## 4 Case Study
### Results and Discussion
#### 4.5.3 Comparison with Other Benchmark Models  
To further verify the superiority of the VMD-SE-ER-PACF-ELM model, its performance is compared with that of some benchmark models in this section. In addition to PACF-ELM and VMD-SE-PACF-ELM prediction models, three other DD prediction models, namely EMD-PACF-ELM, hydrostatic-seasonal-time (HST)-ELM and Arima models, are established. The EMD-PACF-ELM model can be used to verify that the VMD algorithm outperforms the EMD algorithm in DD prediction; the HST-ELM model can validate that the performance of the DD prediction model based on VMD decomposition is better than the ELM model based on statistical optimization, and the Arima model can demonstrate that the proposed model outperforms traditional time series prediction method in terms of prediction accuracy. The prediction results and performance evaluation indicators of each  
Figure 16: Comparison of predicted results before and after considering ER. Figure 15: Analysis process of the ER. model are shown in Figure 17, where the bar graph of the performance evaluation indicators of each model is shown on the left and the curve of the prediction results is shown on the right. As can be seen in Figure 17, both the evaluation indicators and the prediction curves show that Arima has the worst performance. In contrast to Arima, the prediction performance of the HST-ELM model is significantly better. Another important conclusion of Figure 17 is that the prediction results of the models combined with the signal decomposition methods (EMD, VMD) always outperform the other models. Meanwhile, the VMD-SE-ER-PACF-ELM model outperforms the EMD-PACF-ELM model, which indicates that the VMD algorithm used in this study to pre-process the deformation sequence is better than the EMD algorithm. Bar and curve graphs can provide a visual assessment of the prediction ability of the models and the correspondence between the observations and the model predictions. However, performance evaluation indicators can more accurately quantify the predicted performance of each model. Table 6 shows the performance evaluation indicators for the six models including PACF-ELM and VMD-SE-PACF-ELM. The RMSE values of each model are: 0.2755 (VMD-SE-ER-PACF-ELM), 0.2996 (VMD-SE-PACF-ELM), 0.3341 (EMD-PACF-ELM), 0.4400 (HST-ELM), 0.5630 (PACF-ELM) and 0.5955 (Arima). Compared to other models, the RMSEs of the VMD-SE-ER-PACF-ELM model are reduced by 8.04%, 17.54%, 37.39%, 51.07% and 53.74%, respectively. Meanwhile, the values of MAE are: 0.2087 (VMD-SE-ER-PACF-ELM), 0.2343 (VMD-SE-PACF-ELM), 0.2686 (EMD-PACF-ELM), 0.2862 (HST-ELM), 0.4085 (PACF-ELM) and 0.4320 (Arima). The MAE values of the proposed model are reduced by 10.93%, 22.30%, 27.08%, 48.91% and 51.69%, respectively, compared to other models. The above results are consistent with the relatively high R\({}^{2}\) values (0.8912 for the VMD-SE-ER-PACF-ELM, 0.8722 for the VMD-SE-PACF-ELM, 0.8398 for the EMD-PACF-ELM, 0.8047 for the HST-ELM, 0.6888 for the PACF-ELM and 0.6259 for the Arima model). Compared to other models, the R\({}^{2}\) value of the proposed model in this paper has increased by 2.18%, 6.12%, 10.75%, 29.38% and 42.39%, respectively. Through calculation and comparison of the performance evaluation indicators of various models, we find that the VMD-SE-ER-PACF-ELM model is superior to all the other models in the performance of DD prediction. Furthermore, Figure 18 shows the Taylor diagram of the prediction performance of each model. It is clear that Arima has the worst prediction performance and HST-ELM outperforms the traditional time series prediction method. In addition, the prediction performance of the ELM model optimized