Page 292 - Introduction to Statistical Pattern Recognition

274 Introduction to Statistical Pattern Recognition

The optimal k, k*, may be found by minimizing the mean-square error of
(6.94). That is, solving aMSE lak = 0 for k yields [5]

L J

(6.95)

As in the Parzen case, the optimal k is a function of X. Equation (6.95) indi-
cates that k* is invariant under any non-singular transformation. That is,

k*(Z) = k*(X) . (6.96)

Also, k* and I-* of (6.36) are related by

(6.97)

This indicates that both the Parzen and kNN density estimates become optimal
in the same local range of L(X). The resulting mean-square error is obtained
by substituting (6.95) into (6.94).

4
-
MSE* { &X)} = . (6.98)
IA 1''

Note that (6.98) and (6.38) are identical. That is, both the Parzen (with the
uniform kernel) and kNN density estimates produce the same optimal MSE.
The globally optimal k may be obtained by minimizing the integral
mean-square error criterion. From (6.94), with a fixed k,

I 1
.
4
(6.99)
IMSE = -Jp2(X)dX + -~~'"(~)~'~'~~~(X)p~~'"(X)dx
k
N