## 5 Numerical Experiments and Analysis
### Comparative Analysis of the Results
It can be seen from Figure 6 that the prediction accuracy of the AE-PLS-ELM model is significantly better than that of the Bagtree and SVR models at all test sample points. In addition, the residual of the AE-PLS-ELM model is generally smaller than the BP and ELM models at most of the test points. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Characteristics**} & \multirow{2}{*}{**Model**} & \multicolumn{6}{c}{**Training**} & \multicolumn{6}{c}{**Test**} \\ \cline{3-10}  & & **RMSE** & **MAE** & **MAPE** & **NSE** & **RMSE** & **MAE** & **MAPE** & **NSE** \\ \hline \multirow{6}{*}{WH} & Bagtree & 0.01827 & 0.01214 & 0.13402 & 0.99232 & 0.02034 & 0.01404 & 0.16405 & 0.98938 \\  & SVR & 0.01019 & 0.00711 & 0.05071 & 0.99646 & 0.01069 & 0.00749 & 0.05963 & 0.99464 \\ \cline{1-1}  & BP & 0.00317 & 0.00222 & 0.02438 & 0.99977 & 0.00324 & 0.00236 & 0.02524 & 0.99973 \\ \cline{1-1}  & ELM & 0.00290 & 0.00207 & 0.02198 & 0.99981 & 0.00297 & 0.00234 & 0.02467 & 0.99977 \\ \cline{1-1}  & AE-PLS-ELM & 0.00229 & 0.00130 & 0.00599 & 0.99988 & 0.00217 & 0.00142 & 0.00697 & 0.99988 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistics error indices for WH characteristics of different models. Figure 5: 3D surface for WM characteristics based on the proposed AE-PLS-ELM model. The training and test results of the Bagtree, SVR, BP, ELM, and AE-PLS-ELM models for the WM characteristic are shown in Table 2. The comparison of the residuals for the WH characteristics between the AE-PLS-ELM model and the Bagtree, SVR, BP, and ELM models are shown in Figure 7a-d, respectively. The results obtained from Table 2 and Figure 7 are consistent with Table 1 and Figure 7. The neural network models that BP, ELM, and AE-PLS-ELM performed better than the Bagtree and the SVR models. The AE-PLS-ELM model can overcome the instability and multi-collinearity of the single ELM model, and obtain higher modeling accuracy. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Characteristics**} & \multirow{2}{*}{**Model**} & \multicolumn{4}{c}{**Training**} & \multicolumn{4}{c}{**Test**} \\ \cline{3-10}  & & **RMSE** & **MAE** & **MAPE** & **NSE** & **RMSE** & **MAE** & **MAPE** & **NSE** \\ \hline \multirow{4}{*}{WM} & Bagtree & 0.18222 & 0.12422 & 0.14329 & 0.99315 & 0.21125 & 0.15082 & 0.18732 & 0.99001 \\  & SVR & 0.10730 & 0.07476 & 0.05411 & 0.99763 & 0.11193 & 0.07852 & 0.06372 & 0.99719 \\ \cline{1-1}  & BP & 0.03227 & 0.02232 & 0.02680 & 0.99979 & 0.03508 & 0.02601 & 0.03050 & 0.99972 \\ \cline{1-1}  & ELM & 0.02656 & 0.01713 & 0.01582 & 0.99985 & 0.02950 & 0.02081 & 0.02110 & 0.99981 \\ \cline{1-1}  & AE-PLS-ELM & 0.02315 & 0.01269 & 0.00539 & 0.99989 & 0.02027 & 0.01306 & 0.00615 & 0.99991 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistics error indices for WM characteristics of different models. Figure 6: Comparison of the test residuals for WH characteristics between Bagtree, SVR, BP, ELM, and AE-PLS-ELM. (**a**) Bagtree vs. AE-PLS-ELM, (**b**) SVR vs. AE-PLS-ELM, (**c**) BP vs. AE-PLS-ELM, (**d**) ELM vs. AE-PLS-ELM. The regression analysis scatter diagram of the residual and actual value for the _WH_ and _WM_ characteristics is shown in Figure 8. It can be seen from Figure 8 that the scatter plot of the single ELM model is more divergent around the axis, and the points of the AE-PLS-ELM model distributes more closely, which demonstrates the superiority of the AE-PLS-ELM model in modeling the complete characteristic curves of the pump turbine. Figure 8: Comparison of the Scatter diagram for the test residuals between ELM and AE-PLS-ELM. Figure 7: Comparison of the test residuals for WM characteristics between Bagtree, SVR, BP, ELM and AE-PLS-ELM. (**a**) Bagtree vs. AE-PLS-ELM, (**b**) SVR vs. AE-PLS-ELM, (**c**) BP vs. AE-PLS-ELM, (**d**) ELM vs. AE-PLS-ELM.