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94 4 Statistical Classification

Figure 4.15. Loss diagram for two classes of cork stoppers. Correct decisions
have zero loss.

Define:

,Iij = R(a, I w,) as the loss associated with an action aj when the correct class
is w,. In the present case, ai E {SB, NB).
We can arrange the in a loss matrix A, which in the present case is:

Therefore, the risk (loss) associated with the action of using a cork,
characterized by the feature vector x, in special bottles, can be expressed as:

~(a, 1 x)= R(SB I x)= R(SB I S)P(S I X)+A(NB I M)P(A I X) ; (4- 174
~(a, I x)=0.015~(~I x).

And likewise for normal bottles:

~(a, I x)= R(NB I x)= A(NB I S)P(S 1 X)+A(NB I A)P(A I x); (4- 17b)
~(a, I x)= O.O~P(S 1 x).

We assume that in the risk evaluation the only influence is from wrong
decisions. Therefore, correct decisions have zero loss, Aii=O, as in (4-17). If instead
of two classes we have c classes, the risk associated with a certain action a;. is
expressed as follow::

4
It is assumed without loss of generality that only one action corresponds to each class.