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`
^
2
i
ww
S
l
2
l
b
l
(6.41a)
c
D
e
z
x
c
V
i
eqv
eqv
i
i
`
^
3
2
i
ww
S
l
2
l
b
l
(6.41b)
V
D
e
z
x
c
V
i
eqv
eqv
i
i
Now, following the previous technique and realizing that
eqv e can be
considered as the normalized equivalent error and, in addition, taking into account
Equation (6.28) and comparing it respectively with (6.41a) and (6.41b), transposed
Jacobian matrix and the Jacobians
and
T
J
l
,
J
l
T
l
,
J
V for the
l
c
c
J V
i
i
i
i
network free parameters
c and
l
V can be computed as:
l
`
^
2
i
Tl
2
l
l
(6.42a)
Jc
Dz
xc
V
i
eqv
i
i
T
ª
2
º
T
ª
T
l
º
l
l
l
J
l
i
2
¼
(6.42b)
c
J
c
Dz
x
c
V
«
eqv
i
»
i
i
i
¬
¼
¬
3
^
2
i
T
l
2
l
l
(6.42c)
J
V
Dz
x
c
V
i
eqv
i
i
T
ª
3
º
T
2
ª
T
l
º
l
l
l
J
l
i
2
¼
(6.42d)
V
J
V
Dz
x
c
V
«
eqv
i
»
i
i
i
¬
¼
¬
The above equations describe the Jacobian matrices and their transpositions for the
Takagi-Sugeno-type fuzzy logic systems with the adjustable free parameters
i
c
and V when normalized (equivalent) error is considered.
If, however, instead of normalized (equivalent) error only the equivalent error
is considered, then the Jacobian matrices and their transpositions will be the same,
except that in the right-hand sides of Equations (6.42a) - (6.42c) the term z has to
be replaced by normalized degree of fulfilment of the l th rule ,
l
l
b
where
h
z
M
l
b
¦
represents the sum of degree of fulfilment of all rules.
z
l
1
It is to be noted that, while computing the Jacobian matrices, care has to be
taken so that the dimensions of the Jacobians match correctly with
N u ,
where N is the number of training data sets and N p the number of adjustable
parameters in the network's layer considered. In all our simulation experiments
with neuro-fuzzy networks the normalized prediction error has been considered for
the computation of Jacobian matrices for the network's free parameters l T and
i T , so that Equations (6.31a), (6.31b) and Equations (6.34a), (6.34b) delivered the
corresponding transposed Jacobian matrices and their Jacobians respectively. In
contrast, normalized equivalent error has been considered for the computation of
transposed Jacobian matrices and their Jacobians respectively for the mean and
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