## 2 Related Concepts
### SVM System and SVR
If the training sample falls into the interval band, the prediction is considered correct. In order to minimize the loss function, based on the criterion of structural risk minimization [21], determining the constraints \(\mathbf{\omega}\) and \(b\) and optimizing the objective function, the following optimization criterion is constructed:  
\[\min_{\mathbf{\omega},b,\varepsilon,\varepsilon^{*}}\frac{1}{2}\mathbf{\omega}^{T} \mathbf{\omega}+C{\sum_{i=1}^{n}}\Bigl{(}\xi_{i}+\xi_{i}^{*}\Bigr{)}, \tag{4}\]  
\[ \begin{array}{ll}\text{s.t.}&y_{i}-\bigl{(}\mathbf{\omega}^{T}\mathbf{x}_{i}+b \bigr{)}\leq\varepsilon+\xi_{i}\\ &\bigl{(}\mathbf{\omega}^{T}\mathbf{x}_{i}+b\bigr{)}-y_{i}\leq\varepsilon+\xi_{i}^{*}\\ &\xi_{i},\xi_{i}^{*}\geq 0\end{array}  \tag{5}\]  
Figure 1: SVR schematic diagram. The area between the two dashed lines represents the \(\varepsilon\)-interval band; the predicted loss of samples falling into this area is not calculated. This is the convex quadratic programming problem of the original problem of SVR, where \(C\) is the penalty factor and \(C>0\), which aims to find a compromise between generalization performance and training error; \(\varepsilon\) is the maximum tolerance value; and \(\xi_{i}\) and \(\xi_{i}^{*}\) are slack variables to avoid over-training. Combining these variables allows the model a certain degree of fault tolerance [22]. The Lagrange function can be obtained by introducing Lagrange multipliers:  
\[ \begin{array}{l}L\Big{(}\mathbf{\omega},b,\alpha_{i},\alpha_{i}^{*},\xi_{i}, \xi_{i}^{*},\mu_{i},\mu_{i}^{*}\Big{)}\\ =\frac{1}{2}\mathbf{\omega}^{T}\mathbf{\omega}+C\sum\limits_{i=1}^{n}\Big{(}\xi_{i}+ \xi_{i}^{*}\Big{)}-\sum\limits_{i=1}^{n}\mu_{i}\xi_{i}-\sum\limits_{i=1}^{n} \mu_{i}^{*}\xi_{i}^{*}\\ +\sum\limits_{i=1}^{n}\alpha_{i}\Big{(}\mathbf{\omega}^{T}\varphi(\mathbf{x}_{i})+b-y_ {i}-\varepsilon-\xi_{i}\Big{)}+\sum\limits_{i=1}^{n}\alpha_{i}^{*}\Big{(}y_{i }-\mathbf{\omega}^{T}\varphi(\mathbf{x}_{i})-b-\varepsilon-\xi_{i}^{*}\Big{)}\end{array}  \tag{6}\]  
where \(\alpha_{i}^{(*)}\) and \(\mu_{i}^{(*)}\) are Lagrange multipliers and \(0\leq\alpha_{i}^{(*)},\mu_{i}^{(*)}\leq C\). The partial derivatives of \(\mathbf{\omega},b,\alpha_{i}\) and \(e_{i}\) are solved separately. Letting the partial derivatives be equal to zero [23] yields:  
\[ \begin{array}{l}\omega=\sum\limits_{i=1}^{n}\Big{(}\alpha_{i}^{*}-\alpha_{i }\Big{)}\mathbf{x}_{i},\\ \sum\limits_{i=1}^{n}\Big{(}\alpha_{i}^{*}-\alpha_{i}\Big{)}=0,\\ C=\alpha_{i}^{(*)}+\mu_{i}^{(*)}.\end{array}  \tag{7}\]  
Substituting Equation (7) into Equation (6) gives the SVR dual problem as:  
\[ \begin{array}{l}\max\limits_{\alpha,\alpha^{*}}\sum\limits_{i=1}^{n}y_{i} \Big{(}\alpha_{i}^{*}-\alpha_{i}\Big{)}-\varepsilon\Big{(}\alpha_{i}^{*}+ \alpha_{i}\Big{)}-\frac{1}{2}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\Big{(} \alpha_{i}^{*}-\alpha_{i}\Big{)}\Big{(}\alpha_{j}^{*}-\alpha_{j}\Big{)}K\Big{(} \mathbf{x}_{i},\mathbf{x}_{j}\Big{)},\end{array}  \tag{8}\]  
\[ \begin{array}{l}\text{s.t.}\sum\limits_{i=1}^{n}\Big{(}\alpha_{i}^{*}-\alpha _{i}\Big{)}=0.\end{array}  \tag{9}\]  
The above process must satisfy KKT conditions [24], that is:  
\[ \begin{array}{l}\left\{\begin{array}{l}\alpha_{i}(\mathbf{\omega}\mathbf{x}_{i}+b-y _{i}-\varepsilon-\xi_{i})=0\\ \alpha_{i}^{*}\big{(}y_{i}-\alpha\mathbf{x}_{i}-b-\varepsilon-\xi_{i}^{*}\big{)}=0 \\ \alpha_{i}\alpha_{i}^{*}=0,\xi_{i}\xi_{i}^{*}=0\\ (C-\alpha_{i})\xi_{i}=0,\Big{(}C-\alpha_{i}^{*}\Big{)}\xi_{i}^{*}=0\end{array} \right..\end{array}