Page 310 - Introduction to Statistical Pattern Recognition

292 Introduction to Statistical Pattern Recognition

(6.143)

The ai’s are control parameters and must be in the range 0 < ai < 1. The ci’s
can be calculated by
2“- 1
Cj = I: K(XOp(X,)$i(XE) . (6.144)
bO
Two special cases of the above expansion are well known.

The Walsh function: Selecting a; = 0 (i = I, . . . ,n), the basis functions
become

with the kernel
1
K(x) = - (6.146)
2”
This set of basis functions is known as the Wdsh funcfions and is used often
for the expansions of binary functions.

The Bahadur expansion: Let us introduce the following transformation:

(6.147)

That is, x, = + 1 and -1 correspond to yi = 1 and 0. Also, let Pi be the margi-
nal probability of yi = 1,
P, = PI-{Yi = +1) . (6.148)

Then the expected value and variance of yi are given by
E{y;}=lxP;+Ox(l-P;)=P;, (6.149)

If we select ai as