## 2 Experimental Section
### Wavelet Support Vector Machine
#### 2.2.2 Construction of Wavelet Kernel Function  
The kernel function \(k(x,x^{\prime})\) to SVM is the inner product of the image of two input space data points in the spatial characteristic. It has two important features: first, the symmetric function to inner product kernel variables is \(k(x,x^{\prime})=k(x^{\prime},x)\); second, the sum of the kernel function on the same plane is a constant. In general, only if the kernel function satisfies the following two theorems, can it become the kernel of support vector machine [19]. Mercer Lemma  
\(k(x,x^{\prime})\) represents a continuous symmetric kernel, which can be expanded into a series as:  
\[k(x,x^{\prime})=\sum\limits_{i=1}^{\infty}{\lambda_{i}g_{i}(x)g_{i}(x^{\prime})} \tag{35}\]  
where \(\lambda_{i}\) is positive, in order to ensure the above expansion is absolutely uniform convergence, the sufficient and necessary condition is:  
\[\iint{k(x,x^{\prime})g(x)g(x)dxdx^{\prime}}\geq 0,x,x^{\prime}\in R^{n} \tag{36}\]  
For all \(g()\) needs to satisfy the condition: \(\left\{ \begin{array}{c}g()\
eq 0\\ \int_{R^{n}}{g^{2}(\xi)d\xi}<\infty\end{array} \right.,g(x_{i})\) stands for the expansion of the characteristic function, \(\lambda_{i}\) stands for eigenvalue and all are positive, thus the \(k(x,x^{\prime})\) is positive definite. Smola and Scholkopf Lemma  
If the support vector machine's kernel function has meet the Mercer Lemma, then it only needs to prove \(k(x,x^{\prime})\) to satisfy the follow formula:  
\[F(x)(\omega)=(2\pi)^{-n/2}{\int_{R^{n}}{\exp(-J(\omega\cdot x))k(x)dx}}\geq 0,x \in R^{n} \tag{37}\]  
Construction of Wavelet Kernel  
If the wavelet kernel satisfies the condition: \(\psi(x)\in L^{2}(R)\cap L^{\prime}(R)\) and \(\hat{\psi}(x)=\ 0,\hat{\psi}(x)\) is the Fourier transform of \(\psi(x)\), the \(\psi(x)\) can be defined as [20]:  
\[\psi_{\sigma,m}(x)=(\sigma)^{1/2}\psi(\frac{x-m}{\sigma}),x\in R \tag{38}\]  
where \(\sigma\) means contraction-expansion factor, and \(m\) is horizontal floating coefficient, \(\sigma>0,m\in R\). When the function \(f(x),f(x)\in L^{2}(R)\), the wavelet transform to \(f(x)\) can be defined as:  
\[W(\sigma,m)=\sigma^{1/2}{\int_{-\infty}^{+\infty}{f(x)\psi^{*}(\frac{x-m}{ \sigma})dx}} \tag{39}\]where \(\psi^{*}(x)\) is the complex conjugation of \(\psi(x)\). The wavelet transform \(W(\sigma,m)\) is reversible and also can be used to reconstruct the original signal, so:  
\[f(x)=C_{\varphi}^{-1}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}W(\sigma,m )\psi_{\sigma,m}(x)\frac{d\sigma}{\sigma^{2}}dm \tag{40}\]  
Among them \(C_{=}^{\varphi}\int_{-\infty}^{+\infty}\frac{\left|\hat{\psi}(w)\right|^{2}} {\left|w\right|}dw<\infty\), \(\hat{\psi}(w)=\int\psi(x)\exp(-Jwx)dx\). For the above equation, is a constant. The theory of wavelet decomposition is to approximate the function group by the linear combination of the wavelet function. Assuming \(\psi(x)\) is a one-dimensional function, based on tensor theory, the multi-wavelet function can be defined as:  
\[\psi_{l}(x)=\mathop{\Pi}\limits_{i=1}^{l}\psi(x_{i}),x\in R^{lxd},x_{i}\in R^ {d} \tag{41}\]  
The horizontal floating kernel function can be built as:  
\[k(x,x^{\prime})=\mathop{\Pi}\limits_{i=1}^{l}\psi(\frac{x_{i}-x_{i}^{\prime} }{\sigma_{i}}),\sigma_{i}>0 \tag{42}\]  
In support vector machines, the kernel function should satisfy the Fourier transform; therefore, only when the wavelet kernel function satisfies the Fourier transform can it be used for support vector machines. Thus, the following formula needs to be proved. \[F[k](w)=(2\pi)^{-l/2}\int_{R^{l}}\exp(-J(wx))k(x)dx\geq 0 \tag{43}\]  
In order to keep the generality of wavelet kernel function, choose the Morlet mother wavelet as follows:  
\[\psi(x)=\cos(1.75x)\exp(-x^{2}/2) \tag{44}\]  
\(\sigma\) can be figured out. Where \(x\in R^{N}\), \(\sigma,x_{i}\in R^{N}\), the multi-dimensional wavelet function is an allowable multidimensional support vector machine functions. \[