## 3 Methodology
### Establishment of an Optimum BPNN  
Performance evaluation methods should be determined to compare the performance of different BPNNs, and root mean square error (\(RMSE\)) is a commonly used measurement for such comparison [13]. The definition of \(RMSE\) is given as follows:  
\[RMSE=\sqrt{\frac{\sum_{i=1}^{N}{({{o_{i}}-{{t_{i}}}})^{2}}}{N}} \tag{13}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Influencing Factors of Ground Settlement**} & \multicolumn{2}{c}{**Input Value**} \\ \hline Excavation time & & 1,2,3... \\ \hline \multirow{8}{*}{Supporting structure} & & 1: Gravity retaining wall; \\  & Types & 2: Soil mixing wall; \\  & Types & 3: Border pile; \\  & & 4: Drilling bitten pile; \\  & & 5. Slurry wall. \\  & Stiffness & Positive real number. \\  & Depth & Positive real number. \\ \hline \multirow{8}{*}{Construction stage} & Excavation length & Positive real number. \\  & Excavation width & Positive real number. \\  & Excavation depth & Positive real number. \\  & Installation of braces & Positive real number. \\ \cline{1-1}  & Removal of braces & 0: This construction stage is ongoing; \\ \cline{1-1}  & Casting the foundation bed & 1: This construction stage has been finished; \\ \cline{1-1}  & Casting sidewalls & 2: This construction stage is not performed at that time. \\ \cline{1-1}  & Casting the roof slab & 1: Slit; \\ \hline \multirow{8}{*}{Soil effect} & Soil types above the bottom of & 2: Sandy silt; \\ \cline{1-1}  & foundation pits & 3: Clay; \\ \cline{1-1}  & & 4: Slity clay; \\ \cline{1-1}  & & 5: Mucky clay. \\ \hline \multirow{8}{*}{Weather condition} & & 1: Fine; \\ \cline{1-1}  & Weather condition & 2: Overcast; \\ \cline{1-1}  & & 3: Rainy. \\ \hline Previous settlement monitoring data & Data monitored in previous days & Positive real number. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Input parameters of the proposed BPNN model.  
where \(o_{i}\) is the value of the output generated by the BPNN, i.e., the predicted ground settlement; \(t_{i}\) is the value of the target output, i.e., the actual ground settlement monitored at the site; \(N\) is the total number of output data. A small value of \(RMSE\) indicates a high prediction accuracy.  
A key parameter for determining an optimum BPNN is the number of neurons in its hidden layer \(n_{1}\), and such a number is obtained mainly through experience or experimental tests without an exact calculation method [45]. Nevertheless, \(n_{1}\) can be roughly estimated using the following equation [47; 48]:  
\[n_{1}=\sqrt{m+n}+C \tag{14}\]  
where \(m\) and \(n\) are the numbers of neurons in the input layer and the output layer, respectively; \(C\) is a value between 0 and 10. Accordingly, a specific BPNN for the proposed study is established, as shown in Figure 2.  
Furthermore, the proposed study adopts the 10-fold cross-validation method to avoid overfitting of the optimum BPNN model. In this method, a training dataset is divided equally into 10 subsets, and 9 subsets are randomly selected for training BPNN models and the other dataset is used for validation. Accordingly, \(10\)\(RMSE\) can be calculated for a BPNN model, and the average value of \(RMSE\) is considered as the estimate of the real value of \(RMSE\) of the BPNN model. The model with the smallest average value of \(RMSE\) is set as the optimal BPNN model.