Page 347 - Introduction to Statistical Pattern Recognition

7 Nonparametric Classification and Error Estimation 329

The threshold for normal distributions: However, if the p,(X)’s are
normal distributions, the effect of the threshold can be analyzed as follows.
Recall from (6.7) that the expected value of &(X) in the Parzen density esti-
mate is given by p;(x)*~;(X). When both pj(X) and K;(X) are normal with
covariance matrices C; and r2Cj respectively, this convolution yields another
normal density with mean Mi and covariance (l+r2)Ci, as shown in (6.31).
A
For larger values of r, the variance of p;(X) decreases, and the estimate
approaches its expected value. Substituting the expected values into the
estimated likelihood ratio, one obtains

(7.53)

Except for the 1/( l+r2) factors on the inverse covariance matrices, this expres-
sion is identical to the true likelihood ratio, -In p I (X)/p2(X). In fact, the two
may be related by

The true Bayes decision rule is given by - lnpI(X)/p2(X) ><1nPIlP2. Using
(7.54), an equivalent test may be expressed in terms of the estimated densities:

where
1 PI 1 /.? IC,I
t=- (ln-) + -(-)In- . (7.56)
I+/.* P2 2 I+/-, IC, I

In all of our experiments, we assume PI = P2 = 0.5, so the first term of (7.56)
may be neglected. Equation (7.56) gives the appropriate threshold to use when
the Parzen classifier with a normal kernel function is used on normal data.
This indicates that t can be kept at zero if Cl = XI, but t should be adjusted for
each value of I’ if Cl # C2. Otherwise, the classifier based on the Parzen