## 4 Analysis and Verification of Rolling Bearing Fault Diagnosis Tests
### Comparison Test of Diagnostic Models  
The effectiveness of the proposed model is evaluated by comparing it with several currently prevalent fault diagnosis models, which include the following: VMD-PE-LSTM, VMD-PE-IGTO-LSTM, GTO-VMD-PE-LSTM, IGTO-VMD-PE-PE-PSO-LSTM, IGTO-VMD-PE-PSO-LSTM, IGTO-VMD-PE-GWO-LSTM, IGTO-VMD-PE-CDO-LSTM, and IGTO-VMD-PE-GTO-LSTM. Each sample comprised 2048 data points across 480 datasets representing various operational states; 384 datasets were randomly selected for training, while the remaining 96 datasets were utilized for testing. Based on empirical observations from previous studies and preliminary experiments, the hyperparameters for the LSTM model were set as follows: 300 neurons, a maximum \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Fault Label**} & \multirow{2}{*}{**Sample Number**} & \multicolumn{6}{c}{**PE of Different IMFs**} \\ \cline{3-8}  & & **PE1** & **PE2** & **PE3** & **PE4** & **PES** & **PE6** & **PE7** \\ \hline \multirow{8}{*}{L0} & 1 & 0.25127 & 0.374075 & 0.438361 & 0.468658 & 0.464287 & 0.587091 & 0.639929 \\  & 2 & 0.238101 & 0.360905 & 0.438503 & 0.46888 & 0.464293 & 0.587098 & 0.639936 \\  & 3 & 0.267851 & 0.390656 & 0.43066 & 0.464363 & 0.464543 & 0.587348 & 0.640209 \\  & 4 & 0.263381 & 0.386186 & 0.431497 & 0.461794 & 0.465808 & 0.58612 & 0.641587 \\  & 5 & 0.253578 & 0.376382 & 0.436386 & 0.466683 & 0.466355 & 0.589159 & 0.642184 \\ \hline \multirow{8}{*}{L1} & 1 & 0.381256 & 0.393564 & 0.503047 & 0.549674 & 0.631662 & 0.68703 & 0.270641 \\  & 2 & 0.380778 & 0.39307 & 0.495723 & 0.544684 & 0.631899 & 0.687288 & 0.707907 \\  & 3 & 0.386287 & 0.398757 & 0.494779 & 0.543646 & 0.631907 & 0.687296 & 0.707915 \\  & 4 & 0.9393 & 0.405686 & 0.488251 & 0.536473 & 0.632458 & 0.687986 & 0.708533 \\  & 5 & 0.387471 & 0.399978 & 0.496128 & 0.545128 & 0.632482 & 0.687922 & 0.70856 \\ \hline \multirow{8}{*}{L2} & 1 & 0.384635 & 0.392662 & 0.483268 & 0.507432 & 0.605863 & 0.648273 & 0.71331 \\  & 2 & 0.397923 & 0.405882 & 0.479077 & 0.50331 & 0.66197 & 0.646361 & 0.713494 \\ \cline{1-1}  & 3 & 0.370459 & 0.37869 & 0.484 & 0.5082 & 0.606601 & 0.649053 & 0.713969 \\ \cline{1-1}  & 4 & 0.35191 & 0.389222 & 0.488887 & 0.513332 & 0.607032 & 0.649524 & 0.714477 \\ \cline{1-1}  & 5 & 0.425011 & 0.435312 & 0.491682 & 0.516267 & 0.607218 & 0.649723 & 0.714696 \\ \hline \multirow{8}{*}{L3} & 1 & 0.413902 & 0.426319 & 0.500926 & 0.510944 & 0.618285 & 0.649199 & 0.746579 \\ \cline{1-1}  & 2 & 0.4238676 & 0.43692 & 0.516405 & 0.526733 & 0.6185606 & 0.649589 & 0.747028 \\ \cline{1-1}  & 3 & 0.47057 & 0.486487 & 0.50552 & 0.516683 & 0.60933 & 0.639844 & 0.735407 \\ \cline{1-1}  & 4 & 0.441058 & 0.45429 & 0.513628 & 0.5239 & 0.629161 & 0.660619 & 0.759711 \\ \cline{1-1}  & 5 & 0.462693 & 0.476573 & 0.515467 & 0.52577 & 0.619174 & 0.650132 & 0.747652 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: PE values of fault samples extracted by IGTO-VMD. Figure 9: VMD results after parameter optimization. The data presented in Table 5 indicate that Model 3 achieved a test accuracy that is 4.17% higher than that of Model 1, thereby demonstrating that the application of intelligent algorithms significantly enhances the performance of VMD in signal decomposition and fault diagnosis. Furthermore, the comparison between Models 2 and 9 supports this assertion. When comparing Model 3 to Model 4, it is evident that the parameters optimized by the IGTO method yield a higher diagnostic accuracy of 91.98% and a narrower deviation range of [\(-\)2.40, 1.77] compared to those optimized by the GTO method. This finding further substantiates the superiority of IGTO in optimization tasks. Additionally, this conclusion is corroborated by the comparison between Models 7 and 9. Model 9 demonstrated superior performance in rolling bearing fault diagnosis compared to models 5, 6, 7, and 8, achieving the highest accuracy rates of 98.44% during training and 96.67% during testing.