Page 350 - Introduction to Statistical Pattern Recognition

332 Introduction to Statistical Pattern Recognition

This option makes no assumptions about the densities of the data or the shape
of the kernel function. However, since the value of the threshold is customized
to the data being tested, using this option will consistently bias the results low.
This is not objectionable in the case of R errors, since the R error is used as a
lower bound of the Bayes error. However, using this procedure can give
erroneous results for the L error. Options 3 and 4 are designed to alleviate this
problem.

Oprion 3: For each value of r, find the value of t which minimizes the R error,
and then use this value oft to find the L error. Since the selection of the thres-
hold has been isolated from the actual values of the L estimates of the likeli-
hood ratio, using this method does in fact help reduce the bias encountered in
Option 2. Experimental results will show that this method does give reliable
results as long as r is relatively large. When r is small, however, the L esti-
mates of the likelihood ratio are heavily biased as is seen in Fig. 7-9(b), and
use of these estimates to determine the threshold may give far from optimal
results. An advantage of this option is that it requires no more computation
time than Option 2.

Option 4: Under this option, the R error is found exactly as in Option 2, by
finding the value oft which minimizes the R error, and using this error rate. In
order to find the L error, we use an L procedure to determine the value of t to
use for each sample. Hence, under Option 4, we use a different threshold to
test each of the Nl+N2 samples, determining the threshold for each sample
from the other NI+N2-I samples in the design set. The exact procedure is as
follows.
(1) Find the L density estimates at all samples,

i#r (7.57)

(2) To test sample Xf):
(a) Modify the density estimates by removing the effect of Xf) from
all estimates