## 4 Case Study
### Results and Discussion  
#### 4.5.1 Construction of the VMD-SE-PACF-ELM Model  
The ELM models with different structural frameworks are used to train and predict the VMF components separately, and then overlay the prediction results of each component to complete the construction of the VMD-SE-PACF-ELM model. During the prediction process, the result of each step are used to predict the next value until the 30 th prediction result is obtained (Figure 11). The training and prediction results for each VMF are shown in Figures 12 and 13, respectively. The yellow areas in Figure 13 represent the absolute residuals of the prediction results. It can be seen from Figures 12 and 13 that the VMF\({}_{1}\), VMF\({}_{2}\) and VMF\({}_{3}\) components have good fitting capability and low prediction error. In comparison, the fitting and prediction results for VMF4 and VMF5 are somewhat different from the actual values near the curve inflection points, but the overall results are consistent with the trend of the actual values. Therefore, it can be initially concluded that the use of the VMD algorithm can effectively decompose the DD fluctuation information, thereby improving the prediction performance of the model. Here we use RMSE, MAE and R\({}^{2}\) to quantify the performance of the training and testing phases of ELMs, as shown in Table 5. Obviously, ELM\({}_{\text{VMF1}}\) has the smallest RMSE and MAE values and the largest R\({}^{2}\) value in the training and testing phases, followed by ELM\({}_{\text{VMF2}}\) and ELM\({}_{\text{VMF3}}\), and these three sub-models all show strong performance. ELM\({}_{\text{VMF4}}\) and ELM\({}_{\text{VMF5}}\) have relatively large errors during the training and testing phases, but their predictive accuracy is still at a high level so it does not affect the overall performance of the model. Figure 12: Training results for each VMF. To further illustrate the necessity of using the VMD decomposition algorithm in the prediction process of the ELM model, Figure 14 shows the prediction results and evaluation indicators before and after VMD optimization. In Figure 14, 'VMD Optimized ELM' represents the VMD-SE-PACF-ELM model, and 'PACF-ELM' represents the ELM model based on PACF to determine input variables. The left side of the figure is a bar graph of the performance evaluation indicators (RMSE, MAE and R\({}^{2}\)) \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Prediction Model** & \multicolumn{3}{c}{**Training Set**} & \multicolumn{3}{c}{**Testing Set**} \\ \hline  & **RMSE** & **MAE** & **R\({}^{2}\)** & **RMSE** & **MAE** & **R\({}^{2}\)** \\ \hline \(\text{ELM}_{\text{VMF1}}\) & 0.0014 & 0.0011 & 0.9998 & 0.0262 & 0.0201 & 0.9906 \\ \(\text{ELM}_{\text{VMF2}}\) & 0.0015 & 0.0013 & 0.9980 & 0.0431 & 0.0366 & 0.9542 \\ \(\text{ELM}_{\text{VMF3}}\) & 0.0026 & 0.0020 & 0.9989 & 0.0764 & 0.0580 & 0.9820 \\ \(\text{ELM}_{\text{VMF4}}\) & 0.0047 & 0.0035 & 0.9972 & 0.1339 & 0.1034 & 0.9105 \\ \(\text{ELM}_{\text{VMF5}}\) & 0.0056 & 0.0042 & 0.9841 & 0.0843 & 0.0603 & 0.7465 \\ VMD-SE-PACF-ELM & / & / & / & 0.2996 & 0.2343 & 0.8722 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Evaluation indicators of training and testing set for each sub-model. Figure 13: Prediction results for each VMF. of the two models, and the right side is the graph of the predicted results. Obviously, the deformation prediction result of the ELM model optimized based on the VMD algorithm has smaller RMSE and MAE as well as a larger R\({}^{2}\) compared to the single ELM model. Combined with the prediction curves, the following conclusions can be obtained: if the ELM model is used directly to predict the deformation time series with strong fluctuation characteristics, it is not possible to get accurate prediction results; however, the prediction results of the VMD-optimized ELM model are closer to the real values of the DD, which means that the VMD algorithm can decompose the original deformation series into several subsequences with good deformation characteristics.