## 4 Methods
### Least-Squares Support Vector Machine  
SVM is an important branch of machine learning algorithms, minimizing structural risk as a fundamental principle, i.e., achieving a balance between empirical error and confidence intervals to improve the generalization ability of the algorithm. LSSVM was improved by SVM [41], which converts the quadratic inequality constraint of the optimization function into the equation constraints of the linear equation system, and the training time of the algorithm is shortened. At the same time, LSSVM inherits the advantages of SVM in dealing with problems such as small samples, nonlinearity, and local minima. The main principles of LSSVM are as follows:  
Given a training sample dataset \(D=\{(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{N},y_{N})\}\), \(i=1\) to \(N\), where \(x_{i}\in R^{d}\), i.e., a total of N d dimensions for input vectors \(x_{i}\), \(y_{i}\), is the output indicator. A nonlinear function maps the low-dimensional samples to the high-dimensional feature space. The nonlinear function estimation modeling is given as  
\[y_{i}=\mathbf{\omega}\varphi(x_{i})+b \tag{3}\]  
where \(\mathbf{\omega}\) is the weight vector and \(b\) is the bias term.  
In order to evaluate and classify the sample set, the following equality constraint was set as an optimization function in LSSVM.  
\[\begin{split}\min J(\mathbf{\omega},e)=\min(\frac{1}{2}\|\mathbf{\omega }\|^{2}+\frac{1}{2}\gamma\sum\limits_{i=1}^{N}e_{i}^{2})\\ \text{s.t.}& y_{i}=\mathbf{\omega}\varphi(x_{i})+b+e_{ i}\end{split} \tag{4}\]  
Among them, \(\gamma\) is the regularization parameter, which is used to determine the trade-off between model complexity and accuracy; \(e_{i}\) is the error vector, which represents the difference between the actual value of the output indicator and the predicted value.  
Solve the above optimization function by constructing the Lagrangian function and finding partial derivatives:\[L(\mathbf{\omega},b,e,\xi)=\frac{1}{2}{\left\|\mathbf{\omega}\right\|}^{2}+\frac{1}{2} \gamma{\sum\limits_{i=1}^{N}{e_{i}^{2}}}-\sum\limits_{i=1}^{N}{\xi_{i}[\mathbf{ \omega}\varphi(x_{i})+b+e_{i}-y_{i}]} \tag{5}\]  
where \(\xi_{i}\) is the Lagrange multiplier.  
Take the partial derivative for the parameters \(\omega\), \(b\), \(e_{i}\), and \(\xi_{i}\), and set the partial derivative to zero [42]; we have:  
\[\left\{\begin{array}{c}\frac{\partial L}{\partial w}=0\rightarrow\mathbf{\omega }=\sum\limits_{i=1}^{N}{\xi_{i}\mathbf{\omega}\varphi(x_{i})}\\ \frac{\partial L}{\partial b}=0\rightarrow\sum\limits_{i=1}^{N}{\xi_{i}}=0 \\ \frac{\partial L}{\partial e}=0\rightarrow\xi_{i}=\gamma e_{i}\\ \frac{\partial L}{\partial\xi_{i}^{2}}=0\rightarrow\mathbf{\omega}\varphi(x_{i}) +b+e_{i}-y_{i}=0\end{array}\right. \tag{6}\]  
By eliminating \(\omega\) and \(e_{i}\), the four linear equations of Equation (6) are simplified as:  
\[\begin{bmatrix}0&\mathbf{E}^{T}\\ \mathbf{E}&\mathbf{\Omega}+\frac{1}{\gamma}\mathbf{E}\end{bmatrix}\begin{bmatrix}b \\ \xi\end{bmatrix}=\begin{bmatrix}0\\ \mathbf{y}\end{bmatrix} \tag{7}\]  
Among them, \(\mathbf{y}=\left[y_{i},\ldots,y_{N}\right]^{T}\), \(\mathbf{\xi}=\left[\xi_{i},\ldots,\xi_{N}\right]^{T}\), \(\mathbf{E}=\left[1,\ldots,1\right]^{T}\), and \(\mathbf{\Omega}\) is an \(\mathbf{N}\times\mathbf{N}\) symmetry matrix of kernel function, as follows:  
\[\mathbf{\Omega}=K(x_{i},x_{j})=\varphi(x_{i})\varphi(x_{j}) \tag{8}\]  
Equations (7) and (8) lead to the decision function of the LSSVM model:  
\[y(x)=\sum\limits_{i=1}^{N}{\xi_{i}K(x,x_{i})}+b \tag{9}\]  
where the radial basis function (RBF) is widely used as the kernel function. Furthermore, the RBF kernel has also been widely used and shown to achieve the best prediction performance in LSSVM prediction model studies of open-pit blasting vibration [43; 44].  
\[K(x_{i},x_{j})=\exp(-\frac{\left\|x-x_{i}\right\|^{2}}{2\sigma^{2}}) \tag{10}\]  
where \(\sigma\) is the key parameter of the kernel function.