## 2 Methodology
### Implementation of a Data-Driven Prediction Approach Based on Machine Learning  
In general, after defining the input variables and the output variables in Equation (2), the relationship between inputs and outputs can be sought by using the SVM or the BP neural network. However, the machine-learning-based model constructed should have pertinence to the construction method in view of the difference of monitoring items for the cut-and-cover method, shallow-mining method, and shield-tunneling method, etc. The monitoring data need to be pre-processed before establishing the machine-learning model due to the difference in measurement units of the monitoring data. Thus, taking \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Expression** & **Parameters** \\ \hline Linear kernel & \(\text{K}\Big{(}x_{i},x_{j}\Big{)}=x_{i}^{T}x_{j}\) & \\ \hline Polynomial kernel & \(\text{K}\Big{(}x_{i},x_{j}\Big{)}=\Big{(}\gamma x_{i}^{T}x_{j}+b\Big{)}^{d}\) & \(\gamma>0\), \(b<0\), \(d\geq 1\) \\ \hline RBF kernel/Gaussian kernel & \(\text{K}\Big{(}x_{i},x_{j}\Big{)}=exp\Big{(}-\gamma\Big{\|}x_{i}-x_{j}\Big{\|}^ {2}\Big{)}\) & \(\gamma>0\) \\ \hline Laplacian kernel & \(\text{K}\Big{(}x_{i},x_{j}\Big{)}=exp\Big{(}-\frac{\|x_{i}-x_{j}\|}{\sigma} \Big{)}\) & \(\sigma>0\) \\ \hline Sigmoid kernel & \(\text{K}\Big{(}x_{i},x_{j}\Big{)}=\tanh\Big{(}\gamma x_{i}^{T}x_{j}+b\Big{)}\) &
\begin{tabular}{c} tanh is the hyperbolic \\ tangent function, \\ \(\gamma>0,b<0\) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Some common kernel functions. Figure 4: The workflow of a BP neural network algorithm. Figure 3: A structure diagram of a BP neural network. the construction of the machine-learning-based model for predicting the deformation of existing buildings induced by adjacent excavation of a foundation pit as an example, the framework of the data-driven prediction approach based on machine learning is expounded on in this section. Excavation is often accompanied with the uplift of the stratum at the bottom of the foundation pit, the deformation of the retaining structure, and the movement of the stratum around the foundation pit, which will cause the deformation of the surrounding existing buildings and affect their safety. The technical code for the monitoring of building foundation pit engineering (GB50497-2019) [46] recommends the monitoring items of foundation pit engineering according to the safety grade of excavation, as listed in Table 2. In general, some items, such as horizontal displacement at the top of the retaining structure/slope, vertical displacement at the top of the retaining structure/slope, subsurface horizontal displacement, the axial force of bracing/axial load of the anchor bolt, hydraulic head, and ground vertical displacement, must be monitored during excavation. Thus, the monitored items above can be selected as the input variables of a prediction model. Table 3 lists the variables of the prediction model in this article. As shown in Table 3, the measurement units of these variables are different. Meanwhile, each variable is a spatial and temporal variable with different values at a different location and different time. In order to facilitate the construction of the prediction model and reduce the amount of calculation, it is necessary to normalize the model variables and reduce the dimensions of the model variables.