Page 121 - Introduction to Statistical Pattern Recognition

3 Hypothesis Testing 103

Validity of the Bhattacharyya Distance

The Bhattacharyya distance for norma. distributions, (3.152). is a very
convenient equation to evaluate class separability. Even for non-normal cases,
(3.152) seems to be a reasonable equation, measuring in the first term the dis-
tance between MI and M2 normalized by the average covariance matrix, and in
the second term the distance due to the covariance-difference. The question
here is how widely (3.152) can be used. Since we cannot examine all possible
non-normal distributions, we limit our discussion to a family of gamma distri-
butions. Also, in order to avoid complexity, we present only one-dimensional
cases. Note that, if two diagonalized covariance matrices are used, ~(112) of
(3.152) is the summation of the Bhattacharyya distances of individual vari-
ables.

p for gamma densities: When two one-dimensional distributions are
gamma as shown in (2.54), 1-d~ can be computed as

Or, tak-
where ai and pi are the parameters of the gamma distribution for ai.
ing the minus-log of (3.158),