## 2 Materials and Methods
### Adaptive Neuro-Fuzzy Inference System (ANFIS)  
The adaptive neural network based fuzzy inference system (ANFIS) is a hybrid system. It includes both the advantages of the self-adaptability and learning competence of the neural network and the ability of the fuzzy system to take into account the prevailing uncertainty and imprecision of real systems. The neuro-fuzzy modeling approach is concerned with model extraction from numerical data which represents the dynamic behaviour of the scheme. With ANFIS method, an initial fuzzy model is generated with the help of the rules extracted from the input-output data. Next, the neural network is used to tune the rules of the initial fuzzy model to produce the final ANFIS model. The formulations and discussion of ANFIS architecture can be found in [41; 42]. Unlike ANN, it has a higher capability in the learning process to adapt to its environment. Therefore, it can be used to automatically adjust the membership function's parameters and reduce the rate of errors in the determination of rules in fuzzy logic. Figure 4: Takagi-Sugeno type FS system with premise and consequent part. Figure 5: Membership function for Eccentricity feature. The ANFIS architecture shown in Figure 6 is an adaptive network that uses supervised learning algorithm and has a function similar to the model of Takagi-Sugeno fuzzy inference system as discussed in Section 2.5. Let's assume that there are two inputs \(x\) and \(y\), and one output \(f\) of the architecture. Two rules are used in the method of \"If-Then\" for Takagi-Sugeno model, as follows:  
* **Rule 1:** If \(x\) is \(A_{1}\) and \(y\) is \(B_{1}\) Then \(f_{1}=p_{1}x+q_{1}y+r_{1}\). * **Rule 2:** If \(x\) is \(A_{2}\) and \(y\) is \(B_{2}\) Then \(f_{2}=p_{2}x+q_{2}y+r_{2}\). where \(A_{1}\), \(A_{2}\) and \(B_{1}\), \(B_{2}\) are the membership functions of each input \(x\) and \(y\) (premises), while \(p_{1}\), \(q_{1}\), \(r_{1}\) and \(p_{2}\), \(q_{2}\), \(r_{2}\) are linear parameters in the consequent part of Takagi-Sugeno fuzzy inference model. ANFIS architecture has five layers. The first and fourth layers contain an adaptive node, while the other layers are fixed nodes. **Layer 1:** Each node adapts to a function parameter. The output from each node is a degree of membership value that is given by the input of the membership functions. For example, the membership function used in this study is a generalized bell membership function (c.f. Section 2.5). \[\mu_{Ai}(x)=\frac{1}{1+\left|\frac{x-c}{a}\right|^{2b}} \tag{1}\]  
where \(\mu_{Ai}\) is the degree of membership functions for the fuzzy set \(Ai\), and \(\{a,b,c\}\) are the parameters of a membership function which can change the shape of the membership function as shown in Figure 5.  
**Layer 2:** Each node in this layer is fixed or non-adaptive and represented with a product operator \(\Pi\). Each node in this layer represents the firing strength for each rule. **Layer 3:** Each node in this layer is fixed or non-adaptive and labeled as \(N\). It is normalizing the firing strength as \(\bar{w}_{i}=w_{i}/\sum w_{i}\). **Layer 4:** Each node in this layer is an adaptive node with a node function defined as \(\bar{w}_{i}f_{i}=p_{i}x+q_{i}y+r_{i}\). The parameters in this layer are referred to as consequent parameters. **Layer 5:** The single node in this layer is a fixed or non-adaptive node that computes the overall output as the summation of all incoming signals from previous nodes as \(\sum\bar{w}_{i}f_{i}\). In the ANFIS architecture, the first layer and the fourth layer contain the parameters which are tuned during the training phase. The number of training epochs, the membership functions and the number of fuzzy rules should be selected accurately while designing of ANFIS model [41], as it may lead system to overfit the data. This tuning is obtained with a hybrid algorithm combining the least-squares method and the gradient descent method with a mean square error method [44]. Figure 6: ANFIS architecture with input, hidden and output layer [43].