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inputs, one output, and
M
fuzzy rules initially defined by the clustering algorithm,
the
l
th rule has the form:
R
l
: f
x
1
is
l
1
and
x
2
is
l
2
and
...
x
P
is
lP
(7.3)
then
y
is
l
.(Cluster
l
)
lj
=
α
lj
exp
x
ij
−
m
lj
2
2
a
lj
2
−
(7.4)
exp
2
−
(
y
−
n
l
)
l
=
(7.5)
δ
l
2
2
where
m
and
n
are the centers of the Gaussian functions for the inputs and outputs,
a
and
δ
are the widths,
i
=
1
,
2
,...,
N
q
is the index representing the number of
closest neighbours (
N
q
),
j
=
1
,
2
,...,
P
represents the number of input variables,
and
l
M
represents the number of fuzzy rules.
The centers
m
and
n
and the widths
a
and
=
1
,
2
,...,
δ
are obtained from the ECM algorithm,
while the parameter
1) and represents the weight
of each of the input membership functions. These parameters are adjusted with the
back-propagation algorithm, as described in (Song and Kasabov
2006
).
Using the center of area defuzzification method, the output of the TNFIS for an
input vector is calculated as follows:
α
lj
is chosen by design (
α
lj
=
δ
l
2
j
=
1
exp
x
ij
−
m
lj
2
2
a
lj
2
l
=
1
n
l
−
(
x
i
)
=
O
δ
l
2
j
=
1
exp
(7.6)
x
ij
−
m
lj
2
2
a
lj
2
l
=
1
1
−
The resulting error function is stated as a weighted quadratic error function
th
at is
derivable (
7.7
). The system uses input/output data of the closest training data [
x
i
,
Y
i
]
and the goal is to minimize the target function:
1
2
v
i
[
O
Y
i
]
2
E
=
(
x
i
)
−
(7.7)
where
v
i
, with
i
N
q
, indicates the distance weig
ht
(the proximity of each
target to the expected outputs) calculated in the first step,
O
=
1
,
2
,...,
is the defuzzification
function that yields the output of the TNFIS, and
Y
i
is the desired output.
Various clustering and learning algorithms can be applied to this technique. How-
ever, for the sake of simplicity and to demonstrate the role of learning based on a clus-
tering technique, in this chapter is presented solely the ECM and back-propagation
because they both satisfy two of the constraints for real-time applications—speed of
convergence and simplicity.
(
x
i
)
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