Page 96 - Introduction to Statistical Pattern Recognition

78 Introduction to Statistical Pattern Recognition

Reject Option

When r(X) of (3.6) is close to 0.5, the conditional error of making the
decision given X is high. So, we could postpone decision-making and call for
a further test. This option is called reject. Setting a threshold for r(X), we
may define the reject region, LR(t), and reject probability, R (t), as

LR(f) = (XI r(x) 2 f) , (3.80)

R(t) = Pr(r(X) 2 t) =I p(X) dX . (3.81)
LR(r)
Then, the resulting error, &(t), is

=
~(t) &pnrp Ip l~x), dx , (3.82)
where zR is the complementary region of LR. When the minus-log likelihood
test is used, (3.80) can be converted to

inequalities are obtained from
2 t when PIPI(X) > Pzp2(X), and
2 t when PIpI(X) c P2p2(X), respec-
tively. Thus, any sample X which satisfies (3.83) is rejected. On the other
hand, the oI -sample satisfying h (X) > In (1 -f)/t + In P l/P2 and the 9-
sample satisfying h (X) < - In (1 -tyr + In P /P are misclassified.
I
Figure 3-12 shows the relationship between &(f) and R(r) for a simple
one-dimensional example. As seen in Fig. 3-12, as t increases from 0 to 0.5,
&(I) increases from 0 to the Bayes error, and R (t) decreases from 1 to 0.
Error-reject curve: The relation between R(t) and &(t) resembles the
operating characteristics in which and &2 are related with decision threshold
as the parameter. Therefore, the error-reject curve, which plots &(t) as the