## 2 Methods
### Least Square Support Vector Machine  
After decomposing by TVF-EMD, the LSSVM model is built for each subseries. LSSVM has great improvement over the SVM model. The inequality constraints in the standard SVM algorithm are replaced by the equality constraints. On the conditions, the quadratic programming problem is transformed into the problem of solving linear equations [29].  
Considering a set of data \(D\ =\ (x_{i},y_{i}),i\ =\ 1,\cdots,k\), where \(x_{i}\in R\acute{s}\) is input and \(g\) is the dimension of \(x_{i}\) which can be determined by minimizing the root mean square error of the values output by the training part [20]; \(y_{i}\in R\) is  
where \(q_{i}\) denotes error variable; \(\mu\) and \(\varsigma\) denote variable parameters; \(E_{W}=\frac{1}{2}\omega^{T}\omega\); \(E_{D}=\frac{1}{2}\sum\limits_{i=1}^{k-g}q_{i}^{2}=\frac{1}{2}\sum\limits_{i=1} ^{k-g}\left[y_{i}-\omega^{T}\psi(x)+d\right]^{2}\). To solve the above optimization problems, the Lagrange function is constructed as shown in Equation (9).  
\[L(\omega,d,q,\alpha)\ =\ J(\omega,q)-\sum\limits_{i\ =\ 1}^{k-g}\alpha_{i}\left\{ \omega^{T}\psi(x)+d+q-y_{i}\right\} \tag{9}\]  
where \(\alpha_{i}\) is the Lagrange multiplier. According to the Karush-Kuhn-Tucker (KKT) conditions, the optimal solution can be calculated by:  
\[\left\{\begin{array}{rcl}\frac{\partial L}{\partial\omega}&=&0\to\omega\ =\ \sum\limits_{i\ =\ 1}^{k-g}\alpha_{i}\psi(x_{i})\\ \frac{\partial L}{\partial d}&=&0\to\sum\limits_{i\ =\ 1}^{k-g}\alpha_{i}\ =\ 0\\ \frac{\partial L}{\partial q_{i}}&=&0\to\alpha_{i}\ =\ \gamma q_{i}\\ \frac{\partial L}{\partial\alpha_{i}}&=&0\to\omega^{T}\psi(x_{i})+d+q_{i}-y_{ i}\ =\ 0\end{array}\right. \tag{10}\]  
where \(\gamma\ =\ \zeta/\mu\) denotes the penalty coefficient. After eliminating \(q_{i}\) and \(\omega\), the original optimization problem becomes  
\[\left[\begin{array}{cc}0&L^{T}\\ L&\Omega_{ij}+\frac{1}{\gamma^{\prime}}I\end{array}\right]\left[\begin{array}[] {c}d\\ \alpha\end{array}\right]=\ \left[\begin{array}{c}0\\ Y\end{array}\right] \tag{11}\]  
where \(\Omega_{ij}=\psi(x_{i})^{T}\psi\left(x_{j}\right)=K\left(x_{i}x_{j}\right)\); \(L=[1,\cdots,1]^{T}\); \(Y=[y_{1},\cdots y_{N}]^{T}\). Finding out \(\alpha\) and \(d\) through Equation (11), the LSSVM regression model becomes:  
\[f(x)\ =\ \sum\limits_{i\ =\ 1}^{k-g}\alpha_{i}K(x,x_{i})+d \tag{12}\]  
where \(K(x,x_{i})\) is the kernel function which needs to meet Mercer's conditions. Generally, the kernel functions include RBF kernel function, sigmoid kernel function and polynomial kernel function, etc. The RBF kernel function is also called the Gaussian kernel function. It has strong nonlinear learning ability with fewer parameters, which is the most effective kernel function. Therefore, the RBF kernel function is selected in this paper. It can be expressed as  
\[K(x,x_{i})\ =\ \exp\left[-\|x-x_{i}\|^{2}/\left(2\sigma^{2}\right)\right], \sigma>0 \tag{13}\]  
where \(\sigma\) denotes the kernel function parameter. When applying the LSSVM model with RBF kernel function, the selections of the parameter \(\sigma\) and the penalty coefficients \(\gamma\) determine the model's learning and generalization capabilities. Thus, it's vital to search for the most suitable parameters.