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rules. After obtaining both the antecedents fuzzy sets and rule's consequent
parameters, the corresponding fuzzy rule base can be built easily. When
observations have been obtained from a system or a process, an input matrix X and
an output vector y can be constructed as follows:
T
T
Xxx x
ª
,
,
"
,
º
,
yyy y
ª
,
,
"
,
º
¬
¼
¬
¼
12
N
1 2
N
s
s
where N s is the number of training data samples available for fuzzy identification.
Now, for correct selection of input and output variables, the unknown nonlinear
function
can be learnt from the data samples by means of regression
techniques. The variables
yf
X
T
" \ and y \ are called the
regressor and regressand respectively. In order to determine the antecedent fuzzy
sets of the Takagi-Sugeno rules, Babuška and Verbruggen (1995) proposed to
apply either of the fuzzy clustering methods mentioned above in the Cartesian
product space of X u in order to partition the training data into characteristic
regions, where the system's behaviours are approximated by a local linear model
(rules). The pattern matrix Z to be clustered is formed by X and y as follows:
xxx
,
,
,
x
n
12
n
>
@
T ZXy
,
Given the data Z and the number of clusters c , the fuzzy clustering algorithm can
be applied to obtain the partitions of Z into c fuzzy clusters. A fuzzy partition can
be represented as a
> @
g P as described
earlier. For the computation of the fuzzy partition matrix and the corresponding
cluster prototypes (centers) GK clustering algorithm is usually applied, as it applies
adaptive distance norms in order to detect clusters of different geometrical shapes,
unlike the popular fuzzy c -means algorithm, which always identifies spherical-
shape clusters in the data because of it's fixed distance norm. Because each cluster
has it's own distance norm, induced by the fuzzy covariance matrix, that allows to
adapt the local structures of the data. This evidently makes Gustafson-Kessel
clustering superior for identifying subspaces of data (hyperplanes) that can be
effectively modeled by the rules in the Takagi-Sugeno model.
Each cluster represents a certain operating region of the system, and the number
of cluster centers or clusters c sought in the data equals the number of fuzzy rules
implemented. Often, this number is not known a priori ; thus, the optimum number
of clusters is determined using suitable cluster validity measures.
The membership functions of the fuzzy sets in the premise of rules are obtained
from the fuzzy partition matrix U , whose ( g , s )th element
cN
u
matrix U , whose entries are
0,1
s
> @
g P is the
membership degree of the input-output combination in the s th column of Z in
cluster or data group g . To obtain the one-dimensional fuzzy set G gj , the
multidimensional fuzzy sets defined point-wise in the g th row of the partition
matrix U are projected onto the space of input variables x j :
0,1
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