Page 36 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

BAYESIAN CLASSIFICATION 25

function. With our earlier definition of MAP classification we come to
the following conclusion:

Minimum error rate Bayes classification MAP

classification with unit cost function classification

The conditional error probability of a MAP classifier is found by sub-
stitution of (2.11) in (2.13):

e min ðzÞ¼ 1 maxfPð!jzÞg ð2:15Þ
!2O

The minimum error rate E min follows from (2.14):

Z
E min ¼ e min ðzÞpðzÞdz ð2:16Þ
z

Of course, phrases like ‘minimum’ and ‘optimal’ are strictly tied to the
given sensory system. The performance of an optimal classification with
a given sensory system may be less than the performance of a non-
optimal classification with another sensory system.

Example 2.5 MAP classifier for the mechanical parts application
Figure 2.5(c) shows the decision function of the MAP classifier. The
error rate for this classifier is 4.8%, whereas the one of the Bayes
classifier in Figure 2.5(a) is 5.3%. In Figure 2.5(c) four objects are
misclassified. In Figure 2.5(a) that number is five. Thus, with respect
to error rate, the MAP classifier is more effective compared with the
Bayes classifier of Figure 2.5(a). On the other hand, the overall risk of
the classifier shown in Figure 2.5(c) and with the cost function given
in Table 2.2 is $0.084 which is a slight impairment compared with
the $0.092 of Figure 2.5(a).

2.1.2 Normal distributed measurements; linear and quadratic
classifiers

A further development of Bayes classification with uniform cost function
requires the specification of the conditional probability densities. This