Page 116 - Introduction to Statistical Pattern Recognition

98 Introduction to Statistical Pattern Recognition

An upper bound of the integrand may be obtained by making use of the fact
that

min[a, b] I aSbl-’ OIsIl (3.147)

for a, b20. Equation (3.147) simply states that the geometric mean of two
positive numbers is larger than the smaller one. The statement can be proved
as follows. If ad, the left side of (3.147) is a, and the right side can be
rewritten as ax(bla)’-S. Since (bla) > 1 and 1-5 2 0 for 0 I s I 1, the right
side becomes larger than the left side. Likewise, if a>b, the left side of
(3.147) is b, and the right side is rewritten as bx(a/b)’, which is larger than b
because (alh) > 1 and s 2 0. Using the inequality of (3.147), E can be
bounded by

where E, indicates an upper bound of E. This E,, is called the ChernofS bound
[16]. The optimum s can be found by minimizing E~,.
When two density functions are normal as Nx(MI,C,) NX(M2,C2),
and
the integration of (3.148) can be carried out to obtain a closed-form expression
for E,. That is,

where

(3.150)

This expression of p(s) is called the ChernofS disrance. For this case, the
optimum s can be easily obtained by plotting p(s) for various s with given Mi
and Cj. The optimum s is the one which gives the maximum value for p(s).