Page 160 - Introduction to Statistical Pattern Recognition

142 Introduction to Statistical Pattern Recognition

Since s’=q’+o?, f (ql,q2,s:,s$) is a function of ql, q2, o:, and also.
The reason why s’ is used instead of 0’ is that the second order moments are
easier to handle than the central second order moments in which the means
must be subtracted. The variation off, Sf, due to the variation of y(X), 6y(X),
is expressed by

where Sqi and 8s; are computed from (4.48) and (4.49) as

(4.5 1)

(4.52)

Substituting (4.5 1) and (4.52) into (4.50), 6f becomes

In order to make Sf = 0 regardless of Sy(X), the [.] term in the integrand must
be zero. Thus