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This means that they have a zero degree of equality and are, therefore,
considered dissimilar. On the other hand, the two fuzzy sets
F
3
and
F
4
in Figure
7.3(b), although different in shape, have a high degree of similarity or resemblance.
They represent compatible concepts (low temperature) and are considered largely
similar.
7.4.1 Similarity Measure
In the method presented here, two fuzzy sets are considered similar if the two
overlapping membership functions assign approximately the same values of
membership grade to the elements in their universe of discourse. So, the similarity
here is the degree to which they can be considered as equal. Equality is a crisp set
in the classical definition.
Let us now consider two fuzzy sets
F
1
and
F
2
with the membership functions
and
F
P respectively. Then, it holds that the fuzzy sets
F
1
and
F
2
on
X
are equal if
1
P
x
F
, where
X
is the universe of discourse.
Applying this concept of equality to the fuzzy sets in Figure 7.3, we get that
P
x
P
x
and
xX
F
F
2
F F
z because in both cases their membership functions are
different. However,
F
3
and
F
4
can be said to have high degree of equality, and
hence are similar.
As the fuzzy sets allow for gradual transition between full membership and
total non-membership, therefore, the
similarity measure
S
should capture a gradual
transition between equality and non-equality
F
z
F
and
4
,
1
2
3
12
>@
s
SFF
,
,
s
,
0,1
(7.1)
The similarity measure is a function of assigning a similarity value “
s
” to the pair
of fuzzy sets (
F
1
,
F
2
) that indicates the degree to which
F
1
and
F
2
are equal.
7.4.2 Similarity-based Rule Base Simplification
For the purpose of rule base simplification, the fuzzy sets in a rule base that
represent a more-or-less compatible concept should be detected by a similarity
measure. Therefore, the fuzzy sets, representatives of a compatible concept, should
be assigned a high similarity value, whereas more distinct sets should be assigned a
lower similarity value. Furthermore, for a correct comparison of similarity values,
the similarity measure in any case should be independent of the scaling of the
domain on which fuzzy sets are defined. As a consequence, this eliminates the
necessity of normalization of the domains.
Now, let
F
1
and
F
2
be two fuzzy sets on
X
with the membership functions
and
F
P respectively. If the four criteria, as listed below, are satisfied
by the similarity measure, then it can be used as a suitable candidate for an
automated rule base simplification scheme.
1.
P
x
F
Two overlapping fuzzy sets should have a similarity value
s
> 0:
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