Page 195 - Introduction to Statistical Pattern Recognition

4 Parametric Classifiers 177

3. Repeat Experiment 2 for Ni = 50, 100,200,400 and plot the error vs. s.

4. Design the optimum linear classifier by minimizing the mean-square error
of (4.76). Use 100 generated samples per class from Data Z-A for design,
and test independently generated 100 samples per class. Observe the
difference between this error (the error of the holdout method) and the one
of the resubstitution method.

5. Two 8-dimensional normal distributions are characterized by
PI = P2 = 0.5, MI = M2 = 0, Zj = o’Ri where Rj is given in (4.126) with
02- 2-
I - o2 - 1, pI = 0.5, and p2 = -0.5.
(a) Compute the Bayes error theoretically.
(b) Generate N, design samples per class, and compute the sample mean
..
,.
Mi and sample covariance Zi.
,.
(c) Approximate the correlation matrix of Cj by the toeplitz form of
(4.126).
6 A
(d) Design the quadratic classifier with Mi and the approximated Zj.
(e) Generate 1000 test samples, and classify them by the quadratic
classifier designed in (d).
(f) Repeat (b)-(e) 10 times, and compute the average and standard devia-
tion of the error.

(g) Compare the error of (0 with the error of (a) for various N,. Sug-
gested Nj’s are 10, 20, and 40.

Problems

1. Let x,~ (i = 1,. . . ,n) be independent and identically distributed with an
exponential density function
1 sj
p-(r.) = - exp[--1 u(sj) (i = 1,2)
’ -.I
hi hi
where II (.) is the step function.
(a) Find the density function of the Bayes discriminant function h (X).