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6.3.1 ANFIS architecture

For simplicity, we assume the fuzzy inference system under consideration has two inputs $x$ and $y$ and one output $z$. Suppose that the rule base contains two fuzzy if-then rules of Takagi and Sugeno's type:

Rule:1 If $x$ is $A_1$ and $y$ is $B_1$ then $f_1 =
p_1x+q_1y+r_1$,
Rule:2 If $x$ is $A_2$ and $y$ is $B_2$ then $f_2 =
p_2x+q_2y+r_2$.
then the type-3 fuzzy reasoning is illustrated in the Figure 6.2, and the corresponding equivalent ANFIS architecture (type-3 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 $i$ in this layer is a square node with a node function
\begin{displaymath}
O_{i}^{1}(x)=\mu_{A_{i}}(x),
\end{displaymath} (6.14)

where $x$ is the input to node $i$ and $A_i$ is the linguistic label (small, large, etc.) associated with this node function. It other words, $O_{i}^{1}$ is the membership function of $A_i$ and it specifies the degree to which the given $x$ satisfies the quantifier $A_i$. Usually is chosen $\mu_{A_{i}}(x)$ to bell-shaped with maximum equal to 1 and minimum equal to 0, such as the generalized bell function


\begin{displaymath}
\mu_{A_{i}}(x)= \frac{1}{1+[(\frac{x-c_{i}}{a_{i}})^{2}]^{b_{i}}}
\end{displaymath} (6.15)

or the Gaussian function
\begin{displaymath}
\mu_{A_{i}}(x)= e^{-(\frac{x-c_{i}}{a_{i}})^{2}},
\end{displaymath} (6.16)

,where ${a_{i},b_{i},c_{i}}$ is the parameter set. As the values of these parameters change, the bell-shaped functions vary accordingly, thus exhibiting various forms of membership function on linguistic label $A_i$. Parameters in this layer are referred to as premise parameters.
Layer 2 Every node in this layer is a circle node labeled $\Pi$ which multiplies the incoming signal and sends the product out.
Layer 3 Every node in this layer is a circle node labeled N. The $i$-th node calculates the ratio of the $i$-th rules firing strength to the sum of all rules' firing strengths:
\begin{displaymath}
\bar{w_{i}}= \frac{w_{i}}{w_{1}+w_{2}}, i=1,2.
\end{displaymath} (6.17)

For convenience, output of this layer will be called normalized firing strengths.

Layer 4 Every node $i$ in this layer is a square node with a node function
\begin{displaymath}
O_{i}^{4}(x)=\bar{w_{i}}f_{i}=\bar{w_{i}}(p_{1}x+q_{i}y+r_{i})
\end{displaymath} (6.18)

where $\bar{w_{i}}$ is the output of layer 3, and $\{p_i,q_i,r_i\}$ 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 $\sum$ that computes the overall output as the summation of all incoming signals, i.e.,
\begin{displaymath}
O_{1}^{5}(x)= overall\ output=\sum_{i}\bar{w_{i}}f_{i}=
\frac{\sum_{i}w_{i}f{i}}{\sum_{i}w_{i}}
\end{displaymath} (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:

\begin{displaymath}
Rule_n = \prod_{i=1}^{In_n} Mf_i
\end{displaymath} (6.20)

and if $premispara_n$ is the number of all parameters which are necessary for membership function then the number of all parameters is defined as
\begin{displaymath}
para_n = premispara_n\sum_{i=1}{In_n}Mf_i + Rule_n(In_n+1)
\end{displaymath} (6.21)

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