AI - a Guide to Intelligent Systems.pdf - Member of EEPIS

FUZZY EVOLUTIONARY SYSTEMS
293
Figure 8.20
Multiple fuzzy rule tables
How do we interpret the certainty factor here?
The certainty factor specified by Eq. (8.25) can be interpreted as follows. If all the
training patterns in fuzzy subspace A i B j belong to the same class C m , then the
certainty factor is maximum and it is certain that any new pattern in this
subspace will belong to class C m . If, however, training patterns belong to
different classes and these classes have similar strengths, then the certainty
factor is minimum and it is uncertain that a new pattern will belong to class C m .
This means that patterns in fuzzy subspace A 2 B 1 can be easily misclassified.
Moreover, if a fuzzy subspace does not have any training patterns, we cannot
determine the rule consequent at all. In fact, if a fuzzy partition is too coarse,
many patterns may be misclassified. On the other hand, if a fuzzy partition is too
fine, many fuzzy rules cannot be obtained, because of the lack of training
patterns in the corresponding fuzzy subspaces. Thus, the choice of the density of
a fuzzy grid is very important for the correct classification of an input pattern.
Meanwhile, as can be seen in Figure 8.19, training patterns are not necessarily
distributed evenly in the input space. As a result, it is often difficult to choose an
appropriate density for the fuzzy grid. To overcome this difficulty, we use
multiple fuzzy rule tables (Ishibuchi et al., 1992); an example of these is shown
in Figure 8.20. The number of these tables depends on the complexity of the
classification problem.
Fuzzy IF-THEN rules are generated for each fuzzy subspace of multiple fuzzy
rule tables, and thus a complete set of rules can be specified as:
S ALL ¼ XL
K¼2
S K ; K ¼ 2; 3; ...; L ð8:27Þ
where S K is the rule set corresponding to a fuzzy rule table K.
The set of rules S ALL generated for multiple fuzzy rule tables shown in Figure
8.20 contains 2 2 þ 3 3 þ 4 4 þ 5 5 þ 6 6 ¼ 90 rules.
Once the set of rules S ALL is generated, a new pattern, x ¼ðx1; x2Þ, can be
classified by the following procedure:
Step 1:
In every fuzzy subspace of the multiple fuzzy rule tables, calculate the
degree of compatibility of a new pattern with each class:
Cn
KfA i B j g ¼ KfA i gðx1Þ KfBj gðx2ÞCF Cn
KfA i B j g
ð8:28Þ
n ¼ 1; 2; ...; N; K ¼ 2; 3; ...; L; i ¼ 1; 2; ...; K; j ¼ 1; 2; ...; K