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Example 7.1
Five triangular fuzzy sets
12
F F
"
shown in Figure 7.9(a) are used to partition
the universe of
x
. Applying the
Jaccard similarity index
(7.3) a compatibility
relation
C
and the corresponding similarity relation
S
i.e., the
max-min
transitive
closure of C are given as follows:
,
,
,
5
1.0
0.09
0.06
0.05
0.0
1.0
0.09
0.09
0.09
0.09
ª
º
ª
º
«
»
«
»
0.09
1.0
0.73
0.59
0.06
0.09
1.0
0.73
0.73
0.09
«
»
«
»
«
»
, and
«
»
.
C
«
0.06
0.73
1.0
0.73
0.06
S
«
C
0.09
0.73
1.0
0.73
0.09
T
»
»
0.05
0.73
0.59
1.0
0.09
0.09
0.73
0.73
1.0
0.09
«
»
«
»
«
»
«
»
0.0
0.06
0.06
0.09
1.0
0.09
0.09
0.09
0.09
1.0
¬
¼
¬
¼
Applying a threshold
to the similarity relation, we identify a set of similar
O
23
fuzzy sets
^
`
^
`
234
. The linguistic terms (labels) represented
F
FS
!
O
,
l m
z
F F F
,
,
l
lm
by the three fuzzy sets
^
F FF
are merged to create a generalized concept
moderate represented by fuzzy set
F
c
. The resulting fuzzy partition is depicted in
Figure 7.9(b).
`
234
,
,
Mu(x)
Mu(x)
F2 F3 F4
F5
Fc
F5
F1
F1
1.0
1.0
Low
High
Low
High
Moderate
x
x
(a)
(b)
Figure 7.9(a).
Fuzzy sets (initial partition)
Figure 7.9(b).
After merging of fuzzy sets
7.7 Model Competitive Issues: Accuracy versus Complexity
The advantage of transparent representation of the fuzzy model is paid at the cost
of reduced numerical accuracy of fuzzy models compared with that of, say, a
neural-networks-based model, when both models have approximately the same
number of parameters. The reason is that the complexity of fuzzy models grows
with the dimension of input and output spaces, which, as shown by Barron (1993),
is not the case with neural networks. Therefore, for high-order and for
multivariable systems a neural-network-based model might be easier to obtain and
may provide a more compact representation than a fuzzy model. However, fuzzy
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