## 4 Experiments
### Fault classification  
Considering the performance of our fault representations and the limitation of training samples, a classical classifier, SVM model, is employed to implement the fault classification.  
To obtain the SVM model with the best accuracy, we utilize the grid search with cross validation (GSCV) technique [30] to train the SVM models with several hyper-parameters sets. Through the process of the GSCV, the SVM model with the hyper-parameter set {\(C=1.5\), '\(kernel^{\prime}='rbf^{\prime},gamma='auto^{\prime}\)} is selected as the final classifier to implement the fault classification for centrifugal pumps. In addition to that, the formula for accuracy is defined as \(Accuracy=Rights/(Rights+Errors)\times 100\%\), and the corresponding diagnostic results are illustrated in Table 2.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Feature No.** & **Ranking** \\ \hline \(s(q)_{1}\) & 1 \\ \(s(q)_{2}\) & 1 \\ \(s(q)_{3}\) & 1 \\ \(s(q)_{4}\) & 1 \\ \(s(q)_{5}\) & 6 \\ \(s(q)_{6}\) & 5 \\ \(s(q)_{7}\) & 4 \\ \(s(q)_{8}\) & 3 \\ \(s(q)_{9}\) & 2 \\ \(s(q)_{10}\) & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Feature ranking obtained by RFE.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Mode** & **Accuracy** & **Errors** & **Rights** \\ \hline \multirow{8}{*}{TfEn} & Normal & 100.00\% & 0 & 50 \\  & Inner-race fault & 100.00\% & 0 & 50 \\  & Outer-race fault & 100.00\% & 0 & 50 \\  & Ball fault & 100.00\% & 0 & 47 \\  & Impeller fault & 94.34\% & 3 & 50 \\  & Average/Total & 98.80\% & 3 & 247 \\ \hline \multirow{8}{*}{TfEn-PCA} & Normal & 100.00\% & 0 & 50 \\  & Inner-race fault & 100.00\% & 0 & 50 \\  & Outer-race fault & 75.00\% & 13 & 39 \\  & Ball fault & 73.91\% & 12 & 34 \\  & Impeller fault & 94.23\% & 3 & 49 \\  & Average/Total & 88.80\% & 28 & 222 \\ \hline \multirow{8}{*}{TfEn-KPCA} & Normal & 91.11\% & 4 & 41 \\  & Inner-race fault & 100.00\% & 0 & 46 \\  & Outer-race fault & 60.24\% & 33 & 50 \\  & Ball fault & 86.49\% & 5 & 32 \\  & Impeller fault & 94.87\% & 2 & 37 \\  & Average/Total & 82.40\% & 44 & 206 \\ \hline \multirow{8}{*}{TfEn-RFE (proposed)} & Normal & 100.00\% & 0 & 50 \\  & Inner-race fault & 100.00\% & 0 & 50 \\ \cline{1-1}  & Outer-race fault & 100.00\% & 0 & 50 \\ \cline{1-1}  & Ball fault & 100.00\% & 0 & 50 \\ \cline{1-1}  & Impeller fault & 100.00\% & 0 & 50 \\ \cline{1-1}  & Average/Total & 100.00\% & 0 & 250 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results of proposed fault diagnosis.  
Figure 10: Scatters of total TfEn features (**a**); scatters of optimal TfEn features selected by principal component analysis (PCA) (**b**); scatters of optimal TfEn features selected by kernel PCA (KPCA) (**b**); scatters of optimal TfEn features selected by RFE (**d**).  
As shown in Table 2, defining the size of training samples as 50 and using the SVM as the classifier model, the proposed method based on TfEn and RFE brings about the best results among all 4 methods, where its average accuracy is 100.00%, and no testing sample is misclassified.  
Compared with total raw TfEn features, our proposed method improves the average accuracy by 1.20%. Compared with TfEn-PCA and TfEn-KPCA, our proposed method improves the average accuracy by 11.20% and 17.60%. The results preliminarily demonstrate the validity of TfEn, and the superiority of RFE over the mainstream PCA and KPCA-based feature selection methods.