## 3 Background of the Methods Used
### Support Vector Machine (SVM)  
SVM is a machine learning algorithm based on the principle of minimizing structural risk to generalize a limited number of samples better and was proposed by [PERSON], [PERSON] [55]. It can solve both classification and regression problems [56; 57]. Since the objectives of this study was numeric, so, support vector machine for regression problems were applied. In the general regression learning problem, the learning machine-based training data from which it attempts to learn the input-output relationship (dependency, mapping, or function) \(f(x)\). A training data set \(D=\left\{[x(i),y(i)]\in\Re^{n}*\Re,i=1,\ldots,l\right\}\). However, in the case of SVM's regression, the error of approximation was measured rather than the margin used in classification. Therefore, a linear regression hyperplane by minimizing Vapnik's insensitivity loss function is shown in formula (7).  
\[\Re_{w,\psi,\psi^{\star}}=\left[\frac{1}{2}\|W\|^{2}+C\left(\sum_{i=1}^{l}\psi _{i}+\sum_{i=1}^{l}\psi_{i}^{\star}\right)\right] \tag{7}\]  
From the above function it can be seen the slack variables \(\psi_{i}\) and \(\psi_{i}^{\star}\) also related to Lagrange multipliers \(\
u_{i}\) and \(\
u_{i}^{\star}\), and note also the trade-off between an approximation error and the weight vector norm \(\|V\|\) were affected by constant \(C\). While, in the case of nonlinear regression, solving the maximum value of dual Lagrange function is equivalent to solving the learning problem, the dual Lagrangian in input space is:  
\[L_{d}(\
u)=-0.5\
u^{T}H\
u+\sum_{i=1}^{l}\
u_{i} \tag{8}\]  
Subject to : \[\
u=\left[\
u_{1},\
u_{2},\ldots\
u_{l},\
u_{1}^{\star},\
u_{2}^{ \star},\ldots,\
u_{1}^{\star}\right]^{T}\] \[H=\left[G-G;-G\ G\right]\]\[G=\left[\begin{array}{cccc}G_{11}&\ldots&G_{1I}\\ \vdots&G_{ii}&\vdots\\ G_{I1}&\cdots&G_{Il}\end{array}\right]\]  
\[\text{where }G_{ij}=\Phi^{T}(x_{i})\Phi\Big{(}x_{j}\Big{)}\]  
After calculation of Lagrange multipliers \(v_{i}\) and \(v_{i}^{*}\), an optimal weight vector of the regression hyperplane could be expressed with the formula (9).  
\[w_{0}=\sum_{i}^{l}\Big{(}v_{i}-v_{i}^{*}\Big{)}x_{i} \tag{9}\]  
In order to create the best nonlinear regression function, the most crucial method is using the kernel function, there are some formulas kernel functions had been proposed, as listed in Table 1.  
We replaced \(x_{i}\) by the corresponding feature vector in a feature space and made some changes with a transform \(G_{ij}=x_{i}^{T}x_{j}\) to \(G_{ij}=\Phi^{T}(x_{i})\Phi\Big{(}x_{j}\Big{)}\); then it follows that \(G_{ij}=K\Big{(}x_{i},x_{j}\Big{)}\) based the above kernel function \(\big{(}K\Big{(}x_{i},x_{j}\Big{)}=\Phi^{T}(x_{i})\Phi\Big{(}x_{j}\Big{)}\big{)}\).  
Besides, an optimal weighting vector of the kernel's expansion, as shown in formulas (10), can be calculated with Lagrange multiplier.  
\[V_{0}=v-v^{*} \tag{10}\]  
In this way, the regression function combined weighting vector and kernels function as follows,  
\[\begin{split} f(x,w)=w_{0}^{T}\Phi(x)+b=\sum\limits_{i=1}^{l} \Big{(}v_{i}-v_{i}^{*}\Big{)}\Phi^{T}(x_{i})\Phi(x)+b\\ =\sum\limits_{i=1}^{l}\Big{(}v_{i}-v_{i}^{*}\Big{)}K(x_{i},x)+b \end{split} \tag{11}\]