# Research on Hard Rock Pillar Stability Prediction Based on SABO-LSSVM Model
## 2 Materials and Methods
### Introduction of the LSSVM Model  
The least-squares support vector machine (LSSVM) is a variant of the support vector machine (SVM) that aims to simplify the optimization process of the standard SVM [32]. This model reduces computational complexity by transforming the original quadratic programming problem into a system of linear equations. Unlike traditional SVM, LSSVM utilizes a least-squares linear system to solve the optimization objective function, which includes a squared error term. This approach performs well in handling small-scale datasets and offers a higher computational efficiency. In the principle of the LSSVM model, we first need to provide a training set \(\{(x_{i},y_{i})\), I = 1, 2,..., n\}\), where \(x_{i}\) is a d-dimensional vector with dimensions consistent with the input features of the dataset used, such as in the M-1 model mentioned in this study, where d = 7. \(y_{i}\) represents the predicted value of the model, which can be a class label or a regression prediction. The mapping relationship between \(x_{i}\) and \(y_{i}\) is expressed by Equation (4):  
\[y_{i}=\mathbf{\omega}\varphi(x_{i})+b \tag{4}\]  
where \(\mathbf{\omega}\) is the weight vector, b is the bias value, and \(\varphi(x_{i})\) is the mapping function. The LSSVM model transforms an original quadratic programming problem into a linear equation system. It selects the squared error term to represent empirical risk and formulates the evaluation problem as an optimization problem based on the principle of structural risk minimization:  
\[\left\{ \begin{array}{l}min_{\mathbf{\omega},b,\mathbf{c}}J(\mathbf{\omega},\mathbf{e})=\frac {1}{2}\|\mathbf{\omega}\|^{2}+c\frac{1}{2}\sum\limits_{i=1}^{n}e_{i}^{2}\\ \omega\varphi(x_{i})+b+\mathbf{e}_{i}=y\end{array} \right. \tag{5}\]  
where c is the penalty parameter (c > 0), e is the error vector, and \(J\) is the loss function. To solve the aforementioned optimization problem, it is necessary to construct the corresponding Lagrangian function and determine the minimum value of the optimization function, as shown below:  
\[L(\mathbf{\omega},b,\mathbf{e},\mathbf{\xi})=J(\mathbf{\omega},\mathbf{e})-\sum\limits_{i=1}^{n} \mathbf{\xi}_{i}[\mathbf{\omega}\varphi(x_{i})+b+\mathbf{e}_{i}-y_{i}] \tag{6}\]  
where \(\mathbf{\xi}\) represents the Lagrange multiplier, \(\mathbf{\xi}\) = [\(\mathbf{\xi}_{1}\), \(\mathbf{\xi}_{2}\),..., \(\mathbf{\xi}_{n}\)]T.  
By taking the partial derivatives of L(\(\mathbf{\omega}\), \(b\), \(\mathbf{e}\), \(\mathbf{\xi}\)) with respect to \(\mathbf{\omega}\), \(b\), \(\mathbf{e}\), and \(\mathbf{\xi}\), and setting the derivatives of \(L\) to zero, we can obtain the conditions for the optimal solution of the problem:  
\[\left\{ \begin{array}{l}\omega=\sum\limits_{i=1}^{n}\mathbf{\xi}_{i}\varphi(x_{i} )\\ \sum\limits_{i=1}^{n}\mathbf{\xi}_{i}=0\\ \mathbf{\xi}_{i}=c\mathbf{e}_{i}\\ \omega\varphi(x_{i})+b+\mathbf{e}_{i}-y_{i}=0\end{array} \right. \tag{7}\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **W** & **H** & **K** & **C\({}_{\text{pav}}\)** & **UCS** & \(\mathbf{\sigma}\) & \(\mathbf{\sigma_{\text{p}}}\) \\ \hline M-1 & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\ M-2 & \(\surd\) & \(\surd\) & \(\surd\) & & \(\surd\) & \(\surd\) & \(\surd\) \\ M-3 & \(\surd\) & \(\surd\) & \(\surd\By eliminating \(\mathbf{\omega}\) and \(\mathbf{e}\) from the equation, we can obtain:  
\[ \begin{bmatrix}0\\ \mathbf{Y}\end{bmatrix} = \begin{bmatrix}0&\mathbf{Z}^{T}\\ \mathbf{Z}&\mathbf{K}+c^{-1}\mathbf{E}\end{bmatrix}\begin{bmatrix}b\\ \mathbf{\xi}\end{bmatrix}  \tag{8}\]  
where \(\mathbf{Z}=[1,1,\ldots,1]^{\mathrm{T}}\), \(\mathbf{Y}=(y_{1},y_{2},\ldots,y_{\mathrm{n}})^{\mathrm{T}}\), \(\mathrm{E}\) is an n-dimensional identity matrix, and \(\mathbf{K}\) is the kernel function matrix that satisfies [PERSON]'s condition, \(\mathbf{K}=\mathbf{K}(x_{1},x_{2})=\varphi(x_{1})\varphi(x_{2})\).