Page 308 - Introduction to Statistical Pattern Recognition

290 Introduction to Statistical Pattern Recognition

(6.137)
4!c4 = - [*I7 m2 + 1 .3 [:IF
-
4
=
m4
m4
o4 -- 3. (6.138)
o4
Therefore, terminating at i = 4, we have an approximation of a density function
p (x) in terms of &(X) and the moments of p (x) as

L
1 0 1: 1
- [:I
3
1 X2 m3 X
- (2n)1/20 exp -- I+- -
- - [ 202 3!03 I

. (6.139)

Because of the complexity involved in the multivariate case, it is not as
easy to find general basis functions or to calculate the coefficients.

Density Function of Binary Inputs

Basis functions for binary inputs: When the n inputs are binary
numbers +I or -1, it is known that a linear combination of 2" independent
basis functions can yield any density function without error.
2"- I
P(X) = ; =o c;+;(X) . (6.140)

Table 6-5 shows the truth table that specifies p(X). Again, it is hard to say
how we should select the 2" basis functions. However, a typical set of basis