# Vibration Velocity Prediction with Regression and Forecasting Techniques for Axial Piston Pump
## 3 Results and Discussion
### Complexity
#### 3.1.2 Ensemble Learning
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Num Epochs** & **MAPE** & **MAE** & **Test MAPE** & **Test MAE** & **Loss** & **Num Layers** & **Num Neurons** & **Num of Input Timepoints** & **Input Cols** \\ \hline
40 & 7.5436 & 0.1042 & 7.3943 & 0.1071 & 0.0327 & 5 & 10 & 16 & [Outlet pressure/, “Outlet flow/, “Bled pressure”] \\
50 & 6.782 & 0.0802 & 7.9953 & 0.113 & 0.0106 & 5 & 10 & 32 & [Outlet pressure/, “Outlet flow/, “Bled pressure”] \\
34 & 7.198 & 0.1044 & 8.441 & 0.1197 & 0.0226 & 4 & 5 & 4 & [Outlet pressure/, “Outlet flow/, “Bled pressure”] \\
42 & 14.7205 & 0.1729 & 10.6782 & 0.1501 & 0.0435 & 5 & 10 & 16 & [Outlet pressure/, “Outlet flow/] \\
56 & 14.6757 & 0.179 & 10.7966 & 0.1451 & 0.0435 & 5 & 16 & [Outlet pressure/, “Outlet flow/] \\
105 & 13.8929 & 0.1695 & 10.8598 & 0.1486 & 0.0417 & 3 & 10 & 8 & [Outlet pressure/, “Outlet flow”] \\
23 & 14.8037 & 0.1843 & 12.0291 & 0.1674 & 0.0448 & 5 & 10 & 8 & [Outlet pressure/] \\
27 & 15.812 & 0.1699 & 12.1633 & 0.1685 & 0.0248 & 2 & 10 & 32 & [Outlet pressure/] \\
22 & 15.9429 & 0.1915 & 12.3078 & 0.1718 & 0.0465 & 2 & 5 & 16 & [Outlet pressure/] \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of metrics evaluated during training for three best RNN models for each of input variables set. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Num Epochs** & **MAPE** & **MAE** & **Test MAPE** & **Test MAE** & **Loss** & **Num Layers** & **Num Neurons** & **Num of Input Timepoints** & **Input Cols** \\ \hline
58 & 6.9488 & 0.1033 & 7.9439 & 0.1117 & 0.0326 & 5 & 10 & 16 & [Outlet pressure/, “Outlet flow/, “Bled pressure”] \\
71 & 7.4563 & 0.0885 & 7.9897 & 0.1169 & 0.0114 & 2 & 5 & 32 & [Outlet pressure/, “Outlet flow/, “Bled pressure”] \\
112 & 7.4918 & 0.1035 & 8.0788 & 0.1183 & 0.0329 & 5 & 5 & 8 & [Outlet pressure/, “Outlet flow/, “Bled pressure”] \\
104 & 13.8095 & 0.1548 & 9.6899 & 0.1344 & 0.0212 & 4 & 10 & 32 & [Outlet pressure/, “Outlet flow”] \\
74 & 13.9892 & 0.1774 & 10.6833 & 0.1476 & 0.0431 & 4 & 10 & 16 & [Outlet pressure/, “Outlet flow”] \\
56 & 14.0297 & 0.1566 & 0.7201 & 0.1417 & 0.0218 & 3 & 10 & 32 & [Outlet pressure/, “Outlet flow/] \\
40 & 15.5522 & 0.1916 & 11.9598 & 0.1707 & 0.0464 & 5 & 10 & 16 & [Outlet pressure/, “Outlet flow/] \\
32 & 14.1244 & 0.1573 & 12.0584 & 0.1674 & 0.0222 & 5 & 10 & 32 & [Outlet pressure/] \\
25 & 15.777 & 0.1901 & 12.3115 & 0.1681 & 0.0463 & 5 & 10 & 8 & [Outlet pressure/] \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison of metrics evaluated during training for three best LSTM models for each of input variables set. The conclusion for stacking ensemble is that the dense model has the greatest impact on the final value of the ensemble. This conclusion is coherent with the fact that the dense model was the best one in terms of MAPE metrics amongst all of the neural network models. Computing the output of the blender requires the computation of the outputs of models that are part of the ensemble. As there are three models (dense, RNN, LSTM), using the stacking ensemble strategy heavily impacts the computational resources and inference time. The last used ensemble involves GB. Within this approach, we selected one model of each type (dense, RNN, LSTM) and trained a separate linear regression model as a corrector of prediction error of the base model. Three GB ensembles were created, where each of them consists of one base neural network model followed by a linear regression model. The GB approach is less complex than simple strategies in terms of the number of models involved. To calculate the output of the simple strategies ensemble, the outputs of three models (dense, RNN, LSTM) must be computed, while in the GB approach two models must be used: one neural network model and one linear regression model. Among all presented ensembles, in terms of MAPE metrics selected among experiments for non-damaged pumps, which are presented in Table 9, the best one is the median ensemble.