## 2 Methods
### LSSVM Model Parameter Solution Based on Homotopy Continuation Method  
The HC-LSSVM model of soft soil settlement based on homotopy continuation method is mainly expressed as follows: based on the observation data of soft soil settlement, the training sample set and test sample set are established, and on the basis of leave-one-out cross-validation method, the optimization problem of LSSVM model parameters is transformed into solving nonlinear equations problem with the goal of minimizing the sum of squares of prediction error for the whole sample. The homotopy continuation method is used to solve nonlinear equations. The solution results were taken as the optimal parameters of LSSVM model, and the optimal parameters were used to predict the settlement of the test sample set. The construction method of HC-LSSVM model is described as follows. 1. Establish training sample set and test sample set based on soft soil settlement observation data. 2. Construct the homotopy equation: \[ \begin{split}& H((\begin{array}{c}C\\ sig2\end{array}),t)=t(\begin{array}{c}f1\\ f2\end{array})+(1-t)((\begin{array}{c}C\\ sig2\end{array})-(\begin{array}{c}C_{0}\\ sig2_{0}\end{array}))=0\\ & H((\begin{array}{c}C\\ sig2\end{array}),0)=(\begin{array}{c}C\\ sig2\end{array})-(\begin{array}{c}C_{0}\\ sig2_{0}\end{array})=0\\ & H((\begin{array}{c}C\\ sig2\end{array}),1)=(\begin{array}{c}f1\\ f2\end{array})=0\\ & f1=\frac{\partial seg}{\partial C}(C,sig2)\\ & f2=\frac{\partial seg}{\partial sig2}(C,sig2)\end{split} \] (14) where \(C_{0}\) and \(sig2_{0}\) are the initial LSSVM model parameters. When \(t=1\), the obtained solutions of \(H(( \begin{array}{c}C\\ sig2\end{array} ),1)\) are the optimal LSSVM model parameters. Suppose the known solution of \(H(( \begin{array}{c}C\\ sig2\end{array} ),0)=0\) is \(( \begin{array}{c}C_{0}\\ sig2_{0}\end{array} )\). In order to find the solution curve \(( \begin{array}{c}C\\ sig2\end{array} )=( \begin{array}{c}C(t)\\ sig2(t)\end{array} )\) of the homotopy equation, take the derivative of parameter \(t\), then the solution problem of nonlinear equations is transformed into the initial value problem of differential equations, \[ \begin{split}&\frac{\partial t}{\partial C}\frac{dC}{dt}+\frac{ \partial H}{\partial sig2}\frac{dsig2}{dt}+\frac{\partial H}{\partial t}=0\\ &(\begin{array}{c}C\\ sig2\end{array})(0)=(\begin{array}{c}C_{0}\\ sig2_{0}\end{array})\end{split} \] (15)
3. The grid search method was used to select the LSSVM model parameters: the initial regularization parameter \(C_{0}\) and the kernel parameter \(sig2_{0}\). The grid search range of LSSVM parameters was given and the corresponding grid search step was set. Set the grid search stop conditions, including stop the search when the maximum search times are met, stop the search when \(t=1\) is met when the first homotopy path is found, and stop the search when the LSSVM model parameters exceed the grid search range. 4. Set the initial value \(t=0\) of the homotopy parameter \(t\), and set the homotopy parameter step \(t\_step\). Set the iteration stop condition of the [PERSON] forecaster-Newton correction method [23; 24], and repeatedly use [PERSON] forecaster-Newton correction method to solve Equation (15) until the iteration stop condition is satisfied. First, the next approximate point of the homotopy path is obtained by euler estimation method,  
\[( \begin{array}{c}C\\ sig2\end{array} )^{\prime}_{k}=( \begin{array}{c}C\\ sig2\end{array} )_{k-1}+( \begin{array}{c}\frac{dC}{dt}\\ \frac{dsig2}{dt}\end{array} )\Delta t=( \begin{array}{c}C\\ sig2\end{array} )_{k-1}-( \begin{array}{c}\frac{\partial H}{dC}\end{array} \begin{array}{c}\frac{\partial H}{\partial sq2}\end{array} )^{-1}\frac{ \partial H}{\partial t}\Delta t. \tag{16}\]  
Then, [PERSON]'s correction method is used to correct the approximate points,  
\[( \begin{array}{c}C\\ sig2\end{array} )_{k}=( \begin{array}{c}C\\ sig2\end{array} )^{\prime}_{k}-( \begin{array}{c}\frac{\partial H}{\partial C} \end{array}\begin{array}{c}\frac{\partial H}{\partial sq2}\end{array} )^{-1}H (t_{k},(