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In the following, we will suppose that O= O= 0.5. This averaging of the
kernel gives a trade-off between the contributions of the rules in which the fuzzy
sets occur. Figure 7.6 illustrates this method for merging the two fuzzy sets A and
B in order to create the fuzzy set C .
7.5.2 Removing Irrelevant Fuzzy Sets
If the rule base contains irrelevant fuzzy sets, i.e . if a fuzzy set in the premise of a
rule has a membership function 1,
P | then it is similar to the universal
fuzzy set U and can be removed. The similarity of a fuzzy set A to the universal
fuzzy set is to be quantified by S ( A , U ). An example of a fuzzy set quite similar to
the universal fuzzy set is illustrated in the Figure 7.7, where the fuzzy set B 1 that is
highly similar to a universal fuzzy set can be removed and only A 1 is required in
the premise of rule R 1 for distinguishing the associated region in the premise space.
x
x
X
,
If x 1 is ….
and x 2 is ….
then y is ….
f 1
B 1
R 1
A 1
Close to universal
fuzzy set
f2
R 2
A 2
B 2
Close to
singleton fuzzy
set
f3
R 3
A 3
B 3
Figure 7.7. Irrelevant set ( B 1 ) in the rule base and non-similar
AB
z
domain partition
i
i
If the rule base premise consists of all antecedents fuzzy sets similar to the
universal set only, then the corresponding rule can be removed from the rule base.
The activation of such rules is more or less constant for all inputs within the
domain, and the contribution to the output can often be accounted for by re-
estimating the consequents of the other rules. The opposite situation may also
occur. During parameter adaptation of the fuzzy model, the support of one or more
fuzzy sets may become so narrow that they can be almost like fuzzy singletons (see
fuzzy set A 3 in rule R 3 of Figure 7.7), prohibiting the rule from firing. Singleton-
like fuzzy sets have extremely low similarity to the universal fuzzy set, i.e . S ( A , U )
= 0. Rules with such singleton fuzzy sets in their premise are also candidates to be
removed from the rule base. In general, one must be careful, as the rules with
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