## 2 Methodology
### Support Vector Regression Model  
Support vector regression (SVR) is a kind of machine learning algorithm based on the structural risk minimization theory and the statistical learning theory, which has good generalization ability when solving small sample problems [27]. The basic principle is to find an optimal classification surface in the high dimensional space by mapping the nonlinear relation of input variables in the low dimensional space to the high dimensional space through the kernel function, so that the data can be linearly separable in the high dimensional space. When constructing the support vector regression model, it is necessary to introduce the insensitive loss function \(\varepsilon\) to find an optimal classification surface, so that the error of all training samples from this classification surface is minimized. When the deviation between the output value (predicted value) of the model algorithm and the corresponding true value is \([-\varepsilon,\varepsilon]\), the loss is ignored; otherwise, it is included in the loss, as shown in Figure 2.  
Let there be \(n\) training samples \(\{x_{i},y_{i}\}(i=1,2,3,\ldots,m,\)\(x_{i}\in R_{d},y_{i}\in R)\) for regression fitting, where \(x_{i}\) is the input variable of the \(i\)-th sample and \(y_{i}\) is the corresponding output parameter. The linear regression function is established in the high-dimensional space, and the sample is projected into the high-dimensional linear space through the nonlinear mapping relationship:  
\[f(x)=w\varphi(x)+b, \tag{1}\]  
where \(w\) and \(b\) are the parameters to be solved. In order to find the optimal \(w\) and \(b\), the relaxation variables \(\xi_{i}\) and \(\zeta_{i}\) are introduced to construct a minimal objective function, namely:  
\[\left\{ \begin{array}{l}\min\frac{1}{2}\|w\|^{2}+\hat{C}\sum\limits_{i=1}^{n} \left(\xi_{i}+\zeta_{i}\right)\\ s.t.\left\{\begin{array}{l}y_{i}-w\varphi(x_{i})-b\leq\varepsilon+\xi_{i}\\ -y_{i}+w\varphi(x_{i})+b\leq\varepsilon+\xi_{i}\\ \xi_{i}\geq 0,\zeta_{i}\geq 0\\ i=1,2,\cdots,n\end{array}\right.,\end{array} \right. \tag{2}\]  
where \(\hat{C}\) is the penalty factor--the larger the \(\hat{C}\) is, the larger the penalty is for the sample whose training error is greater than \(\varepsilon\); \(\varepsilon\) reflects the error requirement of the regression function, and a smaller \(\varepsilon\) means a smaller error of the regression function. Figure 2: Support vector regression schematic. According to the optimization theory, it is transformed into a dual problem, and a Lagrange function and kernel function \(k(x_{i},x_{j})=\varphi(x_{i})\cdot\varphi(x_{j})\) are introduced to solve it:  
\[\left\{ \begin{array}{l}\max\limits_{\alpha,\alpha^{*}}\left[ \begin{array}{l}-\frac{1}{2}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}(\alpha _{i}-\alpha_{i}^{*})(\alpha_{j}-\alpha_{j}^{*})k(x_{i},x_{j})\\ -\sum\limits_{i=1}^{n}(\alpha_{i}+\alpha_{i}^{*})\varepsilon+\sum\limits_{i=1} ^{n}(\alpha_{i}-\alpha_{i}^{*})y_{i}\\ s.t.\left\{\begin{array}{l}n\\ i=1\end{array}(\alpha_{i}-\alpha_{i}^{*})=0\\ 0\leq\alpha_{i}\leq\hat{C}\\ 0\leq\alpha_{i}^{*}\leq\hat{C}\end{array}\right.\end{array} \right\}\right., \tag{3}\]  
Suppose the optimal solution obtained by solving the above formula is \(\alpha\) and \(\alpha^{*}\), then:  
\[\left\{ \begin{array}{l}w^{*}=\sum\limits_{i=1}^{n}(\alpha_{i}-\alpha_{i}^{*} )\varphi(x_{i})\\ b^{*}=\frac{1}{N_{\text{nsv}}}\left\{\begin{array}{l}\sum\limits_{0<\alpha_{ i}<\hat{C}}\left[y_{i}-\sum\limits_{x_{i}\in\text{SV}}(\alpha_{i}-\alpha_{i}^{*} )k(x_{i},x_{j})-\varepsilon\right]\\ +\sum\limits_{0<\alpha_{i}<\hat{C}}\left[y_{i}-\sum\limits_{x_{j}\in\text{SV}} (\alpha_{i}-\alpha_{i}^{*})k(x_{i},x_{j})+\varepsilon\right]\end{array}\right\} \end{array} \right., \tag{4}\]  
where \(N_{\text{nsv}}\) is the number of support vectors.