# Fuzzy Neural Networks--A Review with Case Study
## 3 The Selected Fuzzy Networks  
### Fuzzy Network FALCON  
[PERSON] and [PERSON] proposed the FALCON fuzzy network [76]. In this network architecture, two sets of parameters are important: the assignment function and the rule parameters (or weights). Learning the network involves modifying just these two sets of parameters. In the learning process, the rule parameters (or weights) are modified first, while the parameters of the membership function remain constant. Then the parameters of just the assignment function are updated with a gradient of decline, while the rule parameters remain at the values set in the earlier step. Unfortunately, this two-step learning process may not give satisfactory results for a specific group of problems, because the learning algorithm assumes that the two mentioned parameter sets are almost independent. However, it turns out that usually, the two sets are strongly dependent on each other [76; 77; 78]. The network architecture is shown in Figure 1. The input layer accepts input data that is processed by the system; it is responsible for data fuzzing. The task of the second layer is to calculate the IF degree of the fuzzy inference, while in layer four, the THEN part is processed. The third layer corresponds to the inference rules. Layer five is responsible for sharpening the result. In this network architecture, the rules are represented through the matrix of combinations of the relevant weights, which are called rule parameters and denoted by \(Q\): the assignment function defined by Gaussian distribution and its parameters (standard deviation and average value) and appropriate to the defined function of the values of the inputs and outputs stored in the matrix \(P\). The task of the network is to calculate the mapping \(F\) of the input vector \(X\) into the output vector \(Y\), using \(P\) and \(Q\):  
\[Y=F\left(X,P,Q\right). \tag{1}\]  
Suppose \(u_{i}{}^{(j)}\) represents the input of \(i\) to layer \(j\), and \(o_{i}{}^{(0)}\) represents the output of \(i\) from layer \(j\), then layer one transmits the inputs to the appropriate place in the network in order to obtain a single entry point. Thus  
\[u_{i}^{(1)}=o_{i}^{(1)} \tag{2}\]  
In the second layer, the assignment function is calculated for each input using the parameters for the type of assignment function. When using a Gaussian function, the  
Figure 1: FALCON network architecture. level of assignment is determined by the following parameters: the mean value \(m_{i}\) and the variance \({\sigma_{i}}^{2}\)  
\[o_{i}^{(2)}=e^{-(u_{i}^{(2)}-m_{i})^{2}/2\sigma_{i}^{2}} \tag{3}\]  
It is not necessary to choose a distribution in the form of a Gaussian function, as other functions can also be used; however, it is necessary to use a finite number of parameters. Taking \(O_{i}\) and \(W_{i}\) as the output vector and weighting connection matrix \(w\) for layer \(i\), respectively, the output value can be calculated as follows:  
\[O_{i}=W_{i}^{T}O_{i-1}\quad da\quad 1\leq i\leq 5, \tag{4}\]  
wherein  
\[O_{0}=X \tag{5}\]  
Further, assuming:  
\[O_{3}=\left(o_{i}^{(3)}\right) \tag{6}\]  
a single output from layer three can be determined by:  
\[o_{i}^{(3)}=f_{i}^{(3)}(P,X) \tag{7}\]  
where \(f_{i}^{(3)}\) is a nonlinear function defining the fuzzing of the input space. According to the adopted assumptions, \(W_{4}\) is responsible for the rule parameters defined by the matrix \(Q\); then, we get:  
\[O_{4}=W_{4}^{T}O_{3}=Q^{T}O_{3} \tag{8}\]  
Based on the specified formulas, the outputs can be calculated as follows:  
\[Y=F\left(X,P,Q\right). \tag{1}\]0  
where  
\[Y=F\left(X,P,Q\right). \tag{1}\]1  
where \(\sum_{0}\) and \(M_{0}\) are defined matrices on \(P\) including standard deviations and average assignments \(\left(W_{5}^{T}\right)_{i}\) and \(\left(\sigma_{i}^{T}\right)_{i}\) represents the column and matrix \(W_{5}^{T}\) and \(\sum_{o}^{T}\). In the learning process, the parameters of the matrix \(Q\) are optimized. The optimization is typically performed using gradient descent or another numerical method.