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A
11
.
x
1
n
1
...
.
N
A
1S
.
.
.
D
A
n1
N
.
x
n
...
x
1
n
A
nS
Fig. 3.1
Fuzzy neural network (Jang et al.
1997
)
Rule
R
j
:
IF
x
1
is
A
x
1
j
,...,
and
x
n
(
k
)
is
A
x
n
j
,
THEN:
y
j
=
g
0
j
+
g
1
j
x
1
+···+
g
nj
x
n
being
x
i
,
y
j
for each rule, the inputs and outputs of the system respectively, and
A
x
i
j
is the fuzzy set respective to
x
i
(
is the output of
the model respective to the operating region associated to that rule. The structure of
antecedents describes fuzzy regions in the inputs space, and the one of consequents
presents non-fuzzy functions of the model inputs.
These models may be formulated as an Adaptive Neuro-Fuzzy Inference System
(ANFIS) (Jang et al.
1997
). In Fig.
3.1
an ANFIS is presented as an example with
n
input variables and one output.
The first layer is composed of membership functions of each
A
ij
, defined by the
membership degree
k
)
on the rule
j
,
g
i
∈ R
,
y
j
(
k
)
μ
A
ij
:
x
i
∈ R −→
μ
A
ij
(
x
i
)
∈ R
(3.1)
, the membership degree of
x
i
.Forthe
definition of these membership functions, some standard types are used, like gaussian
membership functions.
The second layer has nodes labelled with
The output of each node
i
is
μ
A
ij
(
x
i
)
which implement fuzzy inference
machine. Logical operation
AND
may be carried out by multiplication or minimum
value for example, the output of each node
j
of this layer may be:
ω
j
(
x
)
=
μ
A
1
j
(
x
1
)
·
μ
A
2
j
(
x
2
)
·
...
·
μ
A
nj
(
x
n
)
(3.2)
or
ω
j
(
x
)
=
min
{
μ
A
1
j
(
x
1
), μ
A
2
j
(
x
2
),...,μ
A
nj
(
x
n
)
}
(3.3)
The third layer normalizes the inference motor. The output of each node of this
layer is:
)
i
=
1
ω
i
(
ω
i
(
x
a
i
(
x
)
=
(3.4)
x
)
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