## 2 Overview of Soft Computing Approaches
### Adaptive Neuro Fuzzy Inference System  
ANFIS is a combined intelligent system including ideas from neural networks and fuzzy control, combining the advantages of both. Fuzzy logic is a superset of typical logic that has been improved to operate with uncertain data and the theory of partial truth [37; 38]. The most important disadvantage of fuzzy logic is the lack of a systematic approach to choosing membership function variables and designing the fuzzy rules. On the other hand, ANN has the ability to learn its structure from the input-output sets.  
[PERSON] [39] introduced ANFIS as a universal approximation which can estimate any real continued function on a compact data set with the desired precision [40; 41]. In terms of function, ANFIS can be considered as equivalent to fuzzy inference systems, and the ANFIS system used can be considered as comparable to the Sugeno first-order fuzzy model [42]. A simple example is presented below, in which a fuzzy inference system was assumed with two inputs of x and y and one output of z. For this example, the typical rule set of a first-order Sugeno fuzzy model, which possess two simple fuzzy If-Then rules, is as follows:  
\[\text{Rule 1 : IF x is A1, y is B1 and z is C1 THEN }f_{1}=p_{1}x+q_{1}y+r_{1}z+t_{1} \tag{2}\]  
\[\text{Rule 2 : IF x is A2, y is B2 and z is C2 THEN }f_{2}=p_{2}x+q_{2}y+r_{2}z+t_{2} \tag{3}\]  
In the rule set, \(p_{1}\), \(q_{1}\), \(r_{1}\) and \(p_{2}\), \(q_{2}\), \(r_{2}\) are the variables of the THEN-part of the first-order Sugeno fuzzy model.  
The ANFIS system uses a hybrid-learning algorithm to update parameters [43]. This algorithm is composed of two methods: the least squares approach and the gradient descent method. The function of the gradient descent approach is to adjust the variables of premise non-linear membership function, and the function of least squares method is to determine the resultant linear variables {\(p_{i}\), \(q_{i}\), \(r_{i}\)}. The learning process of this system has two steps. The first step includes the identification of consequent variables by the least squares method, while the prior variables are assumed to be fixed for the running cycle by the training set. After that, the error signals will spread backwards. In this part, the function of the gradient descent method includes updating the premise variables by minimizing the cost function, while the resultant variables stay fixed. [PERSON] [39] presented the details of this algorithm and mathematical foundations of the hybrid learning algorithm.  
In the present paper, two different ANFIS methods, including ANFIS with subtractive clustering (ANFIS-SC) and ANFIS with fuzzy c-means clustering (ANFIS-FCM),are utilized as modeling techniques. Subtractive clustering (SC) is an extension of the mountain clustering method suggested by [PERSON] and [PERSON] [44]. In this method, the data are clustered by evaluating the potential of data in the specification space. FCM is the modified K-means algorithm; it has some restrictions and may not operate properly with large data sets. FCM minimizes within cluster variance and the classification of data using the clustering algorithm [45]. FCM works by minimizing the squared error function.