## 2 Experimental Section
### Wavelet Support Vector Machine  
#### 2.2.1 Basic Theory of Support Vector Machine (SVM)  
A support vector machine, proposed by [PERSON], is a kind of feed forward network [18]; its main purpose is to establish a hyper-plane to make the input vector project into another high-dimensional space. Given a set of data \(\{(x_{i},d_{i})\}_{i=1}^{N}\), where \(x_{i}\) is the input vector; \(d\) is the expected output; it is further assumed that the estimate value of \(d\) is \(y\), which is obtained by the projection of a set of nonlinear functions:  
\[y=\sum\limits_{j=0}^{m}w_{j}\phi_{j}(x)=w^{T}\phi(x) \tag{29}\]  
where \(\phi(x)=[\phi_{0}(x),\phi_{1}(x),\cdots,\phi_{m}(x)]^{T}\), \(w=[w_{0},w_{1},\cdots,w_{m}]^{T}\); Let \(\phi_{0}(x)=1;w_{0}\) represent the bias \(b\), and the minimization risk function can be described as follows:  
\[\Phi(w,\xi,\xi^{\prime})=\frac{1}{2}w^{T}w+C{\sum\limits_{i=1}^{N}}\left(\xi_ {i}+\xi^{\prime}_{i}\right) \tag{30}\]  
The minimization risk function must satisfy the conditions:  
\[\left\{\begin{array}{l}d_{i}-w^{T}\phi(x_{i})\leqslant\varepsilon+\xi_{i}\\ w^{T}\phi(x_{i})-d_{i}\leqslant\varepsilon+\xi^{\prime}_{i}\\ \xi_{i}\geqslant 0\\ \xi^{\prime}_{i}\geqslant 0\end{array}\right. \tag{31}\]  
where \(i=1,2,\cdots,N\), \(\xi^{\prime}_{i}\) and \(\xi_{i}\) are slack variables; loss function is a \(\varepsilon\)-insensitive loss function, \(C\) is a constant.  
Establish the Lagrange function and obtain:  
\[\begin{array}{ll}J(w,\xi,\xi^{\prime},\alpha,\alpha^{\prime},\gamma,\gamma^ {\prime})&=C{\sum\limits_{i=1}^{N}}\left(\xi_{i}+\xi^{\prime}_{i} \right)+\frac{1}{2}w^{T}w-{\sum\limits_{i=1}^{N}}\alpha_{i}[w^{T}\phi(x_{i})-d _{i}+\varepsilon+\xi_{i}]\\ &-{\sum\limits_{i=1}^{N}}\alpha^{\prime}_{i}[d_{i}-w^{T}\phi(x_{i})+ \varepsilon+\xi^{\prime}_{i}]-{\sum\limits_{i=1}^{N}}\left(\gamma_{i}\xi_{i} +\gamma^{\prime}_{i}\xi^{\prime}_{i}\right)\end{array} \tag{32}\]  
where \(\alpha_{i}\) and \(\alpha^{\prime}_{i}\) are Lagrange multiplier; take the partial derivative of variables \(w,\xi,\xi^{\prime},\alpha,\alpha^{\prime},\gamma,\gamma^{\prime}\) and obtain:  
\[\begin{array}{l}w={\sum\limits_{i=1}^{N}}\left(\alpha_{i}-\alpha^{\prime}_{i }\right)\phi(x_{i})\\ \gamma_{i}=C-\alpha_{i};\ \gamma^{\prime}_{i}=C-\alpha^{\prime}_{i}\end{array} \tag{33}\]  
The above problem can be converted into a dual problem:  
\[\begin{array}{l}\max Q\left(\alpha_{i},\alpha^{\prime}_{i}\right)={\sum \limits_{i=1}^{N}}d_{i}(\alpha_{i}-\alpha^{\prime}_{i})-\varepsilon{\sum \limits_{i=1}^{N}}\left(\alpha_{i}+\alpha^{\prime}_{i}\right)-\frac{1}{2}{ \sum\limits_{i=1}^{N}}{\sum\limits_{j=1}^{N}}\left(\alpha_{i}-\alpha^{\prime}_ {i}\right)(\alpha_{j}-\alpha^{\prime}_{j})\phi^{T}(x_{i})\phi(x_{j})\\ \\ \left\{\begin{array}{l}{\sum\limits_{i=1}^{N}}\left(\alpha_{i}-\alpha^{\prime}_ {i}\right)=0\\ 0\leqslant\alpha_{i}\leqslant C\\ 0\leqslant\alpha^{\prime}_{i}\leqslant C\\ \\ i=1,2,\cdots,N\end{array}\right.\end{array} \tag{34}\]Solve the equation and obtain: \(w=\sum\limits_{i=1}^{N}{(\alpha_{i}-\alpha_{i}^{\prime})\phi(x)}\)  
Then \(F(x,w)=w^{T}x=\sum\limits_{i=1}^{N}{(\alpha_{i}-\alpha_{i}^{\prime})\phi^{T}(x) \phi(x)}\), let \(K(x^{T},x)=\phi^{T}(x)\phi(x)\) be the kernel function.  
In this paper, we choose the wavelet kernel function to replace the Gaussian kernel function, and the construction of the wavelet kernel function will be introduced in detail in Section 2.2.2.