Page 312 - Introduction to Statistical Pattern Recognition

294 Introduction to Statistical Pattern Recognition

This expansion is called the Bahadur- expansion [24]. In this expansion, we
can see the effects of the correlations on the approximation of a density func-
tion. In general, since the higher-order correlations are usually smaller than
lower-order correlations, we may terminate the expansion with a reasonable
number of terms and reasonable accuracy.

Example 1: Let us calculate the Bahadur expansions for two density
functions, p I (Y) and pz(Y), given in Fig. 6-7. We obtain the same basis func-
tions and the same kernels for both p I(Y) and p2(Y) as

1
1
PI =? and P2=?, (6.160)

(6.161)

(6.1 62)

Y2

Fig. 6-7 An example for the Bahadur expansion.

The correlation coefficients of yI and y2 for p I (Y) and pz(Y), y(,i’ and y(,;), are
different and are calculated by