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variables are assumed to be fuzzy numbers. These fuzzy regression models are
based on the possibility theory instead of the probability theory or they are based
on both possibility and probability theories.
(b) Method proposed by R.J. Hathaway and J.C. Bezdek [215] where first the
fuzzy clusters determined by an fuzzy c-means clustering (FCM) algorithm define
how many ordinary regressions are to be constructed, one for each cluster. Next
each fuzzy cluster is used essentially for switching purposes to determine the most
appropriate ordinary regression that is to be applied for a new input from amongst
a number of ordinary regressions determined in the first place.
(c) Methods proposed by I.B.Turksen [216] and A. Celikyilmaz [217], where
the fuzzy functions approach to system modeling was developed. The new fuzzy
functions approach augments the membership values together with their
transformations to form a new input variable to find local functions. First the
given system domain is fuzzy partitioned into c clusters using fuzzy c-means
clustering algorithm. Then, one regression function is calculated to model the
behavior of each partition. In [216] linear regression function to estimate the
parameters of each function is proposed. A new fuzzy system modeling approach
that identifies the fuzzy functions using support vector machines is proposed in
[218]. This new approach is structurally different from the fuzzy rule base
approaches and fuzzy regression methods. Method support vector machines is
applied to determine the support vectors for each fuzzy cluster obtained by fuzzy
c-means clustering algorithm. Original input variables, the membership values
obtained from the fuzzy c-means clustering algorithm together with their
transformations form a new augmented set of input variables. Methods proposed
in [216-217], were investigated in [219].
In this chapter we have developed a linear and nonlinear regression models,
belonging to group (a). The methods of fuzzy regression from this group have
received a lot of developing in the past years. A major difference between fuzzy
regression and ordinary regression [220 ] is in dealing with errors as fuzzy
variables in fuzzy regression modeling, and in dealing with errors as random
variables in ordinary regression modeling. The researchers have tried to integrate
both fuzziness and randomness into regression model. As a result of this the
hybrid fuzzy least-squares regressions were developed [189, 200—202, 204, 205,
207, 210-212].
In [200, 201] the least squares method is applied to deviations from unity of
possibilities of equality of observable output normal triangular numbers and model
output normal triangular numbers. As known, membership function of a normal
triangular number looks like
a
−
x
a
−
x
⎧
1
−
1
,
0
<
1
≤
1
a
>
0
⎪
L
a
a
⎪
⎪
L
L
x
−
a
x
−
a
()
1
−
1
,
0
<
1
≤
1
a
>
0
μ
x
=
⎨
~
R
a
a
A
⎪
R
R
1
=
;
x
a
⎪
1
⎪
0
x
<
a
−
a
or
x
>
a
+
a
.
⎩
(6.1)
1
L
1
R
