Page 206 - Introduction to Statistical Pattern Recognition

188 Introduction to Statistical Pattern Recognition

However, the third and fourth order moments, (5.14) and (5.15), must be
modified according to (2.53), (2.48), and (2.49), resulting in

1
E{ Amy)Acfj) ] E -COV{ Ax~,Ax~Ax~~)] (5.20)
,
N
I
1
-Var{Axy)Axy)) for i # j, i = k, j =
N
1
E { Ac$)Ac~~) G -Var { Ax!'.)* } for i = j = k = (5.21)
}
N
1
-COV( Ax~)Ax~), Ax~)Ax~)} otherwise
N
,
Subsequently, E(Af} of (5.18) and Var(Af] of (5.19) must be modified. How-
ever, it must be noted that both (5.20) and (5.21) are proportional to 1/N.
Thus, even for non-normal cases, we can isolate the effect of the sample size,
and g(N) of (5.5) becomes 1/N. This means that we can adopt the estimation
procedure off of (5.6).

Bhattacharyya Distance

A popular measure of similarity between two distributions is the Bhatta-
charyya distance

(5.22)

Since p is a function of MI, MZ, C,, and C2, it is a member of the family of
functions discussed previously.