## 2 Materials and Methods
### The Empirical Model--Adaptive Neuro-Fuzzy Inference System (ANFIS)  
Adaptive neuro-fuzzy inference systems (ANFIS) belong to a family of hybrid systems. They use a supervised learning on the learning algorithm which has a function similar to the model of the Takagi-Sugeno fuzzy inference system [11]. An ANFIS combines the advantages of fuzzy logic (FL) and artificial neural networks (ANNs) for modeling purposes. ANNs operate in a black box manner and they have no capability of explaining a particular decision; on the other hand, ANNs have the ability to predict the output of a complex system through a learning process based on the examples provided [12; 13]. FL itself possesses no learning ability but is able to provide mathematical descriptions in the form of linguistic terms and the structure of IF-THEN rules [14], offering an insight into decisions. An ANFIS can be used in a wide range of applications since it possesses the ability to handle large amounts of data in nonlinear and dynamic systems, especially when relationships between data are not fully understood.  
Figure 4 shows the ANFIS scheme. The ANFIS structure consists of five connected network layers, \(l_{1}\) to \(l_{5}\). The first and fourth layers contain adaptive nodes while the other layers contain fixed nodes represented in circular shapes. The adaptive nodes contain parameters that are modified to achieve the desired input-output mapping, while the fixed nodes contain none. The architecture of the ANFIS is presented based on two fuzzy IF-THEN rules and the first order Sugeno model as follows:  
* **if**\(x\) is \(A_{1}\)**and**\(y\) is \(B_{1}\), **then**\(f_{1}=p_{1}x+q_{1}y+r_{1}\).
* **if**\(x\) is \(A_{1}\)**and**\(y\) is \(B_{1}\), **then**\(f_{1}=p_{1}x+q_{1}  
Rule 2: **if**\(x\) is \(A_{2}\)**and**\(y\) is \(B_{2}\), **then**\(f_{2}=p_{2}x+q_{2}y+r_{2}\).  
where:  
* \(x\) and \(y\) are the inputs,
* \(A_{i}\) and \(B_{i}\) are the membership functions (MF),
* \(f_{i}\) is the output, and
* \(p_{i}\), \(q_{i}\) and \(r_{i}\) are the linear design parameters (consequent parameters), determined during the training process [16].  
In \(l_{1}\) all nodes are adaptive nodes. Inputs are presented as \(x\) and \(y\). The output of a node is the membership value \(O_{1,i}\), that specifies the degree of input \(x\) (or \(y\)) belonging to fuzzy set \(A_{i}\) (or \(B_{i}\)):  
\[O_{1,i}=\mu_{A_{i}}(x), \tag{1}\]  
\[O_{1,i}=\mu_{B_{i-2}}(y), \tag{2}\]  
we adopt the Gaussian MF by:  
\[\mu_{A_{i}}(x)=exp\left[-\left(\frac{x-a_{i}}{c_{i}}\right)^{2}\right], \tag{3}\]  
where \(a_{i}\) and \(c_{i}\) are the shape parameters (premise parameters) of an MF.  
Layer \(l_{2}\) is the rule layer with fixed or nonadaptive nodes labeled as \(\pi\). As an output, it computes the firing strength \(w_{i}\) (activation level) of each rule with the AND logical operator (or with other types of T-norm operators with general performance):  
\[O_{2,i}=w_{i}=\mu_{A_{i}}(x)*\mu_{B_{i}}(y). \tag{4}\]  
In \(l_{3}\) is the normalization layer with fixed nodes, labeled as \(N\). The outputs of this layer are normalized representations of each rule:  
\[O_{3,i}=\overline{w_{i}}=\frac{w_{i}}{w_{1}+w_{2}}. \tag{5}\]  
Layer \(l_{4}\) contains adaptive nodes. The output of each node is the product of normalized firing strength and a first order polynomial (for a first order Sugeno model):  
\[O_{4,i}=\overline{w_{i}}*f_{i}=\overline{w_{i}}*(p_{i}x+q_{i}y+r_{i}). \tag{6}\]  
Figure 4: The adaptive neuro-fuzzy inference system (ANFIS) architecture [15].  
Layer \(l_{5}\) contains a single fixed node, labeled as \(\sum\). It computes the summation of all the input signals coming from the nodes in layer \(l_{4}\):  
\[O_{5,i}=\sum_{i}\overline{w_{i}}*f_{i}=\frac{\sum_{i}w_{i}*f_{i}}{\sum_{i}w_{i}}. \tag{7}\]  
Consequent and premise parameters are computed during the training process, based on the back-propagation gradient descent algorithm or with hybrid learning [15].