Page 81 - Introduction to Statistical Pattern Recognition

3 Hypothesis Testing 63

If we select the special set of cost functions

cI1 =c22 and c12=c21, (3.37)
(3.36) becomes

(3.38)
I, Pl(X)dX =I P2(X)dX '
L,
That is, the decision boundary is still determined by the likelihood ratio, but
the threshold is selected to satisfy el = e2.

Operating Characteristics

So far, we have found that the likelihood ratio test is commonly used for
various tests, and only the selection of the threshold varies depending on the
test. Extending this, it is a common practice to plot the relation between el
and by changing the value of the threshold continuously. This curve is
called the operating characteristic [5]. Figure 5 shows an example of the
operating characteristics where el and 1-e2 are used for the x- and y-axes in
log scale. Three curves in Fig. 3-5 show the performance of the likelihood
ratio test for 30, 20, and 9 features which are selected from the same data set.
They indicate that 30 and 20 features give almost identical performance for a
wide range of operating points, while 9 features give much poor performance.
From such curves, the designer of the decision rule can select a proper operat-
ing point and the corresponding threshold, depending on one's need.

Burdick's chart: Various combinations of log and linear scales are used
for operating characteristics. However, the following scale gives a straight line
when h (X) of (3.5) is normally distributed for both o1 and o2 [6].
Let @(a) be a normal error function defined by

(3.39)

If h is distributed as Nh(rn,,o:) for o1 and Nh(m2,0:) for w2, and r is the
value of the threshold as shown in Fig. 3-6, then
t-m2
and e2 =@ [T] (3.40)
.
Or, taking the inverse operation,