## 2 Modeling Method
### Support Vector Regression  
SVR is derived from support vector machine (SVM) and is usually used as classification methodology. This model converts the nonlinear problem in low-dimensional space into high-dimensional space and finds a hyperplane to fit the relation between input and output. Researchers have developed different kinds of SVR models in the previous studies. Among these types, \(\epsilon\)-SVR is considered as the most applied type and is utilized in this paper with an introduction in the following paragraphs. Given a training dataset \(D=\{(\mathbf{\chi}_{i},y_{i}),i=1,2,\ldots,m\}\) with \(m\) samples, where \(\mathbf{\chi}_{i}\) and \(y_{i}\) represent input vector and actual value respectively, a regression function can be built to approximate the target value \(y_{i}\),  
\[f(\mathbf{\chi})=\mathbf{\omega}^{T}\mathbf{\phi}(\mathbf{\chi})+b, \tag{1}\]  
Figure 1: The framework of the hybrid improved SVR. where \(\mathbf{\chi}=(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{m})^{T}\) refers to \(m\) groups of input vectors, \(\phi(\mathbf{x})\) transforms the sample from Euclidean space to high-dimension space, \(\mathbf{\omega}\) is a weight vector, superscript \(T\) denotes transpose, and \(b\) is a bias value. \(\varepsilon\)-SVR method adopts a new type of loss function, which creates an insensitive zone by maximum \(\varepsilon\) value. \(\varepsilon\) refers to the maximum deviation from optimal hyperplane. When the absolute value of deviation between hyperplane function \(f(\mathbf{\chi})\) and target \(\mathbf{y}=(y_{1},y_{2},\ldots,y_{m})^{T}\) is less than \(\varepsilon\), the loss is acceptable. Based on that, the SVR problem can be formalized as  
\[\underset{\mathbf{\omega},b}{min}\frac{1}{2}\|\mathbf{\omega}\|^{2}+C\sum_{i=1}^{m}l_ {\varepsilon}(f(\mathbf{x}_{i})-y_{i}), \tag{2}\]  
where \(C\) is the penalty factor regulating the tolerance for samples beyond insensitive zone, and \(\ell_{\varepsilon}\) represents the \(\varepsilon\)-insensitive loss function in Equation (3):  
\[\ell_{\varepsilon}(f(\mathbf{x}_{i})-y_{i})=\left\{ \begin{array}{ll}0,&if\mid f( \mathbf{x}_{i})-y_{i}\mid\leq\varepsilon,\\ \mid f(\mathbf{x}_{i})-y_{i}\mid-\varepsilon,&otherwise.\end{array} \right. \tag{3}\]  
We introduce slack variables \(\xi_{i}\) and \(\xi_{i}^{*}\) to consider the training samples outside \(\varepsilon\)-insensitive zone, then the Equation (3) can be rewritten as  
\[\underset{\mathbf{\omega},b,\xi_{i},\xi_{i}^{*}}{min}\frac{1}{2}\|\bm {\omega}\|^{2}+C\sum_{i=1}^{m}(\xi_{i}+\xi_{i}^{*}), \tag{4}\] \[s.t\left\{ \begin{array}{ll}f(\mathbf{x}_{i})-y_{i}\leq\varepsilon+ \xi_{i},\\ y_{i}-f(\mathbf{x}_{i})\leq\varepsilon+\xi_{i}^{*},\\ \xi_{i},\xi_{i}^{*}\geq 0,&i=1,2\ldots m.\end{array} \right. \tag{5}\]  
By introducing a set of Lagrange multipliers, including \(\mu_{i}\geq 0\), \(\mu_{i}^{*}\geq 0\), \(\alpha_{i}\geq 0\), \(\alpha_{i}^{*}\geq 0\), a Lagrange can be created as  
\[L(\mathbf{\omega},B,\mathbf{\alpha},\mathbf{\alpha}^{*},\mathbf{\xi},\mathbf{\xi}^{*},\mathbf{\mu},\mathbf{\mu}^{*})= \frac{1}{2}\|\mathbf{\omega}\|^{2}+C\sum_{i=1}^{m}(\xi_{i}+\xi_{i}^{* })-\sum_{i=1}^{m}(\mu_{i}\xi_{i}+\mu_{i}^{*}\xi_{i}^{*})+ \tag{6}\] \[\sum_{i=1}^{m}\alpha_{i}(f(\mathbf{x}_{i})-y_{i}-\varepsilon-\xi_{i}) +\sum_{i=1}^{m}\alpha_{i}^{*}(y_{i}-f(\mathbf{x}_{i})-\varepsilon-\xi_{i}^{*}).\]  
Let the partial derivativesfor \(L\) with respect to \(\omega\), \(b\), \(\xi_{i}\), and \(\xi_{i}^{*}\) equal zero.