Page 209 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 191

TABLE 5-1

SAMPLE BIAS EXPRESSIONS FOR THE
BHATTACHARYYA DISTANCE

hi h; = 1 hi = 1 hi =4

PI 0 0.82 0
P2 0 0 0.11 n
E 50% 10% Depends on n
0.25
0.35
0.25
1 - - -
k k k
n+l n+l n+l
E{ApZJ 0.125- 0.125- 0.08-
k k k

TABLE 5-2

VALUES OF k AND N REQUIRED TO MAINTAIN E { i] 0.223
I

I n I 4 8 16 32 64 128 1
lL=nk 1 3.9 6.2 10.7 19.6 2407 9396
39.6
50
628
172
16

A
E{pJ = 0.223 [4]. In this example, two sets of samples are drawn from the
same source. Therefore, if an infinite number of samples is generated, the dis-
tributions of two sets are identical (the Bayes error of 50% or p = 0). How-
ever, with a finite number of samples, E ( p} is no longer equal to zero, indicat-
ing that there exists a difference between two generated distributions. Accord-
ing to (3.151), E(p} = 0.223 means that the overlap between them (the Bayes
error) is less than 40%. The larger E { p) is, the smaller the overlap becomes.