## 2 Methods
### LSSVM Model for Soft Ground Settlement Prediction
According to Mercer's condition [14], there is a kernel function \(K(x_{k},x_{l})\):  
\[K(x_{k},x_{l})=\Phi^{T}(x_{k})\Phi(x_{l}). \tag{6}\]  
Solve the optimization condition (Equation (5)), eliminate \(w\), and \(e_{i}\), and replace \(\Phi^{T}(x_{k})\Phi(x_{l})\) with \(K(x_{k},x_{l})\). Finally, the parameter optimization problem is transformed into the problem of solving a system of linear equations. \[\left[ \begin{array}{cccc}K(x_{1},x_{1})+\frac{1}{C}&\cdots&K(x_{1},x_{n})&1 \\ \vdots&\ddots&\vdots&\vdots\\ K(x_{n},x_{1})&\cdots&K(x_{n},x_{n})+\frac{1}{C}&1\\ 1&\cdots&1&0\end{array} \right]\left[ \begin{array}{c}\alpha_{1}\\ \vdots\\ \alpha_{n}\\ b\end{array} \right]=\left[ \begin{array}{c}y_{1}\\ \vdots\\ y_{n}\\ 0\end{array} \right]. \tag{7}\]  
The LSSVM algorithm uses the least square method to solve the linear equations represented by Equation (7), avoiding the convex quadratic programming problem in the standard support vector machine algorithm, and has better prediction performance than the traditional support vector machine method. LSSVM has the same principle as SVM. In order to conduct sample training and estimation, some model parameters must be defined first, such as those in Equation (4). Meanwhile, the prediction accuracy of LSSVM is greatly affected by the selection value of model parameters. Therefore, this study adopts the fast retention method proposed by [PERSON] et al. [22] to optimize LSSVM parameters. Suppose  
\[A=\left[ \begin{array}{cccc}K(x_{1},x_{1})+\frac{1}{C}&\cdots&K(x_{1},x_{n}) &1\\ \vdots&\ddots&\vdots&\vdots\\ K(x_{n},x_{1})&\cdots&K(x_{n},x_{n})+\frac{1}{C}&1\\ 1&\cdots&1&0\end{array} \right],\]\[S=\left[ \begin{array}{c}\alpha_{1}\\ \cdots\\ \alpha_{n}\\ b\end{array} \right],\]  
\[Y=\left[ \begin{array}{c}y_{1}\\ \cdots\\ y_{n}\\ 0\end{array} \right].\]  
Then, Equation (7) can be expressed as  
\[A\bullet S=Y. \tag{8}\]  
Then, \(S=A^{-1}\bullet Y\) is the LSSVM coefficient solution of the whole sample, and the leave-one-out cross-validation method is adopted for the training sample set, then the LSSVM coefficient solution of the \(p\) th time can be expressed as [22]  
\[S_{p}=S(p^{-})-\frac{S(p)}{A^{-1}(p,p)}A^{-1}(p^{-},p), \tag{9}\]  
where \(S(p)\) is the \(p\) th element in \(S\), \(S(p^{-})\) is the column vector of \(S\) after removing the \(p\) th element, \(A^{-1}(p,p)\) is the element of the \(p\) th row and the \(p\) th column in \(A^{-1}\), and \(A^{-1}(p^{-},p)\) is the column vector of the \(p\) th column in \(A^{-1}\) after removing the \(p\) th element. It is assumed that the kernel function takes the radial basis function (RBF function) [12],  
\[K(x_{i},x_{j})=\exp(-\frac{\left\|x_{i}-x_{j}\right\|^{2}}{\sigma^{2}}), \tag{10}\]  
where \(\sigma^{2}\) is the kernel parameter, denoted as _sig2_. Note:  
\[K^{\prime}=\left[ \begin{array}{cccc}K(x_{1},x_{1})+\frac{1}{C}&\cdots&K(x_{ 1},x_{n})&1\\ \vdots&\ddots&\vdots&\vdots\\ \vdots&&\ddots&\vdots\\ K(x_{n},x_{1})&\cdots&K(x_{n},x_{n})+\frac{1}{C}&1\end{array} \right].\]  
Meanwhile, as the LSSVM model parameters are (\(C\),_sig2_), the error of the LSSVM model for the \(p\) th individual can be expressed as  
\[e_{p}=K^{\prime}(p,p^{-})\bullet S_{p}-y_{p}, \tag{11}\]  
where \(K^{\prime}(p,p^{-})\) is the row vector of the \(p\) th row in the matrix \(K^{\prime}\) after removing the \(p\) th element. Therefore, the leave-one-out error of the whole sample can be expressed as  
\[sse(C,sig2)=\frac{1}{2}\sum_{p=1}^{n}{e_{p}}^{2}. \tag{12}\]  
In order to obtain the optimal parameters of LSSVM model, the key is to minimize _sse_(\(C\),_sig2_) value represented by Equation (12). Therefore, the optimal parameters of LSSVM model should meet Equation (13),  
\[\left\{ \begin{array}{c}\frac{\partial sse}{\partial C}(C,sig2)=0\\ \frac{\partial sse}{\partial s\partial S2}(C,sig2)=0\end{array} \right.. \tag{13}\]In order to solve Equation (13), this study mainly constructs the homotopy equation of Equation (13) and uses the homotopy continuation method to solve Equation (13).