## 3 Modeling Method
### Support Vector Machine
#### 3.2.1 Support Vector Regression  
The principle of SVR is briefly introduced here. In SVR, the main goal is to obtain a function that differs at most \(\varepsilon\) from the actual targets for all training data, while being as flat as possible [45]. This means that our attention is no longer focused on the errors because small residuals are inevitable, but we focus on avoiding errors larger than a certain value. Suppose that the training data are given as \(\{(\mathbf{x_{1}},y_{1}),(\mathbf{x_{2}},y_{2}),\ldots,(\mathbf{x_{n}},y_{n})\}\), where \(n\) is the number of the training datasets, \(\mathbf{x_{i}}\) is the input vector and \(y_{i}\) is the output target value. Then, a function \(f\) (\(\mathbf{x}\)) corresponding to the lowest expected risk is constructed, and the \(\varepsilon\)-insensitive loss function is expressed as:  
\[L_{\varepsilon}(y,f(\mathbf{x}))=\max\{0,|y-f(\mathbf{x})|-\varepsilon\},\varepsilon>0 \tag{6}\]  
For the linear regression, function \(f(\mathbf{x})\) can be represented by  
\[f(x)=\langle\mathbf{w},\mathbf{x}\rangle+b \tag{7}\]  
where \(\mathbf{w}\in\mathbb{R}^{n}\) is the weight vector, determining the orientation of a discriminating plane, and \(\mathbb{b}\in\mathbb{R}\) is the scalar threshold, determining the offset of the discriminating plane from the origin [16]. In terms of the nonlinear regression, the process proceeds with mapping of the data into a high feature dimension space through a non-linear mapping procedure, and then solving a linear regression in this high feature dimension space. Then, a nonlinear mapping \(\Phi\) is introduced and can be expressed as:  
\[f(x)=\langle\mathbf{w},\Phi(\mathbf{x})\rangle+b \tag{8}\]  
Figure 10: Final network configuration used in present study.  
To ensure the flatness of Equation (8), a smaller value of \(\mathbf{w}\) is required. In real problems, it is impossible for all data points to have an error lesser than \(\varepsilon\) for the one function. For this reason, the slack variables \(\xi_{i}\) and \(\xi_{i}^{*}\) are introduced. Therefore, the SVR can be formulated as an optimization problem:  
\[\text{Minimize }\frac{1}{2}\|\mathbf{w}\|^{2}+C\sum\limits_{i=1}^{n}\left(\xi_{i} +\xi_{i}^{*}\right)\text{\ \ subjected to \ \ }\left\{\begin{array}{l}y_{i}-\langle\mathbf{w},\Phi(\mathbf{x}_{i})\rangle-b \leq\varepsilon+\xi_{i}\\ \langle\mathbf{w},\Phi(\mathbf{x}_{i})\rangle+b-y_{i}\leq\varepsilon+\xi_{i}^{ *}\\ \xi_{i},\xi_{i}^{*}\geq 0\end{array}\right. \tag{9}\]  
where \(C\) is the regularized constant specified by the user. It is defined as the penalty factor to indicate the trade-off between the flatness of the function and the empirical error.  
The optimization problem in Equation (9) can be solved by introducing the Lagrangian multipliers \(\alpha_{i},\alpha_{i}^{*},\eta_{i},\eta_{i}^{*}\) and the above optimization problem can be transformed into a dual quadratic programming problem:  
\[\begin{array}{l}\text{Maximizing}-\frac{1}{2}\sum\limits_{i=1}^{n}\sum \limits_{j=1}^{n}\left(\alpha_{i}-\alpha_{i}^{*}\right)(\alpha_{j}-\alpha_{j}^ {*})K(x_{i},x_{j})-\varepsilon\sum\limits_{i=1}^{n}\left(\alpha_{i}+\alpha_{i} ^{*}\right)+\sum\limits_{i=1}^{n}y_{i}(\alpha_{i}-\alpha_{i}^{*})\\ \text{Subjected to }\sum\limits_{i=1}^{n}\left(\alpha_{i}-\alpha_{i}^{*} \right)=0\text{ and }\alpha_{i},\alpha_{i}^{*}\in[0,C]\end{array} \tag{10}\]  
where \(K(\mathbf{x}_{i},\mathbf{x}_{j})=\Phi(\mathbf{x}_{i})\cdot\Phi(\mathbf{x}_{j})\) is defined as the kernel function, which is an important analysis technique in SVR. The values of \(\alpha_{i}\) and \(\alpha_{i}^{*}\) can be obtained by solving Equation (10), and the function \(f(\mathbf{x})\) is finally written as:  
\[f(\mathbf{x})=\sum\limits_{i=1}^{n}\left(\alpha_{i}-\alpha_{i}^{*}\right)K(x_ {i},x_{j})+b \tag{11}\]