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m
A
{
¦
D
e
D
e
D
e
"
D
e
(6.36)
j
1
2
m
j
1
2
m
j
1
Our objective is to find suitable terms
D
eqv
and
e
eqv
such that their product is
equal to
A
{
De
DeDe
"
De
(6.37)
eqv
1
2
m
eqv
1
2
m
where the term
e
eqv
is such that it contributes the same amount of sum squared
error value
S
of equation (6.11b) as that can be obtained jointly by all the
{
from the multiple-input multiple-output network. Therefore,
f
e
d
j
j
j
2
2
2
p
p
p
"
p
m
,
(6.38)
e
e
e
e
1
2
eqv
where,
p =
1, 2, 3, …,
N
; corresponding to
N
training samples. This results in
1
A
e
(6.39a)
D
eqv
eqv
This can be written in matrix form using the pseudo inverse as
1
T
T
A
(6.39b)
E
D
E
E
eqv
eqv
eqv
eqv
containing
eqv
where
E
eqv
is the equivalent error vector of size
1
p
e
as its
elements for all (
N
) training samples. Similarly,
D
eqv
and
A
are matrices of size
N
u
and
1
u
M
respectively. Once the matrix
D
eqv
and the equivalent error
vector
E
eqv
are known, we can replace matrix
A
with their product. Therefore,
M
u
N
A
DE
(6.40a)
eqv
eqv
or, equivalently as,
A
(6.40b)
e
D
eqv
eqv
can be calculated. In the case of a multiple-input single-output neuro-fuzzy
network,
i.e
. for
m =
1 and
hold. This means
that, in this case, Equations (6.37) - (6.40b) need not be computed.
However, for the multiple-input multiple-output case, where
A
1
,
and
D
D
e
e
D
e
1
eqv
1
eqv
1
m
t , using
2
(6.37) we can write Equations (6.15c) and (6.15d) as
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