## 2 Methodology
### IPSO-LSSVM Model  
#### 2.2.1 LSSVM Principle  
[PERSON] and [PERSON] initially proposed the LSSVM [27], which is an improvement of SVM. LSSVM uses fewer hyper-parameters than SVM. The quadratic programming issue is converted into a linear system of equations using LSSVM, which substitutes equality constraints for inequality constraints in optimization problems. Reduced computational complexity, accelerated convergence, and improved solution correctness are all benefits of LSSVM. The principle of LSSVM is as follows:  
Given a dataset \(\{(x_{i},y_{i}),i=1,\cdots,l\}\), \(\mathbf{x}_{i}\in R^{d}\) is a \(d\)-dimensional input vector and \(y_{i}\in R\) is a scalar output. A regression function can be used to represent the nonlinear relationship between the input and output as:  
\[f(\mathbf{x})=\mathbf{\omega}^{T}\varphi(\mathbf{x})+b \tag{5}\]  
where \(\mathbf{\omega}\) is the weight vector; \(\varphi(\mathbf{x})\) is a nonlinear mapping from input space to high-dimensional feature space; and \(b\) is the deviation. The purpose of quadratic optimization in LSSVM is based on the structural risk reduction concept and is as follows:  
\[\begin{split}\min\frac{1}{2}\|\mathbf{\omega}\|^{2}+\frac{1}{2} \gamma\sum\limits_{i=1}^{l}{e_{i}}^{2}\\ s.t.\ \ \mathbf{\omega}^{T}\varphi(\mathbf{x_{i}})+b+e_{i}=y_{i},i=1, \cdots,l\end{split} \tag{6}\]  
where \(e_{i}\) is the error variable and \(\gamma\) is the regularization parameter. The Lagrange multiplier \(\lambda\in R^{l\times 1}\) is used to convert constrained optimization into unconstrained optimization r to resolve the optimization problem.  
\[\begin{split}\min/=\frac{1}{2}\|\mathbf{\omega}\|^{2}+\frac{1}{2} \gamma\sum\limits_{i=1}^{l}{e_{i}}^{2}-\sum\limits_{i=1}^{l}{\lambda_{i}(\mathbf{ \omega}^{T}\varphi(\mathbf{x}_{i})+b+e_{i}-y_{i})}\end{split} \tag{7}\]  
The Karush-Kuhn-Tucker (KKT) conditions for optimality are given by:  
\[\left\{\begin{array}{l}\frac{\partial l}{\partial\mathbf{\omega}}=0\to\sum \limits_{i=1}^{l}{\lambda_{i}\varphi(\mathbf{x}_{i})}=\mathbf{\omega}\\ \frac{\partial l}{\partial b}=0\to\sum\limits_{i=1}^{l}{\lambda_{i}}=0\\ \frac{\partial l}{\partial c_{i}}=0\to\lambda_{i}=\gamma e_{i},i=1,2, \ldots,l\\ \frac{\partial l}{\partial\lambda_{i}}=0\to\mathbf{\omega}^{T}\varphi(\mathbf{x}_{i}) +b+e_{i}-y_{i}=0,i=1,2,\ldots,l\end{array}\right. \tag{8}\]  
After eliminating \(\mathbf{e}_{i}\) and \(\mathbf{\omega}\), the following linear equation set is obtained:  
\[\left[\begin{array}{cc}0&\mathbf{Q}_{l}{}^{\mathrm{T}}\\ \mathbf{Q}_{l}&K+\gamma^{-1}\mathbf{I}\end{array}\right]\left[\begin{array}{c}b\\ \mathbf{A}\end{array}\right]=\left[\begin{array}{c}0\\ \mathbf{Y}\end{array}\right] \tag{9}\]  
where \(\mathbf{Q}_{l}=[1,1,\cdots,1]^{\mathrm{T}}\) is an \(l\)-dimensional column vector; \(\mathbf{I}\) is the identity matrix; \(\mathbf{Y}=[y_{1},y_{2},\cdots,y_{m}]^{\mathrm{T}}\) is the output vector of the training set; \(\mathbf{A}=[\lambda_{1},\lambda_{2},\cdots,\lambda_{m}]^{\mathrm{T}}\); \(K\) is the kernel function, \(K(x_{i},x_{j})=\varphi(x_{i})^{T}\varphi(x_{j})\).  
The dot product operation in the high-dimensional feature space is replaced by using \(K\) to transfer the original input space to the high-dimensional space. The regression function can be expressed as:  
\[y=\sum\limits_{i=1}^{l}{\lambda_{i}K(x_{i},x_{j})}+b \tag{10}\]