Geology Reference
In-Depth Information
equivalent ANFIS architecture can be found in Fig.
4.14
. In the ANFIS, nodes in
the same layer have similar functions as described below.
1. Layer 1(Input nodes): Nodes of this layer generate membership grades of the
crisp inputs which belong to each of the convenient fuzzy sets using the
membership functions. The generated bell-shaped membership function given
below was used:
1
l
A
i
ð
x
Þ
¼
ð
4
:
50
Þ
2b
i
1
þðð
x
c
i
Þ=
a
i
Þ
l
A
i
is the appropriate membership function for Ai fuzzy set, and {a
i
,b
i
,
c
i
} is the membership function
where
is parameter set (premise parameters) which
changes the shape of membership function from 1 to 0.
'
2. Layer 2(Rule nodes): In this layer, the rule operator (AND/OR) is applied to get
one output that represents the results of the antecedent for a fuzzy rule. The
outputs of the second layer, known as
firing strengths O2i
i
, are the products of the
incoming signals obtained from the layer 1, named as w below:
O
i
¼
w
i
¼
l
A
i
ð
x
Þl
B
i
ð
y
Þ
i
¼
1
;
2
ð
4
:
51
Þ
3. Layer 3(Average nodes): In this layer, the nodes calculate the ratio of the ith
rule
'
s
firing strength to the sum of all rules
'
firing strengths:
w
i
P
i
w
i
O
i
¼
;
ð
4
:
52
Þ
w
i
¼
i
¼
1
2
4. Layer 4(Consequent nodes): In this layer, the contribution of ith rule towards
the total output or the model output and/or the function calculated as follows:
O
i
¼
w
i
f
i
¼
w
i
ð
p
i
x
þ
q
i
y
þ
r
i
Þ
i
¼
1
;
2
ð
4
:
53
Þ
where
cients of a linear
combination in the Sugeno inference system. These parameters of this layer are
referred to as consequent parameters.
w
i
is the output of Layer 3 and {pi,
i
,q
i
,r
i
} are the coef
5 Layer 5(Output nodes): This layer is called the output nodes. This single
xed
node computes the
final output as the summation of all incoming signals.
P
i
w
i
f
i
O
i
¼
P
i
w
i
f
ð
x
;
y
Þ
¼
ð
4
:
54
Þ
