Page 32 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

BAYESIAN CLASSIFICATION 21

The integral extends over the entire measurement space. The quantity R
is the overall risk (average risk, or briefly, risk) associated with the
decision function ^ !(z). The overall risk is important for cost price
!
calculations of a product.
The second prerequisite mentioned above states that the optimal
classifier is the one with minimal risk R. The decision function that
minimizes the (overall) risk is the same as the one that minimizes the
conditional risk. Therefore, the Bayes classifier takes the form:

!
!
!
^ ! ! BAYES ðzÞ¼ ^ ! i such that: Rð^ ! i jzÞ Rð^ ! j jzÞ i; j ¼ 1; ... ; K ð2:6Þ
This can be expressed more briefly by:

^ ! ! BAYES ðxÞ¼ argminfRð!jzÞg ð2:7Þ
!2O

The expression argminfg gives the element from O that minimizes
R(!jz). Substitution of (2.3) and (2.4) yields:

( )
K
X
^ ! ! BAYES ðzÞ¼ argmin Cð!j! k ÞPð! k jzÞ
!2O k¼1
( )
K
X pðzj! k ÞPð! k Þ
¼ argmin Cð!j! k Þ ð2:8Þ
!2O pðzÞ
k¼1
K
( )
X
¼ argmin Cð!j! k Þpðzj! k ÞPð! k Þ
!2O k¼1
Pattern classification according to (2.8) is called Bayesian classification
or minimum risk classification.

Example 2.4 Bayes classifier for the mechanical parts application
Figure 2.5(a) shows the decision boundary of the Bayes classifier
for the application discussed in the previous examples. Figure
2.5(b) shows the decision boundary that is obtained if the prior
probability of scrap is increased to 0.50 with an evenly decrease of
the prior probabilities of the other classes. Comparing the results
it canbeseenthatsuchanincreaseintroducesanenlargement of
the compartment for the scrap at the expense of the other com-
partments.