## 2 Methodology
### Feature Selection by the Relieff Method  
The Relieff algorithm [45; 46] is a feature weighting algorithm that assigns different weights to features based on the relevance of the features and categories and removes features with weights that are less than a certain threshold. This method can be used to deal with multi-category problems and regression problems where the target attribute is a continuous value. The Relieff algorithm performs the feature selection work by randomly selecting a sample point \(R\) from the training set, and then identifies \(k\) nearest neighbor samples(\(H_{j}\)) in the data of the same class as \(R\). The algorithm also identifies \(k\) nearest neighbor samples \(M_{j}(C)\) in the data of the different class as \(R\). These processes are repeated to update the feature weights and ultimately to obtain feature weight scores. The equation is shown as follows. \[W(A)=W(A)-\sum_{j=  
an optimal decision hyperplane that maximizes the distance between the two different classes of samples that are closest to the hyperplane. In contrast to SVM, LSSVM [22; 23] applies a linear least squares criterion to the loss function instead of equational constraints. The optimization problem is formulated as follows.  
\[\min_{\omega,\varepsilon}\left[\frac{1}{2}\|\omega\|^{2}+\frac{1}{2}C\sum_{i=1 }^{N}\varepsilon_{i}^{2}\right] \tag{9}\]  
\[\text{s.t.}\ y_{i}=\omega^{T}\phi(x_{i})+b+e_{i},i=1,\ldots,N \tag{10}\]  
where \(e_{i}\) is the error between the actual output of the \(i\)th sample and the predicted output of the \(i\)th sample. To solve the above optimization problem, construct the Lagrangian function as follows. \[L(\omega,b,e,\alpha)=R(\omega,e)-\sum_{i=1}^{N}\alpha_{i}\left[\omega^{T}\phi (x_{i})+b+e_{i}-y_{i}\right],i=1,\ldots,N \tag{11}\]  
where \(a_{i}\) is the \(i\)th Lagrange multiplier. The regression equation for the LSSVM can be obtained by taking partial derivatives of each parameter in the respective equation and making them equal to 0. The equation is shown as follows. \[y(x)=\omega^{T}\phi(x)+b=\sum_{i=1}^{N}\alpha_{i}k(x,x_{i})+b \tag{12}\]  
Two significant parameters that determine the performance of LSSVM are \(C\) and \(\gamma\). In order to ascertain the most appropriate parameters and prevent overfitting of the model, this study uses the PSO to optimize the kernel function parameter in the LSSVM to improve the model prediction accuracy. The PSO [31; 32] relies on the comparison of the extreme values of the particles with the overall optimal solution of the particle swarm. This process is repeated, with the speed and position being constantly adjusted until the single optimal extreme value is obtained as the current global optimal solution of the entire particle swarm. The formula employed to calculate the updated speed and position is as follows. \[\left\{ \begin{array}{l}V_{i}^{(t+1)}=\omega V_{i}^{(t)}+c_{1}rand_{1}\left(P_ {i}^{(t)}-X_{i}^{(t)}\right)+c_{2}rand_{2}\left(P_{g}^{(t)}-X_{i}^{(t)}\right)\\ X_{i}^{(t+1)}=X_{i}^{(t)}+V_{i}^{(t+1)}\end{array} \right. \tag{13}\]  
where \(V_{i}\) is the velocity corresponding to the \(i\)th particle, \(P_{i}\) is the individual extremum of the population, \(P_{g}\) is the global extremum of the population, \(\omega\) is the inertia weight, \(X_{i}\) is the position corresponding to the \(i\)th particle, \(c_{1}\) and \(c_{2}\) are non-negative learning factors, \(rand_{1}\) and \(rand_{2}\) are random numbers between \([0,1]\), and \(t\) is the number of current iterations. Compared with the standard PSO, traditional PSO typically employs a fixed velocity upper limit. In this paper, the velocity boundaries are automatically calculated based on the search range, thereby avoiding the issue of unreasonable velocity settings in problems of different scales and achieving a more balanced search. For the \(i-th\) particle, the velocity constraint is as follows:  
\[ \begin{array}{l}V_{i,\max}=k\cdot\left(X_{\max}-X_{\min}\right)\\ V_{i}\in[-V_{i,\max},V_{i,\max}]\end{array}