Module: EEEM007 ADVANCED SIGNAL PROCESSING
Year: 2018/2019. Examiner: M D Plumbley
Date Due: 4pm Tue 12 March 2019 (Week 6)
ASSIGNMENT
Consider two equiprobable normally distributed classes,1 and 2 with the same covariance matrix Σ.
The mean of class and the covariance matrix Σ are given as
1. Sketch the classes, marking the orientation of axes of the equiprobable density ellipses of the two
distributions, and the length of their major and minor axes. [Hint: Use eigenvalues/eigenvectors.]
2. Find the classification boundary for the Bayes optimal decision rule [Hint: Equiprobable, same
covariance]. Calculate the Bayesian error for both distributions, and hence the total Bayes error.
3. Assume that the true covariance matrix Σ is known, and the mean 2 of class 2 is known.
Assume however that the mean 1 of class 1 is not known, but instead we will have to make a
decision using an estimate 1 calculated as the sample mean of = 25 samples from class 1.
(a) Give the new decision rule for this case.
(b) For a sample = [1 2]
from class 1, consider the variance of the marginal distribution of
each component 1 and 2. Hence for the sample mean 1 = [11 12]
of = 25 samples
from class 1, calculate the variance and standard deviation of each component 11 and 12.
Suppose that each component 11 and 12 of the estimated mean ???1 is lower than its true value by
one standard deviation of the distribution of sample means, as calculated in part 3(b) above.
(c) Sketch the decision boundary in this case, and calculate its location.
(d) Estimate the expected error of the resulting classifier.
4. The classifier designed in Step 3 is tested on a test set containing 25 samples from each class. The
samples in the test set are normally distributed with known covariance Σ as above, and the mean
of the test set for class 1 is the same as the true mean 1. However, the test set class for class 2
as a slightly different mean. Both components of the mean for the test set for class 2 are either
(a) higher than the true mean 2 by one standard deviation of the distribution of sample means; or
(b) lower than the true mean 2 by two standard deviations of the distribution of sample means.
For each case (a) and (b), what will be the estimated error in this case?
What would be the effect of changing the size of the test set?
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5. Comment on the relationships between the errors obtained in Steps 2-4. How do the results using
estimated class parameters and test sets compare with optimal and expected errors?
Marking Scheme
1. Analysis and sketch of class distributions 20%
2. The Bayes optimal decision rule [10%] and the decision rule designed in Step 3 [10%] 20%
3. Errors obtained in Steps 2 - 4 [Step 2: 10%; Step 3: 10%; Step 4: (a): 10%, (b) 10%] 40%
4. Comments 20%
In order to gain full marks, all the steps leading to the final answer must be clearly indicated, explained and
justified. The length of the completed assignment is expected to be 4-6 pages.
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