## 4 Experiment and Discuss
### Comparative Analysis and Experimental Results
#### 4.3.3 Performance Evaluation of the Diagnostic Framework
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Contrastive Models**} & \multicolumn{2}{c}{**PNN**} & \multicolumn{2}{c}{**SVM**} & \multicolumn{2}{c}{**ELM**} & \multicolumn{2}{c}{**KELM**} & \multicolumn{2}{c}{**HMLC-SBELM**} \\ \cline{2-10}  & \(\varepsilon^{*}\) & \(F_{me}\) & \(\varepsilon^{*}\) & \(F_{me}\) & \(\varepsilon^{*}\) & \(F_{me}\) & \(\varepsilon^{*}\) & \(F_{me}\) & \(\varepsilon^{*}\) & \(F_{me}\) \\ \hline \(\varepsilon^{*}\), \(F_{me}\) & 0.71 & 0.863 & 0.68 & 0.829 & 0.69 & 0.857 & 0.69 & 0.903 & 0.71 & 0.923 \\ \hline \hline \end{tabular} (3) Performance evaluation of the diagnostic model  
The performance of various diagnostic models is evaluated by using the testing set Dtesting which contains both single and simultaneous fault samples. The comparison results are listed in Table 5 and shown in Figure 9 in which H-SBELM represents the HMLC-SBELM-based model.  
\end{table} 
Table 4: The optimal decision threshold and \(F_{me}\). Figure 9: Comparison of fault diagnostic result. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Various Models**} & \multirow{2}{*}{
\begin{tabular}{c} **Multi-Label** \\ **Classification Strategy** \\ \end{tabular} } & \multicolumn{4}{c}{**Accuracy (\%)**} \\ \cline{2-6}  & One-to-all & 88.45 (\(\pm\)1.52) & 78.15 (\(\pm\)1.63) & 82.79 (\(\pm\)1.49) \\ \cline{2-6}  & HMLC strategy & 92.88 (\(\pm\)1.31) & 81.26 (\(\pm\)1.49) & 86.37 (\(\pm\)1.72) \\ \hline \multirow{2}{*}{SVM} & One-to-all & 90.12 (\(\pm\)1.25) & 76.23 (\(\pm\)1.75) & 81.94 (\(\pm\)1.44) \\ \cline{2-6}  & HMLC strategy & 91.69 (\(\pm\)1.18) & 79.81 (\(\pm\)1.52) & 84.52 (\(\pm\)1.29) \\ \hline \multirow{2}{*}{ELM} & One-to-all & 85.61 (\(\pm\)1.46) & 74.08 (\(\pm\)1.91) & 82.91 (\(\pm\)1.55) \\ \cline{2-6}  & HMLC strategy & 91.19 (\(\pm\)1.37) & 77.35 (\(\pm\)1.68) & 85.44 (\(\pm\)1.59) \\ \hline \multirow{2}{*}{KELM} & One-to-all & 93.26 (\(\pm\)1.54) & 83.62 (\(\pm\)1.62) & 89.17 (\(\pm\)1.77) \\ \cline{2-6}  & HMLC strategy & 96.38 (\(\pm\)1.71) & 86.31 (\(\pm\)1.43) & 91.93 (\(\pm\)1.65) \\ \hline \multirow{2}{*}{HMLC-SBELM} & One-to-all & 93.62 (\(\pm\)1.05) & 85.38 (\(\pm\)1.45) & 88.24 (\(\pm\)1.23) \\ \cline{2-6}  & HMLC strategy & **98.03 (\(\pm\)0.96)** & **88.71 (\(\pm\)1.01)** & **94.43 (\(\pm\)1.14)** \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The contrastive results of diagnostic models. As indicated in Figure 9, among the above-mentioned five diagnostic models, the HMLC-SBELM-based diagnostic model outperforms other models in recognizing single-fault modes and simultaneous failure modes. Unlike other models by employing regression and fitting to solve the problem, the SBELM-based model improves the generalization performance by the probability distribution estimation. For seven single failure modes and five simultaneous failure modes, the average testing accuracy of HMLC-SBELM reach 98.03% and 88.71%. The diagnostic accuracy is improved by 5% to 15% over several other diagnostic models. In the proposed HMLC-SBELM-based model, the HMLC strategy is closely related to the improvement of performance. By comparing the diagnostic accuracy of the above five diagnostic models with the HMLC strategy and traditional one-to-all multi-class classification strategy, the effectiveness of the HMLC strategy is validated and the results are listed in Table 5.  
As illustrated in Table 5, for the same diagnostic model, the classification accuracy of diagnostic models with the HMLC strategy is 2% to 6% higher than that of models with a one-to-all strategy. The main reasons are that the hierarchical pairing framework proposed in this paper fully considers the correlation between multiple single failure modes when diagnosing multiple faults.