Page 45 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

34 DETECTION AND CLASSIFICATION

error over all measurements except those that fall inside the reject
region:
Z
E min ¼ e min ðzÞpðzÞdz ð2:33Þ
fzjC rej e min ðzÞg
Comparison of (2.33) with (2.16) shows that the error rate of a classi-
fication with reject option is bounded by the error rate of a classification
without reject option.

Example 2.6 The reject option in the mechanical parts application
In the classification of bolts, nuts, rings and so on, discussed in the
previous examples, it might be advantageous to manually inspect
those parts whose automatic classification is likely to fail. We
assume that the cost of manual inspection is about $0.04. Table
2.3 tabulates the cost function with the reject option included (com-
pare with Table 2.2).
The corresponding classification map is shown in Figure 2.11. In
this example, the reject option is advantageous only between the
regions of the rings and the nuts. The overall risk decreases from
$0.092 per classification to $0.093 per classification. The benefit
of the reject option is only marginal because the scrap is an expensive
item when offered to manual inspection. In fact, the assignment of an
object to the scrap class is a good alternative for the reject option.

Listing 2.5 shows the actual implementation in MATLAB. Clearly it is very
similar to the implementation for the classification including the costs. To
incorporate the reject option, not only the cost matrix has to be extended,
but clabels has to be redefined as well. When these labels are not
supplied explicitly, they are copied from the data set. In the reject case, an
extra class is introduced, so the definition of the labels cannot be avoided.

Table 2.3 Cost function of the mechanical part application with the reject option
included

C( ^ w i w i jw k ) in $ True class
! 1 ¼ bolt ! 2 ¼ nut ! 3 ¼ ring ! 4 ¼ scrap
class ^ ! ! ¼ bolt 0.20 0.15 0.07 0.07
0.07
0.07
^ ! ! ¼ ring
0.07
0.07
Assigned ^ ! ! ¼ nut 0.16 0.11 0.05 0.07
0.07
0.07
0.03
^ ! ! ¼ scrap
0.03
0.03
0.03
0.07
^ ! ! ¼ ! 0 ¼ rejection
0.01