## 4 Results
### Performance Comparisons between DPGA and Grid-Search Approaches  
We compare the prediction performance of our method to the traditional exhaustive grid-search (EGS) approaches applied to MLP, kNN, SVM, and RF (we call them EGS-MLP, EGS-kNN, EGS-SVM, and EGS-RF, respectively). In an EGS approach, \(5\)-fold cross-validation is conducted to find an optimal set of parameters. In addition, we further compared the performance of the ensemble of the prediction models of all EGS approaches (we call this EGS-E). As our method, the voting ensemble and the Kalman filter ensemble for the fault-type classification and the RUL prediction, respectively, were used for EGS-E. We first scatter-plotted the relation of the performance between the training and test sets by DPGA and five EGS approaches (Figure 4). Unfortunately, the positive relation was not observed among the results of EGS approaches in all figures. This implies that a better solution in the training set can show a worse performance over the test set. Therefore, it is not efficient to simply select a best grid-search approach based on the training set. Interestingly, our approach (DPGA) was best over the test set, whereas it was not best over the training set in all datasets. Figure 5 shows the result where \(Y\)-axis values mean the average and the standard deviation of the F1-score or MSE values in the test dataset over \(50\) trials. As shown in the figure, our DPGA achieved significantly better results among the examined methods in all datasets of fault-type classification and the RUL prediction problems (all \(p\)-values \(<0.02\)). Specifically, the second-best methods were EGS-kNN, EGS-E, and EGS-MLP for the C-MAPSS, steel-plate, and SECOM datasets, respectively. This implies that the performance of a learning algorithm is varied across the given dataset, but our method stably overwhelmed the EGS approaches. In addition, we investigated the best solutions found by EGS (Table 5) and DPGA (Tables 6 and 7), and observed that they are very different to each other. EGS approaches have almost found the best solution, which only includes an algorithm of the FFE (filter-based feature extraction) part and the RD (rebalancing data) part in the RUL prediction and the fault-type classification, respectively. In other words, the RFE and the PCA part were not useful in the search. On the other hand, the best solutions found by DPGA have included valid algorithms in all RD, FFE, RFE, and PCA parts. Specifically, the RD (in fault-type classification), FFE (in RUL prediction), and PCA parts were effective in all best-found solutions. This means that DPGA has efficiently searched a variety of combinations of all subtasks in the fault-type classification and the RUL prediction. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sub-Dataset** & **FD001** & **FD002** & **FD003** & **FD004** \\ \hline Train trajectories & 100 & 260 & 100 & 248 \\ Test trajectories & 100 & 259 & 100 & 248 \\ No. of operational modes & 1 & 6 & 1 & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: NASA C-MAPSS dataset details. Figure 4: Relations of the performance between the training and the test data by DPGA and the exhaustive grid-search approach. (**a–d**) Results in remaining useful life (RUL) prediction datasets. (**e,f**) Results in fault-types classification datasets. Figure 5: Performance comparison of our DPGA approach with the exhaustive grid-search approaches on different machine-learning methods. (**a–d**) Results in RUL prediction datasets. (**e,f**) Results in fault-types classification datasets.