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96 4 Statistical Classification

Figure 4.16. Implementation of the Bayesian decision rule for two classes with
different loss factors for wrong decisions.

For the cork stopper losses &=0.015 and /22,=0.01, using the previous
prevalences, one obtains ~*(w~)=0.308 and ~*(@)=0.692. The higher loss
associated with a wrong classification of a @ cork stopper leads to an increase of
P*( @) compared with P*( wl). The consequence of this adjustment is the decrease
of the number of @ cork stoppers wrongly classified as y. This is shown in the
classification matrix of Figure 4.17.

DISCRIM. Rows: Observed classifications
ANALYSIS Columns: Predicted classifications
G-1.1 G-2.2
Group p=. 30800 p=. 69200
54 00000
G-2 2 90 00000
Total 3 2 6 8

Figure 4.17. Classification matrix of two classes of cork stoppers with adjusted
prevalences.

We can now compute the average risk for the 2-class situation, as follows:

where R, and R are the decision regions for and 132 respectively, and Peg is the
error probability of deciding class mi when the true class is wj.