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10.5.2.3 Fuzzy Model Identification Using Entropy-based Fuzzy Clustering
In this section,
entropy-based fuzzy clustering
(EFC) will be presented to construct
a fuzzy model for predicting values of output variables. The fuzzy modelling
approach presented here is proposed by Yao
et al.
(2000) and differs slightly from
the other modelling approach described in Chapter 4 and elsewhere in the topic.
In the EFC-modelling approach Takagi-Sugeno-type rules with singleton
consequents are considered. A fuzzy rule is based on a fuzzy partition of the input
space. In each fuzzy subspace one input-output relation is formed. For a data point
with an unknown value of output variable the values of input variables of the data
point are applied to all rules and each rule gives a value by fuzzy reasoning. The
predicted output value is then obtained by aggregation of all the values given by
the rules.
Consider now a set of
c
cluster centres (
v
1
*,
v
2
*, ...,
v
c
*) in
M
-dimensional
hyperspace that is generated by the EFC algorithm. Now, suppose that the last
L
dimensions of a
k
th cluster centre (
v
k
*) are output dimensions, whereas the first
(
M
-
L
) dimensions are input dimensions. Then, each cluster centre
v
k
* can be
decomposed into two vectors:
x
k
* in (
M
-
L
)-dimensional input space and
y
k
* in
L-
dimensional output space. Then, a fuzzy model is a collection of
c
rules of the form
Rule k: IF X is close to x
k
* THEN Y is close to y
k
*,
where
X
is the input vector consisting of (
M-L
) input variables [
x
s1
,
x
s2
,
...
,
x
s(M-L)
]
and
Y
is the output vector consisting of
L
output variables [
y
s1
,
y
s2
, ...,
y
sL
] of a data
point
z
s
, with
s
= 1, 2, ...,
N
, training (input-output) samples. The membership
function, representing the degree to which rule
k
is satisfied, is given as
2
P
,
exp
V
xx
*
k
k
k
where
x
is the input vector,
X
=
x
, and V is automatically calculated from the
data. In the above, the symbol ||.|| denotes the Euclidean distance. The output
vector,
Y
=
y
, is calculated as
c
P
y
*
¦
kk
k
1
y
.
c
P
¦
k
k
1
We can now write a fuzzy rule in a more specific form as
IF x
1
is A
k1
and x
2
is A
k2
and … and x
(M-L)
is A
k(M-L)
THEN Y is y,
for k = 1, 2,…, c.
where
x
j
is the
j
th input variable and
A
kj
is given by
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