## 5 Test and Verification
### Condition Detection via the Proposed Method
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{**Symptom Parameters**} \\ \cline{2-8}
**DI Values** & \multicolumn{2}{c}{**RMS (Root**} & \multicolumn{2}{c}{**Skewness**} & \multicolumn{2}{c}{**Kurtosis**} & \multicolumn{2}{c}{_ApEn1_} & \multicolumn{2}{c}{_ApEn2_} & \multicolumn{2}{c}{_ApEn3_} & \multicolumn{2}{c}{_ApEn4_} \\ \cline{2-8}  & **Mean Square** & \multicolumn{1}{c}{**Skewness**} & \multicolumn{1}{c}{**Kurtosis**} & \multicolumn{1}{c}{_ApEn1_} & \multicolumn{1}{c}{_ApEn2_} & \multicolumn{1}{c}{_ApEn3_} & \multicolumn{1}{c}{_ApEn4_} \\ \hline \(\text{DI}_{\text{N-O}}\) & 1.26 & 1.536 & 1.284 & 1.662 & 2.119 & 5.376 & 6.931 \\ \(\text{DI}_{\text{N-I}}\) & 0.828 & 1.344 & 2.868 & 1.332 & 8.675 & 2.098 & 9.341 \\ \(\text{DI}_{\text{N-R}}\) & 1.232 & 0.864 & 1.356 & 11.813 & 8.103 & 6.769 & 6.280 \\ \(\text{DI}_{\text{O-I}}\) & 2.108 & 1.176 & 2.412 & 0.918 & 5.572 & 8.268 & 2.779 \\ \(\text{DI}_{\text{O-R}}\) & 1.136 & 0.696 & 1.656 & 3.079 & 24.526 & 3.918 & 2.468 \\ \(\text{DI}_{\text{I-R}}\) & 2.352 & 0.348 & 1.92 & 1.256 & 17.051 & 8.059 & 0.527 \\ \hline \hline \end{tabular}  
* Here, N, O, I, R indicate normal, outer ring defect, inner ring defect and roller element defect states, respectively. RMS: root mean square.  
\end{table} 
Table 2: Detection index (DI) values of each symptom parameter (SP). Figure 12: Inner ring defect state signal VMD. Figure 13: Roller defect state signal VMD. and condition identification. The calculated \(ApEn\) sample is randomly partitioned into five subsamples. Of the five subsamples, three subsamples are retained as the training data, one subsample is used as test data for parameter optimization, and the remaining one subsample is used as verification data for condition identification of the bearing. As an example, some PSO-KELM training data are listed in Table 4.  
After parameter optimization, the bearing condition identification is performed by the optimal KELM and 5-CV method. The verification result shown in Table 5 demonstrates that the normal state average diagnosis accuracy reaches 100%, the outer ring defect state average fault diagnosis accuracy reaches 90.3%, the outer ring defect state average fault diagnosis accuracy reaches 85.7%, the roller element defect state average fault diagnosis accuracy reaches 96.3%, and the overall accuracy reaches  
Figure 14: PSO-KELM (particle swarm optimization-kernel extreme learning machine) algorithm procedure. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Normal State**} & \multicolumn{2}{c}{**Outer Ring Defect**} & \multicolumn{2}{c}{**Inner Ring Defect**} & \multicolumn{2}{c}{**Roller Element Defect**} \\ \cline{2-9} _Apen_ & **Average** & **Standard** & **Average** & **Standard** & **Average** & **Standard** & **Average** & **Standard** \\  & & **Deviation** & & **Deviation** & & **Deviation** & & **Deviation** \\ \hline \(ApEn1\) & 0.731 & 0.010 & 0.145 & 0.021 & 0.405 & 0.083 & 0.434 & 0.012 \\ \(ApEn2\) & 0.712 & 0.012 & 0.221 & 0.030 & 0.841 & 0.013 & 0.790 & 0.051 \\ \(ApEn3\) & 0.691 & 0.031 & 0.458 & 0.031 & 0.578 & 0.029 & 1.005 & 0.061 \\ \(ApEn4\) & 0.670 & 0.005 & 0.997 & 0.012 & 1.044 & 0.050 & 1.066 & 0.104 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Distribution information of the \(ApEn\). 93.08%. These diagnosis results demonstrate the effectiveness of the fault diagnosis method proposed in this paper. To further verify the effectiveness of the PSO-KELM algorithm proposed in this paper, the backpropagation (BP) neural network, conventional ELM and SVM algorithms, and the PSO-KELM algorithm proposed in this paper are compared. When the BP neural network and conventional ELM and KELM algorithms are used for bearing state identification and diagnosis, the vibration data are identical to those for the PSO-KELM algorithm. Additionally, vibration signal features are extracted via VMD and the \(ApEn\) method introduced in Sections 2 and 3. In the SVM, a one-against-one method [27] is used to establish a multiclass SVM system.