Page 343 - Computational Statistics Handbook with MATLAB

332 Computational Statistics Handbook with MATLAB

C
(
(
PFA) = ∫ Px ω )P ω( 2 )d , x
2
– ∞
where C represents the value of x that corresponds to the decision boundary.
We can factor out the prior, so

C
∫
(
(
(
PFA) = P ω 2 ) Px ω 2 )d . x
– ∞
We then have to find the value for C such that

C PFA)
(
∫ Px ω 2 ) xd = ---------------- .
(
(
P ω 2 )
– ∞
From Chapter 3, we recognize that C is a quantile. Using the probabilities in
(
(
Example 9.3, we know that P ω 2 ) = 0.4 and Px ω 2 ) is normal with mean 1
and variance of 1. If our desired PFA( ) = 0.05 , then
C
∫ Px ω 2 ) xd = 0.05 0.125 .
(
---------- =
0.40
– ∞
We can find the value for C using the inverse cumulative distribution func-
tion for the normal distribution. In MATLAB, this is
c = norminv(0.05/0.4,1,1);
This yields a decision boundary of x = – 0.15 .

9.3 Evaluating the Classifier
Once we have our classifier, we need to evaluate its usefulness by measuring
the percentage of observations that we correctly classify. This yields an esti-
mate of the probability of correctly classifying cases. It is also important to
report the probability of false alarms, when the application requires it (e.g.,
when there is a target class). We will discuss two methods for estimating the
probability of correctly classifying cases and the probability of false alarm:
the use of an independent test sample and cross-validation.

© 2002 by Chapman & Hall/CRC