Information Technology Reference
In-Depth Information
numerical data can contain unnecessary redundancy in the form of highly
overlapping and compatible membership functions. Also, when modelling
approaches such as fuzzy clusterings are applied, this redundancy is predominant
because the rules defined in the multidimensional premise are overlapping in one
or more dimensions. As a result, more membership functions will be required to
describe the same concept adequately in the final rule base.
Another common fuzzy modelling approach, such as neuro-fuzzy approach
proposed by (Wang and Mendel, 1992), and it's modification by (Palit and
Popoviü, 1999b), and (Palit and Babuška, 2001), is based on parameter adaptation.
In this approach, an initial partition of the input space is usually given by randomly
generated fuzzy sets or by a number of equidistant symmetrical fuzzy sets defined
for all the premise variables of the system. This partition can be seen as a uniform
grid in the premise space. Thereafter, the parameters of the membership functions
are adapted using the steepest descent method (backpropagation algorithm) (Wang
and Mendel, 1992b; Palit and Popoviü, 1999b) or by it's superior form, such as
Levenberg-Marquradt algorithms (Palit and Popoviü, 1999b), (Palit and Babuška,
2001). An undesired effect of adaptation is that antecedent Gaussian fuzzy sets can
move closer to each other and may end up in overlapping positions. Also, some
sets may grow to cover the whole space (
universal fuzzy set
), or diminish to non-
influential
singletons
. As illustrated in Figure 7.1, an initially transparent fuzzy
model may become unreadable after parameter adaptation.
Highly
overlapping
Close to
Singleton
Three distinct and
interpretable fuzzy sets
Low
Medium
High
Noninterpretable
fuzzy sets
x
x
Figure 7.1.
Fuzzy sets before adaptation (left) and after adaptation (right)
Undesired redundancy in the form of similarity between fuzzy sets can manifest
itself in three different ways:
x Similarity of a particular fuzzy set
A
with other fuzzy sets in the model.
x Similarity of a fuzzy set
A
to the universal fuzzy set
U
:
1,
P |
x Similarity of a fuzzy set
A
to a singleton fuzzy set such that,
x
x
X
.
A
and
P
x
1,
if x
x
;
P
x
z
0,
x
x
,
x
X
.
A
0
A
0
As similar fuzzy sets represent compatible concepts in the rule base, a model with
many similar fuzzy sets becomes redundant, unnecessarily complex and
computationally less efficient. Linguistic interpretation of such a model is also
difficult, as it is not trivial to assign qualitatively meaningful labels to highly
similar fuzzy sets. As an illustration of the latter, consider the Figure 7.1 (right),
Search WWH ::
Custom Search