Module: EEEM007 ADVANCED SIGNAL PROCESSING

Year: 2019/2020. Examiner: M D Plumbley

Date Due: 4pm Tue 17 March 2020 (Week 6)

[Note: This is a theoretical assignment: use of Matlab or similar is not required. It is recommended to do the

assignment by hand, then scan to PDF for submission to SurreyLearn.]

ASSIGNMENT

Consider two equiprobable normally distributed two dimensional classes, 1 and 2 with the same

covariance matrix Σ and equal priors with the following gaussian distribution:

where = 2 is the dimensionality, Σ1 = Σ2 = Σ = [] and the mean of classes 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,

1-dimensional classification problem]

2. Find the classification boundary for the Bayes optimal decision rule defined as:

Calculate the Bayesian error for both distributions such that, and hence the

total Bayes error [Hint: Equiprobable, same covariance].

3. Assume that the true covariance matrix Σ is known, and the true mean 2 of class 2 is known.

Assume however that the mean 1 of class 1 is not known for the purpose of designing a decision

rule, but instead we will have to make a decision using an estimate

1 calculated as the sample

mean of = 25 samples from class 1.

2 and use it to calculate the variance and standard deviation from = 25]

(b) Suppose that a particular = 25 samples are taken from class 1, and that each component

of the average of those samples (the sample mean) is a slightly lower than the true mean 1,

such that each component

11 and

12 of the estimated mean

1 is lower than its true mean 1

by one standard deviation of the distribution of sample mean, as calculated in part 3(a) above.

Calculate the estimated mean

1, and give the new decision rule.

(c) Sketch the decision boundary in this case and calculate its location.

(d) Estimate the expected e???? ?f ?he ?e??l?i?g cla??ifie?.

[Hint ? The meaning of different symbols is shown in the table below]

NOTE: Use your personalized assignment values

for to do the assignment.

Estimated value Sample mean /Average

4. Assume that the performance of the cla??ifie? de?ig?ed in Step 3 is estimated by using a test set

containing test = 25 samples from each class. The samples in the test set for both class 1 and 2

are normally distributed with known covariance Σ as above. The test set for class 1 has a sample

mean which is the same as the true mean 1. However, the test set for class 2 has a sample mean

that is slightly different to the true mean 2. Consider two possible cases:

(a) Both components of the sample mean for the test set for class 2 are higher than the true mean

2 by one standard deviation of the distribution of sample means; or

(b) Both components of the sample mean for the test set for class 2 are lower than the true mean

2 by two standard deviations of the distribution of sample means.

For each case (a) and (b), estimate the expected value of the test set error.

(c) What would be the effect of changing the size of the test set, test?

[Hint: From Question 3,

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. Question 1 ? 3

2. Question 2 ? 4

3. Question 3 ? 5

4. Question 4 ? 5

5. Question 5 ? 3

Total marks – 20

I? ??de? ?? gai? f?ll ?a?k?? all ?he ??e?? leadi?g ?? ?he fi?al a??∥e? ???? be clea?l? i?dica?ed? e??lai?ed

a?d j???ified? The le?g?h ?f ?he c???le?ed assignment is expected to be 4-6 pages.

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