Page 276 - Introduction to Statistical Pattern Recognition

258 Introduction to Statistical Pattern Recognition

Therefore, the variance of p(X) is

Approximations of moments: In order to approximate the moments of
A
p(X), let us expand p(Y) around X by a Taylor series up to the second order
terms as
1
+
p(~) E~(x) V~'(X)(Y-X) + -~~{v~~(x)(Y-x)(Y-x)'] (6.10)
.
2
Then, p (X)*K(X) may be approximated by
p(X)*K(X) = jp(Y)K(Y-x)dY

gp (X)jK(Y-X)dY
1
+ -tr{ v2p (X)j(Y -X)(Y -X)'K(Y -X)dY ) , (6.1 1)
2
where the first order term disappears because K(.) is a symmetric function.
Since ~K(Y-x)~Y = 1 and ~(Y-x)(Y-x)'K(Y-x)~Y = r.2~ for K(.) of (6.3),
(6.1 1) can be expressed by

(6.12)

where

(6.13)

Similarly,
p(X)*d(X) Ep(X)jt?(Y-X)dY

1
+ -~~(v~~(x)~(Y-x)(Y-x)~(Y-x)~Y
.
1
2 (6.14)
Although K(.) is a density function, K*(.) is not. Therefore, ld(Y)dY has a
value not equal to 1. Let