Page 157 - Introduction to Statistical Pattern Recognition

4 Parametric Classifiers 139

Procedure I1 to find s (the resubstitution method):

A 1
(1) Compute the sample mean, Mi, and sample covariance matrix, Zj.
(2) Calculate V for a given s by V = [siI + (I-S)%J’($~-$~).

(3) Using the V obtained, compute yy) = VTXf) (i = 1,2;
j = 1, . . . ,N), where Xy’ is the jth oj-sample.
(4) The y;” and Y)~”S, which do not satisfy y;” c -v, and
yj (2) > -v,, are counted as errors. Changing v, from -- to -tm,
find the v, which gives the smallest error.
(5) Change s from 0 to 1, and plot the error vs. s.

Note that in this process no assumption is made on the distributions of X.
Also, the criterion function, fi is never set up. Instead of using an equation for
ft the empirical error-count is used. The procedure is based solely on our
knowledge that the optimum V must have the form of
[SEI + (l-s)x*]-yM*-M,).
In order to confirm the validity of the procedure, the following experi-
ment was conducted.

Experiment 1: Finding the optimum s (Procedure 11)
Data: I-A (Normal, n = 8, E = 1.9%)
Sample size: N = N2 = 50, 200
I
No. of trials: z = 10
Results: Fig. 4-8

Samples were generated and the error was counted according to Procedure 11.
The averaged error over 10 trials vs. s is plotted in Fig. 4-8. Note that the
error of Procedure I1 is smaller than the error of Procedure I. This is due to the
fact that the same samples are used for both designing and testing the classifier.
This method of using available samples is called the resubstitution merhod, and
produces an optimistic bias. The bias is reduced by increasing the sample size
as seen in Fig. 4-8. In order to avoid this bias, we need to assure independence
between design and test samples, as follows: