Page 152 - Introduction to Statistical Pattern Recognition

134 Introduction to Statistical Pattern Recognition

hi
ao:
-- --=Mi, (4.23)
av
av -2x;v,
(4.24)

Substituting (4.23) and (4.24) into (4.21) and (4.22), and equating (4.21) and
(4.22) to zero,

(4.25)

(4.26)

Substituting (4.26) into (4.25), and solving (4.25) for V, the optimum V can be
computed. However, it should be noted that the error in the h-space depends
only on the direction of V, and not on the size of V. Therefore, for simplicity,
we eliminate any constant term (not a function of Mi and E;) multiplying to V,
resulting in

v = [SEI + (l-S)Z2]--1(M2 - M,) (4.27)
,
where

(4.28)

Note that the optimum V has the form of (4.27) regardless of the selection off.
The effect off appears only in s of (4.28). In (4.2), the Bayes classifier for
normal distributions with the equal covariance matrix X has V = C-I(M2-MI).
Replacing this C by the averaged covariance matrix [sCl+(l-s)Z2], we can
obtain the optimum V of (4.27).

Once the functional form off is selected, the optimum v, is obtained as
the solution of (4.26).

Example 2: Let us consider the Fisher criterion which is given by