Page 278 - Introduction to Statistical Pattern Recognition

260 Introduction to Statistical Pattern Recognition

(6.20)

Uniform kernel: For a uniform kernel with the covariance matrix r2A,
llv inside L(X)
K(Y) = (6.22)
0 outside L(X) .
where

<
L(X) = (Y: ~(Y,x) r4n+2 1 , (6.23)
d2(Y,X) = (Y-X)TA-'(Y-X), (6.24)

and

(6.25)

Then, K~(X) is also uniform in L(X) with the height llv2. Therefore,
w = [($Y)dY = -
1
(6.26)
V
Also, since the covariance matrix of K(X) is r2A, the covariance matrix of
d(X)lw is also r2A as
1
[(x,(Y -X)(Y -X)'-dY = r2A (6.27)
v
Therefore, for the uniform distribution of (6.22),
B =A and p(X) = a(X) . (6.28)

Note that w's for both normal and uniform kernels are proportional to
I' -n or v-' . In particular, w = l/v for the uniform kernel from (6.26). Using
this relation, the first order approximation of the variance can be simplified
further as follows: