## 2 Methodology
### Discussions  
Figure 9a,b show that good prediction results can be obtained using the settlement prediction method proposed in this article, which means that the proposed approach is reasonable and feasible. However, due to the operation difference between SVM and BP neural networks, the results obtained by the two models are different. The maximum difference between the prediction results based on SVM model and the equivalent settlement of the building occurs in the fourth issue of the testing sets: the maximum difference value is 0.4817 mm. However, the maximum difference between the prediction results based on the BP model and the equivalent settlement of the building appears in the third issue of the testing sets: the maximum difference value is \(-0.4161\) mm. Therefore, in order to minimize the impact caused by the deficiency of the BP neural network and SVM in the prediction process, it is suggested to assign different weights to the BP neural network and SVM according to game theory [54] and combine the BP neural network and SVM for prediction, which is called the BP-SVM model. The construction of the BP-SVM model is shown in Equation (11). \[x=\omega_{1}x_{1}+\omega_{2}x_{2}+\cdots+\omega_{n}x_{n} \tag{11}\]  
Subject to the constraints: \(\omega_{1}+\omega_{2}+\cdots+\omega_{n}=1\), where, \(x\) is the combined predicted value, \(x_{i}\) is the predicted value obtained by the \(i\)th algorithm, \(\omega_{i}\) is the weight of the \(i\)th algorithm and \(\omega_{i}\geq 0,(i=1,2,\cdots,n)\), and \(n\) is the number of algorithms. Assuming that the prediction error of the \(i\)th algorithm is \(e_{it}\) (\(i\) represents the \(i\)th algorithm, \(t\) represents the issue number), the prediction error of the combined prediction method is \(e_{t}=\sum\limits_{i=1}^{n}\omega_{i}e_{it}\), \((t=1,2,\cdots,\text{m})\). The objective function \(\mathbf{M}\) is constructed as  
\[\mathbf{M}=\sum\limits_{t=1}^{m}{{{e_{t}}^{2}}}=\mathbf{e}^{\mathsf{T}} \mathbf{e}=\mathbf{W}^{\mathsf{T}}\mathbf{E}_{n}\mathbf{W} \tag{12}\]  
\[\text{where},\mathbf{e}= \begin{bmatrix}\mathbf{E}_{1}&\mathbf{E}_{2}&\cdots& \mathbf{E}_{n}\end{bmatrix} ,\mathbf{E}_{i}= \begin{bmatrix}e_{i1}&e_{i2}&\cdots& e_{im}\end{bmatrix} ^{\mathsf{T}},\mathbf{W}= \begin{bmatrix}\omega_{1}&\omega_{2}&\cdots& \omega_{n}\end{bmatrix} ^{\mathsf{T}},\text{and}\]  
\[\mathbf{E}_{n}= \begin{bmatrix}E_{11}&E_{12}&\cdots&E_{1m}\\ E_{21}&E_{22}&\cdots&E_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ E_{n1}&E_{n2}&\cdots&E_{nm}\end{bmatrix} \]  
Taking matrix R as \(n\times 1\), \(\mathbf{R}_{n}= \begin{bmatrix}1&1&\cdots&1\end{bmatrix} ^{\mathsf{T}}\), the  
\[{\mathbf{R}_{n}}^{\mathsf{T}}\mathbf{W}=1\]  
Then, the optimal weighting coefficient vector \(\mathbf{W}\) can be obtained from Equation (13). \[min\{\mathbf{M}\}=min\Big{\{}\mathbf{W}^{\mathsf{T}}\mathbf{E}_{n}\mathbf{W} \Big{\}} \tag{13}\]  
Based on the above predicted results from the SVM model and BP model, the corresponding prediction errors are obtained, and according to the optimal weight criterion, the optimal weights \(\omega_{\text{SVM}}\) and \(\omega_{\text{BP}}\) are gained, which are 0.4444 and 0.5556. Figure 9c shows the building settlement predicted by the BP-SVM model. Comparing Figure 9a-c, it is found that the results predicted by the BP-SVM model are closer to the real results. According to the MSPE and the \(R^{2}\), the BP-SVM model effectively reduces the difference fluctuation, improves the utilization of data, enhances the reliability of the prediction model, and improves the accuracy of the prediction results compared with the single algorithms. In addition to gradually increasing with the advance of excavation, the settlement of existing buildings is also related to the plane position and the excavation process. Generally, the settlement of buildings facing the middle of excavation is large. Thus, the proposed approach is also used to predict the settlement of another building at JGY-22.