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the mean (
c
i
) and variance (
V
i
) parameters of the Gaussian membership
functions, so that the performance function (6.11a) is minimized. For convenience,
we replace
j
p
p
j
d
,, in the above definition of error by
f
j
, d
j
, and
e
j
respectively, so that the individual error becomes
f
and
x
e
j
.
We recall that the
steepest descent rule
used for training of neuro-fuzzy
networks is based on the recursive expressions
f
d
e
j
j
j
l
(
k
)
l
( )
k
ww
K
S
l
(6.12a)
T
T
T
0
j
0
j
0
j
l
(
k
)
l
( )
k
ww
K
S
l
(6.12b)
T
T
T
ij
ij
ij
l
(
k
ww
)
l
( )
k
K
S
l
(6.12c)
c
c
c
i
i
i
l
(
k
)
l
( )
k
ww
K
S
l
(6.12d)
V
V
V
i
i
i
where
S
is the performance function (6.11b) at the
kth
iteration step and
T T V
are the free parameters of the network at the
same iteration step, the starting values of which are, in general, randomly selected.
In addition, K is the constant step size or learning rate (usually
l
j
k,
l
k,
l
k
,and
l
k
c
0
ij
i
i
K ),
i =
1,
2, ...,
n
(with
n
as the number of inputs to the neuro-fuzzy network);
j =
1, 2, ...,
m
(with
m
as the number of outputs from the neuro-fuzzy network); and
l =
1, 2, 3,
...,
M
(with
M
as the number of Gaussian membership functions selected, as well as
the number of fuzzy rules to be implemented).
From Figure 6.6, it is evident that the network output
f
j
and hence the
performance function
S
j
and, therefore, finally
S
depends on
1
l
l
T T
only
through
y
j
l
.
Similarly, the network output
f
j
and, thereby, the performance functions
S
j
and
S
depend on
and
0
j
ij
V
only through
z
l
, where,
f
j
,
y
j
l
,
b
,and
z
l
are
l
and
l
c
i
i
represented by
M
j
l
l
f
y
(6.13a)
¦
h
j
l
1
l
l
l
l
l
nj
y
T
"
(6.13b)
T
x
T
x
T
x
j
1
2
n
0
j
2
j
1
j
M
l
l
l
b
,
and
b
¦
(6.13c)
h
z
z
l
1
2
§
·
§
l
·
xc
n
i
i
¨
¸
¨
¸
l
exp
(6.13d)
z
¨
¨
i
¸
¸
V
i
1
©
¹
©
¹
Therefore, the corresponding chain rules
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