, determine the above optimal decision rule that detects if the position has been changed by ∆ along this road. The decision rule must be based on the normalised mea- surement quantity,
s(t) = y(t) − θ0 σ
(c) If the level of the change is unknown, discuss how you would select the threshold with the above decision rule.
3. A change detection system is required to make decisions on whether the system has changed or not. The system will need to make this decision based on a collection of measurement data.
(a) If the signal measurements are y(1), y(2), · · · , y(t), · · · , y(T ), then determine the mean and
the variance of the term
t
g(t) = s(τ) τ=1
where s(τ) = y(τ)−θ0 and θ0,σ0 are the mean and standard deviation of the signal mea- σ0
surements when the system is operating normally. (b) For the data record given below:
t
1
2
3
4
5
6
7
8
y(t)
1252
1246
1253
1248
1246
1262
1266
1264
1
apply the one-sided positive CUSUM test under the assumption that the normal opera- tional value of the system is 1250 with a variance of 4. You may also ignore the leakage term γ and choose a threshold of 10.
4. Aspeedsensorinstalledbyaroadprovidesnoisymeasurementsofvehiclespeedswithrandom variations arising from a number of environmental factors.
(a) Assuming the speed measurement obtained by the sensor is corrupted by noise modelled as a random variable, provide a formal definition of the speed estimation problem.
(b) Data from previous experiments has been used to produce a random model of the vehicle speed. In particular, the speed is modelled as random variable X with distribution PX and zero mean. Derive the optimal estimator when the performance criteria is the mean square error (MSE) of the estimate.
(c) The sensor manufacturer has provided an additive white Gaussian noise (AWGN) model for the noise introduced by the sensor. Specifically, the noise is modelled as a random variable Z ∼ N (0, σ2) which results in measurement model Y = X + Z. Because of computational limitations the chief software engineer has requested a linear estimator of the form Xˆ = αY to run on the processor of the sensor. Derive the linear minimum mean square error (LMMSE) estimator using the orthogonality principle.
5. Define the following system health states:
(a) Normal. (b) Anomaly.
(c) Fault.
(d) Malfunction.
(e) Failure.
6. A post-operative patient’s health is monitored with EEG signal measurements.
(a) What functionalities should extracted features have?
(b) Suggest a suitable set of features for monitoring the patient health.
7. A health monitoring decision support system reeives multiple signal data that a human op- erator can visually assess for changes to the underlying health of a vehicle. The visualisation is to be achieved by using principal component analysis (PCA) applied to the data to extract a two-dimensional feature vector that can be projected on to a screen.
(a) Describe the steps involved in the feature extraction process, with mathematical details but without derivation.
(b) Based on the historical collection of data from three vehicle sensors under normal condi- tions, the data mean was found to be 0 and data covariance matrix to be
5 −2 3 C=−2 10 −2
3 −2 5
Use PCA to identify the data transformation that extracts informative features and deter-
mine the percentage of variation captured by the data.
(c) For the following data vectors, determine the feature vectors that result from the data transformation above.
131 √√3√
3 3 √3 3
1 −3 2 √√√
y(1)=2,y(2)=2,y(3)= 0 ,y(4)=2 5 −3 0 −2
√√√
666
2
8. Define the following functions of a health monitoring decision support system:
(a) Novelty detection. (b) Fault detection.
(c) Fault Diagnosis. (d) Prognosis.
9. Anoveltydetectionsystemistobedesignedtodetectifpatientsareinnormaloranomalousstate during a surgical operation.
(a) Outline the simple discordancy test, defining the decision rule and all relevant terms.
(b) A sensor measures the deviations from normal of a measure associated with depth of anaesthesia with the expected measure being zero and variance of 1. Apply the discor- dancy test at 95% level of confidence to the different patients data below.
(c) What is the disadvantage of applying the discordancy test to data from multiple sensors (multivariate data)?
10. A complex system is to be monitored using multiple sensor signals to detect if the system operation has deviated from normal or not.
(a) OutlinetheMahalanobisdiscordancytestbasednoveltydetectionandthemaximumlikelihood based novelty detection methods including the relevant mathematical details.
(b) Statetheconditionsunderwhichtheybecomeequivalentbyshowingthismathematically.
(c) Suggest an improved method of novelty detection if the assumptions made above are not true.
11. Adecisionsupportsystemisrequiredtoclassifythedatafromtwosensorsfromaircraftengines into specific fault classes.
(a) What are the two assumptions that need to be made for a pattern classification approach to be used for fault diagnosis.
(b) Describe the nearest neighbour approach to pattern classification.
(c) Comment on the advantages and disadvantages of the nearest neighbour method when compared with the linear discriminant method.
12. A multi target tracking system uses multisensor fusion methodologies to obtain estimates of the targets.
(a) Describe the group sensor method of sensor fusion.
(b) Describe the track to track sensor fusion method of sensor fusion.
(c) Compare the two methods in terms of their advantages and disadvantages.
k
1
2
3
4
5
6
7
8
r(k)
1.1
-0.3
-2.0
2.1
-0.9
1.8
0.9
2.4
3