## 2 Data-Driven Modeling
### Support Vector Machine (SVM)  
Support vector machine (SVM) is one of supervised learning models. SVM initially developed by [PERSON] [73], weights are calculated based on input data, known as training data, in order to learn the governing function from a set of inputs. SVM selects a limited number of input sample vectors, which are always a fraction of the total number of samples. These input vectors are referred to as support vectors. Using these input vectors, the parameter values that minimize a cost function are calculated. As a result, SVM requires much less data than similar methods such as ANN, and as a result, it takes much less time and has a lower cost.  
The goal of this method, is to find a classifier that separates the data and maximizes the distance between these two classes. Figure 2 shows a plane cloud, that separates two sets of points.  
Figure 1: Classification of the AI modeling techniques (adapted from [PERSON] et al. [63] and [PERSON] et al. [64]).  
According to Figure 3, the closest points used to determine the hyperplane, are called support vectors. As is clear, there are many hypermaps that can divide samples into two categories. The principle of SVM, is to choose something that maximizes the minimum distance between the hypermap and the training samples (that is, the distance between the hypermap and the support vectors), this distance is called the margin.  
The kernel function, as an important part of the SVM method, receives data as input and transforms them into the required form. Various functions are provided for this purpose. Some of these functions, including linear, nonlinear, polynomial, radial basis function (RBF), and sigmoid, are given in Equations (1)-(5). According to Equations (1)-(5), the kernel function is a series of mathematical functions that provide a window for data manipulation. With the help of this transformation, a complex and nonlinear level of decision making becomes a linear equation, but in a larger number of dimensional spaces.  
\[\text{Polynomial kernel}:\ k=k\big{(}x_{i},x_{j}\big{)}=\big{(}x_{i}.x_{j}+1 \big{)}^{d} \tag{1}\]  
\[\text{Gaussian kernel}:\ k=k\big{(}x_{i},x_{j}\big{)}=exp\Bigg{(}-\frac{ \big{\|}x_{i}-x_{j}\big{\|}^{2}}{2\sigma^{2}}\Bigg{)} \tag{2}\]  
Figure 3: Support vectors.  
Figure 2: The hyperplane H, that separates the two groups of points.  
\[\text{Radial basis function }(\text{RBF}):\ k=k\big{(}x_{i},x_{j}\big{)}= exp\Big{(}-\gamma\big{\|}\,x_{i}-x_{j}\big{\|}^{\,2}\Big{)} \tag{3}\] \[\text{Hyperbolic }\tan\text{gent kernel}:\ k=k\big{(}x_{i},x_{j} \big{)}=tanh\big{(}kx_{i}.x_{j}+c\big{)}\] (4) \[\text{Sigmoid kernel}:\ k=k\big{(}x_{i},x_{j}\big{)}=tanh\Big{(} ax^{T}y+c\Big{)} \tag{5}\]  
where \(x_{i}\) and \(x_{j}\) are vectors in the input space, the \(\gamma\) and \(\sigma\) parameters define the distance influence of a single sample, \(d\) is the degree of the polynomial, \(c\in R\) is the relative position to the origin, \(\kappa\)' is a real-valued positive type function/kernel, and \(\alpha\), \(T\), and \(c\) are certain values.  
In this study, two important functions, i.e., the radial basis function (RBF) and Gaussian kernel, which have been successful in past research, were evaluated, along with two other functions, i.e., the sigmoid kernel and polynomial kernel functions.