## 2 Methodology
### Support Vector Machine  
Support vector machine is proposed by [PERSON] et al. It is a small sample machine learning method on the basis of VC dimension theory [24; 25]. The basic idea of using the SVM algorithm to estimate the regression function is to nonlinearly map data to a high-dimensional feature space, and then carry out linear regression in this high-dimensional space [26]. Set n training sample sets: \(\{(\mathrm{x_{i}},\mathrm{y_{j}})|\mathrm{i=1,2,\cdots,n}\}\). Among them, \(\mathrm{x_{i}}\in\mathrm{R^{d}}\) are training sample inputs of d dimension, \(\mathrm{y_{i}}\in\mathrm{R}\) are training sample outputs. The nonlinear mapping \(\psi(\mathrm{x})=(\phi(\mathrm{x_{1}}),\phi(\mathrm{x_{2}}),\cdots,\phi( \mathrm{x_{n}}))\) is used to map input samples from the original space \(\mathrm{R^{d}}\) to the high-dimensional feature space \(\mathrm{R^{k}}(\mathrm{k>d})\). The optimal linear regression function is constructed in the high-dimensional feature space  
\[\mathrm{f(x_{i})}=\omega^{\mathrm{T}}\phi(\mathrm{x_{i}})+\mathrm{b} \tag{6}\]  
In the equation: \(\omega\) is the weight vector in the high-dimensional feature space, \(\omega\in\mathrm{R^{k}}\). b is the bias constant, \(\mathrm{b}\in\mathrm{R}\). Determine \(\omega\) and b in the light of the principle of structural risk minimization  
\[\mathrm{minR}=\frac{1}{2}\|\omega\|^{2}+\mathrm{c}\cdot\mathrm{R_{emp}} \tag{7}\]  
In the equation, \(\|\omega\|^{2}\) controls the model complexity, c is the regularization parameter and \(\mathrm{R_{emp}}\) is the error control function. If \(\mathrm{R_{emp}}\) selects different loss functions, different forms of SVM can be constructed. The loss function of standard SVM in the optimization objective is the linear term of the error. In the light of the principle of structural risk minimization, Equation (7) can be transformed into  
\[\mathrm{minJ}=\tfrac{1}{2}\|\omega\|^{2}+\mathrm{c}\sum\limits_{i=1}^{n}\left( \xi_{i}+\xi_{i}{}^{*}\right) \tag{8}\] \[\mathrm{s.t. }\left\{ \begin{array}{l}\mathrm{y_{i}}-\omega^{ \mathrm{T}}\phi(\mathrm{x_{i}})-\mathrm{b}\leq\varepsilon+\xi_{i}\\ \omega^{\mathrm{T}}\phi(\mathrm{x_{i}})+\mathrm{b}-\mathrm{y_{i}}\leq \varepsilon+\xi_{i}\\ \xi_{i},\xi_{i}{}^{*}\geq 0\ (\mathrm{i=1,2,\cdots,n})\end{array} \right.\]  
In the equation, \(\varepsilon\) is the insensitivity coefficient, \(\xi_{i}\) and \(\xi_{i}{}^{*}\) are relaxation factors. Solve optimization problem with Lagrange function,  
\[\mathrm{L}(\omega,\xi_{i},\xi_{i}{}^{*},\alpha,\alpha^{*}, \mathrm{c},\beta,\beta^{*})= \tfrac{1}{2}\|\omega\|^{2}+\mathrm{c}\sum\limits_{i=1}^{n}\left( \xi_{i}+\xi_{i}{}^{*}\right) \tag{9}\] \[-\sum\limits_{i=1}^{n}\alpha_{i}\big{[}\omega^{\mathrm{T}}\phi( \mathrm{x_{i}})+\mathrm{b}-\mathrm{y_{i}}+\varepsilon+\xi_{i}\big{]}\] \[-\sum\limits_{i=1}^{n}\alpha_{i}{}^{*}\big{[}\mathrm{y_{i}}- \omega^{\mathrm{T}}\phi(\mathrm{x_{i}})-\mathrm{b}+\varepsilon+\xi_{i}{}^{*} \big{]}\] \[-\sum\limits_{i=1}^{n}\left(\beta_{i}\xi_{i}+\beta_{i}{}^{*}\xi_{i }{}^{*}\right)\]In the equation, \(\alpha_{i}\), \(\alpha_{i}\)*, \(\beta_{i}\), \(\beta_{i}\)* are all Lagrange multipliers and \(\alpha_{i}>0\), \(\alpha_{i}\)* \(>0\), \(\beta_{i}\)* \(>0\) (\(i=1,2,\cdots,n\)). According to Karush-Kuhn-Tucker optimization condition,  
\[\left\{ \begin{array}{l}\frac{\partial L}{\partial\omega}=0\rightarrow\omega= \sum\limits_{i=1}^{n}{(\alpha_{i}-{\alpha_{i}}^{*})\phi(x_{i})}\\ \frac{\partial L}{\partial b}=0\rightarrow\sum\limits_{i=1}^{n}{(\alpha_{i}- \alpha_{i}^{*})}=0\\ \\ \frac{\partial L}{\partial\lambda_{i}}=0\rightarrow\mathrm{c}-\alpha_{i}-\beta _{i}=0\\ \\ \frac{\partial L}{\partial\lambda_{i}^{*}}=0\rightarrow\mathrm{c}-{\alpha_{i}}^{ *}-{\beta_{i}}^{*}=0\end{array} \right. \tag{10}\]  
Define \(\mathrm{K}(x_{i},x_{j})=\phi(x_{i})^{T}\phi(x_{j})\) as the symmetric kernel function of the Mercer condition. Then the optimization problem can be expressed as:  
\[