## 3 QPSO-RBF-SVM Model  
### RBF-SVM Modeling  
Support vector machine (SVM) is a statistical-based machine learning algorithm that effectively addresses the challenges of small sample sizes and nonlinearity [23]. SVM achieves linear separability in high-dimensional space by nonlinearly mapping the input samples from low-dimensional space to high-dimensional space and then constructs classification or regression functions.  
The set of nonlinear samples is assumed to be \(\{x_{i},y_{i}\}(i=1,2,\ldots,m)\), where \(x_{i}\) represents the input vector of the \(i\)-th sample and \(y_{i}\) is the output; a loss function is introduced to address the equation:  
\[f(x,\omega)=(w,\varphi(x))+b \tag{15}\]  
The weight vector is denoted by \(w\) in the formulations, while the bias is denoted by \(b\).  
To solve the optimization problem of finding the maximum hyperplane with respect to the loss function, the problem description formula is as follows:  
\[\min\left\{\frac{1}{2}||w||^{2}+c\sum_{i=1}^{m}\left(\xi_{i}+\xi_{i}^{\prime} \right)\right\},\]  
\[s.t.\left\{\begin{array}{l}y_{i}-wk\varphi(x)-b\leq\xi_{i}+\varepsilon\\ wk\varphi(x)+b-y_{i}=\xi_{i}^{\prime}+\varepsilon\\ \xi_{i}\geq 0,\xi_{i}^{\prime}\geq 0\end{array}\right., \tag{16}\]  
where \(c\) represents the penalty factor, \(\xi_{i},\xi_{i}^{\prime}\) is the relaxation factor, \(\varepsilon\) denotes the loss function, and \(k\) stands for the kernel function.  
By introducing the Lagrangian factor \(\alpha_{i}\), \(\beta_{i}\) and kernel function \(k\), we formulated the optimization problem as follows:  
\[\max W(\alpha_{i},\beta_{i})=\sum_{i=1}^{n}y_{i}(\alpha_{i}-\beta_{i})- \varepsilon\sum_{i=1}^{n}\left(\alpha_{i}+\alpha_{i}^{\prime}\right)-\frac{1 }{2}\sum_{i=1,j=1}^{n}\left(\alpha_{i}-\alpha_{i}^{\prime}\right)\left(\alpha _{j}-\alpha_{j}^{\prime}\right)k\left(x_{i}-x_{j}^{\prime}\right)\]  
\[s.t.\left\{\begin{array}{l}\sum\limits_{i=1}^{n}\alpha_{i}=\sum\limits_{i=1 }^{n}\alpha_{i}^{\prime}\\ \alpha_{i}\geq 0,\alpha_{i}^{\prime}\leq c\end{array}\right. \tag{17}\]  
In this equation, \(W\) represents the dual problem corresponding to the original optimization problem.  
Revised sentence: Support vector machine regression (SVR) is a crucial component of support vector machines, which utilizes kernel tricks to transform a low-dimensional linear regression problem into a high-dimensional nonlinear regression problem. It aims to minimize the loss function in order to fit the data while maximizing the margin between the predicted value and the actual value. This transformation results in a convex optimization problem that needs to be solved. The generalization ability of SVR heavily relies on the selected kernel function [24]. Common types of kernel functions include Linear Kernel Function, Polynomial Kernel Function, Multi-layer Perception Kernel Function, Radial Basis Function Kernel Function, and Convex Optimization Problem.  
The RBF kernel function was utilized in this paper due to its superior capability for local approximation and better handling of nonlinearities. Its expression is given by the following:  
\[K\big{(}x,x^{\prime}\big{)}=\exp\left(-\gamma\|x-x^{\prime}\|^{2}\right), \tag{18}\]  
where \(x\) and \(x^{\prime}\) are sample points, and \(\gamma\) represents the width parameter of the RBF kernel.