Page 85 - Introduction to Statistical Pattern Recognition

3 Hypothesis Testing 67

I-@) = 41(x) + . . . + qn.-~(X) + qk+l(X) + . . . + qL(X) = 1 - qk(X), and the
Bayes error is the expected value of r(X) over X. That is,

r(X) = 1 - max qi(X) and E = E(r-(X)} . (3.45)
I
When cost functions are involved, the decision rule becomes

rk(X) = min ri(X) + X E ok (3.46)
I
where ri(X) is a simple extension of (3.20) to L classes as
L
ri(X> = Ccijqj(X) (3.47)
j=l
and ci, is the cost of deciding X E wi when X E mi. Substituting (3.47) into
(3.46) and using the Bayes theorem,
L L
CckjPjp,(X) = min xcijPjpj(X) + X E ok . (3.48)
j=l i j=1
The resulting conditional cost given X and the total cost are

r(X) = min rj(X) and I‘ = E(r(X)} . (3.49)
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Example 5: When cii = 0 and cij = 1 for i#j, ri(X) of (3.47) becomes

(3.50)

Therefore, the decision rule of (3.46) and the resulting conditional cost of
(3.49) become (3.43) and (3.43, respectively.

Single Hypothesis Tests

So far, we have assumed that our task is to classify an unknown sample
to one of L classes. However, in practice, we often face the problem in which
one class is well defined while the others are not. For example, when we want
to distinguish targets from all other possible nontargets, the nontargets may
include trucks, automobiles, and all kinds of other vehicles as well as trees and
clutter discretes. Because of the wide variety, it is almost impossible to study
the distributions of all possible nontargets before a decision rule is designed.