Page 284 - Introduction to Statistical Pattern Recognition

266 Introduction to Statistical Pattern Recognition

In the transformed Z-space, IMSE$ of (6.41) becomes

(6.54)

where c1 and c2 are positive constants. IMSE; can be minimized by minimiz-
ing t? 1. with respect to A. Since A is normalized by I A I I" such that

(6.55)

the sca : of the matrix has no effect. Thus, we will minimize t?(.) with
respect to hi's with the constraint

n
Ihl = Ilkj = 1 (6.56)
i=l

Now, tr( .] can be evaluated as

(6.57)

where

Thus, the criterion to be optimized is

where p is a Lagrange multiplier. Taking the derivative of J with respect to hk
and setting the result equal to zero,