Page 186 - Introduction to Statistical Pattern Recognition

168 Introduction to Statistical Pattern Recognition

However, with approximation, the effect of the sample size is significantly
reduced.

Experiment 4: Error of the quadratic classifier
Data: RADAR
Dimension: n = 66
Sample size: N = N2 = 4400,720, 360 (Design)
I
N = N2 = 4400 (Test) ,.
I
Approximation: Toeplitz approximation for Zj (Design only)
No. of trials: z = 1
Results: Table 4- 1

A A
In this experiment, Mi and the approximated C; were used to design the
quadratic classifier of (4. l), and independent 4400 samples per class were
tested. The results were compared with the error of the quadratic classifier
designed without the approximation. The error of the approximated case is
somewhat larger than the error without approximation. However, with approx-
imation, the effect of the sample size is virtually eliminated.
The performance evaluation of the toeplitz approximation can be carried
out experimentally as seen in Experiments 3 and 4. That is, the means and the
parameters of the covariance matrices are estimated from design samples, and
the quadratic classifier based on these estimated parameters is tested by
independent test samples.
However, when the distributions of X are normal with given Mi and C;,
the performance of the quadratic classifier with the toeplitz approximation can
be evaluated theoretically as follows.

(1) Average the first off-diagonal terms of Ri from the given Xj and
form the toeplitz approximation as in (4.143).

(2) Using the given Mi and approximated Zj, design the quadratic
classifier of (4.1).

(3) Compute the error by testing the original distributions of
Nx(M;,Z;)’s. Since Xi’s used for design (the toeplitz approximations) are dif-