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In time series modelling or in data-driven identification of dynamic systems,
when the similarity of the partitioning as a whole of two or more inputs occurs,
another type of redundancy may be encountered. For instance, as illustrated in
Figure 7.6, when delayed samples of the same variables are used as input, say
x
(
k
)
and
x
(
k
-1), they may have a highly similar influence in the model's premise. In this
case, the degree of firing of the various rules can be determined by one such input
only, which reduces the dimensionality (feature) of the rule base premise.
7.5.1 Merging Similar Fuzzy Sets
In general, when two fuzzy sets are considered to be similar, the rule base can be
simplified by
x replacing
A
by
B
x replacing
B
by
A
, or
x replacing both
A
and
B
by a new fuzzy set
C
.
When the rule base represents a system model, two important aspects of the
simplified rule base are to be considered: the model accuracy and it's coverage of
the premise space. Here, owing to the rule base simplification, the uncovered
regions should not occur in the premise space. Assuming that the model's accuracy
is measured by the sum of squared errors
J
, the effect of replacing
A
and
B
by
C
should be as small as possible with respect to
J
. Finding the fuzzy set
C
best suited
to replace
A
and
B
becomes a question of evaluating
J
. Considering the
nonlinearity of fuzzy models and the possible interplay between the rule
antecedents and the rule consequents, optimizing the fuzzy set
C
based on
J
becomes a computationally intensive search problem. In general, if the model is
more sensitive to changes in
A
than to the changes in
B
, then the fuzzy set
A
should
replace the fuzzy set
B
, or the common fuzzy set
C
should resemble
A
more than
B
.
In particular cases, some additional aspects like model granularity (number of
linguistic terms per variable), interpretability or physical relevance may be
important.
For a better understanding of merging fuzzy sets, we define a trapezoidal fuzzy
set
A
using parametric membership functions
1234
P
xa a a a
;, , , ;
a
ddd,
a
a
a
A
1
2
3
4
-
0,
for
xa
d
, or
xa
t
1
4
°
P
xa a a a
;
,
,
,
1,
for
and
a
d d
x
a
(7.4)
®
°
A
1234
2
3
>@
P
x
P
x
0,1
¯
A
A
One way to merge the fuzzy sets is to take the support of
A
* as the support of
the new fuzzy set
C
. This guarantees preservation of the coverage of the whole
premise space when
C
replaces
A
and
B
in the premise of the rule base. The kernel
(cardinality) of
C
is given by aggregating the parameters describing the kernels of
A
and
B
. Thus, merging
A
and
B
, defined by
P
xa a a a
;, ,,
and
A
1234
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