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104 4 Statistical Classification

Let us now stipulate that the posterior probabilities of a cork stopper must be
higher than a certain reject threshold A,, otherwise it is classified in a special reject
class w,. The Bayes rule is now reformulated as:

When comparing the likelihood ratio with the prevalence ratio, one now has to
multiply this ratio by (1- A)/ 5. Notice that for a c class problem there is never a
rejection if 5 <(c- l)/c, therefore R, E [(c- l)lc, I].
Let us illustrate the concept of a reject class using the cork stoppers data.
Suppose that a reject threshold of 2~0.7 is stipulated. In order to compute decision
borders for the reject class, it is enough to determine the discriminant function for
the new prevalences P(o,)=(l- Ar)=0.3, P(@)= A,.=0.7 and vice-versa. The decision
lines have the same slope, and intersect the vertical axis at PRTIO=15.5 and
PRT10=20.1, respectively. Notice that these lines are therefore symmetrically
disposed around the decision line determined in section 4.1.3. (crossing at 17.8).
Figure 4.23 shows the scatter plot with the new decision lines. The area between
the solid lines is the reject region.

Figure 4.23. Discriminant analysis for two classes of cork stoppers with reject
region between the solid lines corresponding to reject threshold &=0.7.

Let us look now to the classification matrices shown in Figure 4.24. A bit of
thought will reveal that 4 patterns of class I, and 5 patterns of class 2 fall into the