Page 351 - Computational Statistics Handbook with MATLAB

340 Computational Statistics Handbook with MATLAB

2. Leave k points out of the non-target class to form a set of test cases
2 ()
denoted by TEST. We denote cases belonging to class ω 2 as x i .
3. Estimate the class-conditional probabilities using the remaining
n – k non-target cases and the n 1 target cases.
2
4. For each of those k observations, form the likelihood ratios

P x ( 2 () ω ) 2 ()
2 ()
(
i
1
L R x i ) = -------------------------; x i in TEST .
2 ()
(
P x i ω 2 )
5. Repeat steps 2 through 4 using all of the non-target cases.
6. Order the likelihood ratios for the non-target class.
7. For each probability of false alarm, find the threshold that yields
that value. For example, if the P(FA) = 0.1, then the threshold is
ˆ of the likelihood ratios. Note that higher
given by the quantile q 0.9
values of the likelihood ratios indicate the target class. We now
have an array of thresholds corresponding to each probability of
false alarm.
8. Leave k points out of the target class to form a set of test cases
1 ()
denoted by TEST. We denote cases belonging to ω 1 by x i .
9. Estimate the class-conditional probabilities using the remaining
n 1 – k target cases and the n 2 non-target cases.
10. For each of those k observations, form the likelihood ratios

( 1 () ω 1 )
1 ()
P x i
(
1
L R x i ) = -------------------------; x i in TEST .
1 ()
P x ( ω )
i 2
11. Repeat steps 8 through 10 using all of the target cases.
12. Order the likelihood ratios for the target class.
13. For each threshold and probability of false alarm, find the propor-
tion of target cases that are correctly classified to obtain the
1 ()
(
(
PCC Target ). If the likelihood ratios L R x i ) are sorted, then this
would be the number of cases that are greater than the threshold.
This procedure yields the rate at which the target class is correctly classified
for a given probability of false alarm. We show in Example 9. 8 how to imple-
ment this procedure in MATLAB and plot the results in a ROC curve.

Example 9.8
In this example, we illustrate the cross-validation procedure and ROC curve
using the univariate model of Example 9.3. We first use MATLAB to generate
some data.
© 2002 by Chapman & Hall/CRC