Page 333 - Introduction to Statistical Pattern Recognition

7 Nonparametric Classification and Error Estimation 315

Finally, combining (7.27), (7.29), and (7.34), and taking the expectation
with respect to X [SI,

E{;NN~ EE~N -tP,E,( IA l-””tr{ABI(X)]} (7.35)
where

1
I
1
B 1 (XI = P-*”’(X)[q*(x)-q (X)l[Vp(X)VTq (X)p-?x)+p2q 1 (X)] 1
(7.36)

n +2
- r21n (-)r( 1 +2in )
2
- N-2111 (7.37)
-
nn
The second line of (7.37) is obtained by approximating l-(x+a)lT(x) by xa for
a large integer x and a small a.

Effect of Parameters

Several observations may be made at this point. First, note that the
value of PI is completely independent of the underlying densities. It depends
only on the dimensionality of the data and the sample size, and does not
depend on the particular distributions involved. The term inside the expecta-
tion in (7.33, on the other hand, does not depend on the sample size. For any
given set of distributions this term remains fixed regardless of the number of
samples. This term does, however, depend heavily on the selection of the
metric, A. These equations, therefore, yield much information about how the