Page 153 - Introduction to Statistical Pattern Recognition

4 Parametric Classifiers 135

(rll -112)2
f= o:+o; (4.29)
This criterion measures the difference of two means normalized by the aver-
aged variance. The derivatives off with respect to 0: and 0; are

(4.30)

Therefore, s = 0.5 and the optimum V is
1 1
v = [--XI + -22]-'(M2 -MI) . (4.3 1)
2 2
The h(X) with V of (4.31) and the resulting linear classifier are called the
Fisher discriminant function and Fisher linear classifier, respectively [2]. The
Fisher criterion does not depend on v,, because the subtraction of q2 from ql
eliminates v, from (4.19). Therefore, we cannot determine the optimum v, by
maximizing this criterion.

Example 3: Another possible criterion is
p1d + p2rl; (4.32)
f= Plo:+P,o; .

This criterion measures the between-class scatter (around zero) normalized by
the within-class scatter, and will be discussed in Chapter 10. For this criterion,

(4.33)

Thus, s = P I and the optimum V is
V=[P,& +P22J1(M2 -MI). (4.34)

On the other hand,

(4.35)

Substituting (4.35) into (4.26), and rewriting (4.26) by using (4.19)

VT[PIMI + P2M2]+ v, = 0, (4.36)
or