## 2 Basic Method Principles
### Least-Square Support Vector Machine  
LSSVM is a variant of support vector machine that has unique advantages in handling small samples and nonlinear problems [28; 29]. SVM solves a quadratic programming problem with inequality constraints. LSSVM simplifies the problem by replacing the inequality constraints with equality constraints and changing the quadratic programming task into a linear one to reduce the complexity and significantly accelerate the solving process.  
Suppose that the training set consists of \((x_{i},y_{i})\), where \(i=1,2\ldots,n\). Here, \(x_{i}\in R^{m}\) is the m-dimensional input vector of the system, and \(y_{i}\in R^{n}\) is the corresponding output value. If the problem is nonlinear, the input vectors can be mapped to a high-dimensional space using a nonlinear mapping, which transforms the problem into a linear-like problem.  
The minimization function for the optimization problem of the LSSVM is:  
\[\underset{w,b,\xi}{min}\frac{1}{2}\parallel w\parallel^{2}+\frac{C}{2}\ \sum_{i=1}^{k}\xi_{i}^{2} \tag{1}\]  
where \(w\) is the weight vector; \(C\) is the penalty factor, also known as the regularization constant. The constraints are as follows:  
\[y_{i}-w\varphi(|x_{i})|=b+\xi_{i}| \tag{2}\]  
For the problem of LSSVM, according to Equations (1) and (2), the Lagrange solving equation is defined as follows:  
\[L(|w,b,\xi,a)|=\frac{1}{2}\parallel w\parallel^{2}+C\sum_{i=1}^{k}\xi^{2}|- \sum_{i=1}^{k}a_{i}|[w\varphi(|x_{i})|+b|+\xi_{i}-y_{i}|] \tag{3}\]  
where \(a_{i}\) is the Lagrange multiplier.  
The optimal values of parameters \(a\) and \(b\) can be obtained through Karush-Kuhn-Tucker (KKT) conditions.  
\[\frac{\partial L}{\partial b}=0|,\frac{\partial L}{\partial\xi}=0|_{i},\frac {\partial L}{\partial w}=0|_{i},\frac{\partial L}{\partial a}=0 \tag{4}\]From Equation (4), we obtain:  
\[\sum_{i=1}^{k}a_{i}=0;a_{i}=C\xi_{i};w=\sum_{i=1}^{k}a_{i}\varphi(x_{i})| \tag{5}\]  
\[w\varphi(x_{i})|+b|+\xi_{i}|-y_{i}|=0| \tag{6}\]  
By eliminating \(w\) and \(\xi\) from Equation (6), we transform the optimization problem into solving the following equation:  
\[\begin{bmatrix}0&\Theta^{T}\\ \Theta&\Omega+C^{-1}I\end{bmatrix}\begin{bmatrix}\mathbf{b}\\ \mathbf{a}\end{bmatrix}=\begin{bmatrix}0\\ y\end{bmatrix} \tag{7}\]  
In the equation, \(y=\big{[}|y_{1}|y_{2}|,\cdots|y_{k}\big{]}^{T};|a=\)\([|a_{1}|\mu_{2}|,\cdots|\mu_{k}|]^{T};|\Theta=[1,\cdots,1]^{T}\).  
Here, \(\Omega\) is a matrix, and \(\Omega_{ij}=\varphi(x_{i})^{T}(x_{j})\). Its inner product can be equivalently replaced by a kernel function in the original input space. Therefore, transform is not required to handle nonlinear problems. The inner product calculation directly adopts the kernel function, which is denoted as \(K(x_{i},x_{j})=\varphi(x_{i})^{T}(x_{j})\). After obtaining the values of parameters \(a\) and \(b\), we can derive the regression model of the LSSVM:  
\[f(|x)|=\sum_{i=1}^{k}a_{i}K(|x|,x_{i})|+b \tag{8}\]  
Kernel function \(K(x,x_{i})\) is any symmetric function that must satisfy Mercer condition. Commonly used kernel functions include linear functions, radial basis functions (RBF), and polynomial functions. The LSSVM introduced in this paper uses the RBF kernel [30]. Parameter \(\sigma^{2}\) of the RBF kernel and penalty factor \(C\) in the expression of the LSSVM significantly affect the operation results of the LSSVM. Therefore, the selection of these two parameters determines the classification accuracy and fitting quality. Kernel parameter \(\sigma^{2}\) reflects the distribution characteristics of the training sample dataset, whereas penalty factor \(C\) adjusts the balance between the confidence range of LSSVM and the empirical risk. Therefore, finding the optimal parameters will improve the performance of LSSVM.