Page 30 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

BAYESIAN CLASSIFICATION 19

An erroneous assignment of a class to an object causes some damage, or
some loss of value of the misclassified object, or an impairment of its
usefulness. All this depends on the application at hand. For instance, in the
application ‘sorting tomatoes into classes A, B or C’, having a class B
tomato being misclassified as ‘class C’ causes a loss of value because a
‘class B’ tomato yields more profit than a ‘class C’ tomato. On the other
hand, if a class C tomato is misclassified as a ‘class B’ tomato, the damage
is much more since such a situation may lead to malcontent customers.
A Bayes classifier is a pattern classifier that is based on the following
two prerequisites:

. The damage, or loss of value, involved when an object is erro-
neously classified can be quantified as a cost.
. The expectation of the cost is acceptable as an optimization criterion.

If the application at hand meets these two conditions, then the develop-
ment of an optimal pattern classification is theoretically straightforward.
However, the Bayes classifier needs good estimates of the densities of the
classes. These estimates can be problematic to obtain in practice.
The damage, or loss of value, is quantified by a cost function (or loss
function) C(^ !j! k ). The function C(:j:): O O ! R expresses the cost that is
!
involved when the class assigned to an object is ^ !, while the true class of that
!
object is ! k . Since there are K classes, the function C(^ !j! k ) is fully specified
!
by a K K matrix. Therefore, sometimes the cost function is called a cost
matrix. In some applications, the cost function might be negative, expressing
the fact that the assignment of that class pays off (negative cost ¼ profit).

Example 2.3 Cost function of the mechanical parts application
In fact, automated sorting of the parts in a ‘bolts-and-nuts’ box is an
example of a recycling application. If we are not collecting the
mechanical parts for reuse, these parts would be disposed of. There-
fore, a correct classification of a part saves the cost of a new part, and
thus the cost of such a classification is negative. However, we have to
take into account that:
. The effort of classifying and sorting a part also has to be paid. This
cost is the same for all parts regardless of its class and whether it has
been classified correctly or not.
. A bolt that has been erroneously classified as a nut or a ring causes
more trouble than a bolt that has been erroneously misclassified as
scrap. Likewise arguments hold for a nut and a ring.