## Combined Predictive Models
### Least-Squares Support Vector Machine  
The support vector machine (SVM) is a machine learning algorithm based on statistical principles and the principle of structural risk minimization proposed by [PERSON] et al. [18] in 1995. Its basic idea is to map the original input space data into a high-dimensional feature space via non-linear mapping so that a non-linear regression problem in the original space is transformed into a linear regression problem in the feature space. Support vector machines have a great advantage in solving small sample, non-linear, and high-dimensional problems and can avoid becoming trapped in local optima. [PERSON] et al. [19] proposed the least squares support vector machine (LSSVM) to improve the support vector machine algorithm, where the inequality constraints in the original SVM training data set are changed to equation constraints in the LSSVM. In the LSSVM model, let the training dataset be denoted by Equation (12).  
\[(x_{i},y_{i})\in R^{n}\times R(i=1,2,\ldots,l) \tag{12}\]  
where \(x_{i}\) denotes the input vector, \(y_{i}\) denotes the output vector, and \(l\) denotes the number of samples. In the feature space, the regression function \(f(x)\) can be expressed as:  
\[f(x)=\omega^{T}\varphi(x)+b \tag{13}\]  
where \(\omega\) denotes the weight vector; \(b\) denotes the amount of bias and \(\varphi(x)\) denotes the kernel function that projects the input vector to the higher dimensional features. Based on the principle of minimizing the risk of the structure of the algorithm, the function optimization problem can be expressed as:  
Figure 1: Diagram of irrational escape methods.  
\[\left\{\begin{array}{l}\min_{\omega,b,\xi}(\omega,\xi)=\frac{1}{2}\omega^{T} \omega+\gamma\frac{1}{2}\sum\limits_{i=1}^{l}\xi^{2}\\ y=\omega^{T}\varphi(x_{i})+b+\xi_{i},i=1,2,\ldots,l\end{array}\right. \tag{14}\]  
where \(J(\omega,\xi)\) is the construction function; \(\gamma\) denotes the penalty parameter and \(\xi_{i}\) denotes the \(i\)th error variable, constructing Lagrange-style solutions.  
\[R^{n}L(\omega,b,\zeta,a)=\frac{1}{2}\omega^{T}\omega+\gamma\frac{1}{2}\sum \limits_{i=1}^{l}\xi_{i}^{2}\sum\limits_{i=1}^{l}\alpha_{i}\Big{(}\omega^{T} \varphi(x_{i})+b+\xi_{i}y_{i}\Big{)} \tag{15}\]  
Figure 2: Irrational escape strategy optimization sparrow algorithm flow chart.  
where \(R^{n}\to R^{nth}\) denotes mapping to a vector space of a higher dimension; \(\alpha_{i}\) is a Lagrange multiplier and \(L(\omega,b,\xi,\alpha)\) is a Lagrange function. This is subject to the following requirements for a partial differential with \(\omega,b,\xi,\alpha\).  
\[\frac{\partial L}{\partial\omega}=0,\frac{\partial L}{\partial b}=0,\frac{ \partial L}{\partial\xi_{i}}=0,\frac{\partial L}{\partial\alpha_{i}}=0 \tag{16}\]  
It can be found that:  
\[\left\{\begin{array}{l}\omega=\sum\limits_{i=1}^{l}\alpha_{i}\varphi(x_{i}) \\ \sum\limits_{i=1}^{l}\alpha_{i}\\ \alpha_{i}=\gamma\xi_{i}\\ \omega^{T}\varphi(x_{i})+b+\xi_{i}\cdot y_{i}=0\end{array}\right. \tag{17}\]  
According to the Karush-Kuhn-Tucke optimality condition, a linear equation can be obtained as:  
\[\begin{bmatrix}0&1&\cdots&1\\ 1&K(x,x_{i})+1/\gamma&\cdots&K(x,x_{i})\\ \vdots&\vdots&\ddots&\vdots\\ 1&K(x,x_{i})&\cdots&K(x,x_{i})+1/\gamma\end{bmatrix}\begin{bmatrix}b\\ \alpha_{1}\\ \vdots\\ \alpha_{1}\end{bmatrix}=\begin{bmatrix}0\\ y_{1}\\ \vdots\\ y_{l}\end{bmatrix} \tag{18}\]  
where \(K(x,x_{i})=\varphi(x_{i})^{T}\varphi(x)\) is a symmetry function that satisfies the Mercer condition.