For simplicity, we assume the fuzzy inference system under
consideration has two inputs
and
and one output .
Suppose that the rule base contains two fuzzy ifthen rules of
Takagi and Sugeno's type:
Rule:1 If
is
and
is
then
,
Rule:2 If
is
and
is
then
.
then the type3 fuzzy reasoning is illustrated in the Figure
6.2, and the corresponding equivalent ANFIS
architecture (type3 ANFIS) is shown in the Figure 6.3.
The node functions in the same layer are are of the same function
family as described below:
 Layer 1 Every node
in this layer is a square node
with a node function

(6.14) 
where
is the input to node
and
is the linguistic
label (small, large, etc.) associated with this node function. It
other words,
is the membership function of
and
it specifies the degree to which the given
satisfies the
quantifier .
Usually is chosen
to
bellshaped with maximum equal to 1 and minimum equal to 0, such
as the generalized bell function

(6.15) 
or the Gaussian function

(6.16) 
,where
is the parameter set. As the values
of these parameters change, the bellshaped functions vary
accordingly, thus exhibiting various forms of membership function
on linguistic label .
Parameters in this layer are referred
to as premise parameters.
 Layer 2 Every node in this layer is a circle node
labeled
which multiplies the incoming signal and sends the
product out.
 Layer 3 Every node in this layer is a circle node
labeled N. The th node calculates the ratio of the th
rules firing strength to the sum of all rules' firing strengths:

(6.17) 
For convenience, output of this layer will be called
normalized firing strengths.
 Layer 4 Every node
in this layer is a square node
with a node function

(6.18) 
where
is the output of layer 3, and
is the parameter set. Parameters in this layer
will be referred to as consequent parameters.
 Layer 5 The single node in this layer is a circle
node labeled
that computes the overall output as the
summation of all incoming signals, i.e.,

(6.19) 
Consider using all possible parameters which the number is function
of both, the number of inputs and the number of membership function then
can be defined number of all rules as:

(6.20) 
and if
is the number of all parameters which are
necessary for membership function then the number of all
parameters is defined as

(6.21) 