## 4 Analysis and Verification of Rolling Bearing Fault Diagnosis Tests
### Additional Validation Experiments
To mitigate the randomness introduced by intelligent optimization algorithms in the fault diagnosis process, each model was executed 10 times. The classification accuracies obtained in the testing phase are summarized in Table 9 below. To compute the average accuracy, highest accuracy, lowest accuracy, standard deviation of accuracy, and the 95% confidence intervals for the 10 runs in the training phase, the following results are presented in Table 10 below. \begin{table}
\begin{tabular}{c c c c} \hline
**Bearing Status** & **Labels** & **Rotation Speed** & **Sampling Frequency** & **Number of Data** \\ \hline Regular state & n & 600 rpm & 50 KHz & 480 \\ Inner race faults & ib & 600 rpm & 50 KHz & 480 \\ Outer race faults & ob & 600 rpm & 50 KHz & 480 \\ Rolling element faults & tb & 600 rpm & 50 KHz & 480 \\ \hline \end{tabular}
\end{table} 
Table 8: Labeling and information on bearing status. Figure 12: (**a**) Structural diagram of the test bench; (**b**) schematic diagram of the test bench [45]. From Table 10 above, it is evident that the models with hyperparameter optimization achieve higher fault diagnosis accuracy. A comparison of models 5, 6, 7, 8, and 13 shows that IGTO provides the best hyperparameter optimization for LSTM. Similarly, a comparison of models 9, 10, 11, 12, and 13 demonstrates that IGTO offers the best hyperparameter optimization for VMD. Based on the analysis of standard deviation and 95% confidence interval, Model 13 performs the best among all models. Its diagnostic accuracy has the smallest standard deviation, only 0.54, and the 95% confidence interval is [98.47%, 99.24%], indicating that the model performs very stably across different datasets and achieves a high level of accuracy. Therefore, Model 13 exhibits the best stability and precision. The test phase accuracy boxplots for each model were then plotted and the following results were obtained as shown below in Figure 13.  
As shown in Figure 13, the median accuracy of Model 13 is 98.96%, which is significantly higher than the medians of the other models, indicating that Model 13 has the most stable and superior diagnostic performance. Additionally, although most models exhibit some fluctuation in accuracy (as indicated by the interquartile range and the length of the whiskers in the boxplot), the boxplot for Model 13 is more compact with a smaller fluctuation range, suggesting its greater stability. The absence of outliers in Model 13 further indicates that it is not influenced by extreme individual cases, and its performance remains consistent across different test sets or operating conditions, demonstrating good reliability for practical applications. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Model** & **Avg/\%** & **Max/\%** & **Min/\%** & **Std/\%** & **95\% Confidence Intervals/\%** \\ \hline
1-VMD-PE-LSTM & 86.28 & 88.28 & 83.33 & 1.7281 & (85.04, 87.51) \\
2-VMD-PE-IGTO-LSTM & 89.04 & 92.45 & 86.20 & 1.7386 & (87.79, 90.28) \\
3-GTO-VMD-PE-LSTM & 91.98 & 94.27 & 89.58 & 1.5916 & (90.84, 93.12) \\
4-IGTO-VMD-PE-LSTM & 94.09 & 95.83 & 91.41 & 1.5457 & (92.98, 95.19) \\
5-IGTO-VMD-PE-GW-LSTM & 95.94 & 97.40 & 92.97 & 1.6019 & (94.79, 97.08) \\
6-IGTO-VMD-PE-PSO-LSTM & 96.43 & 98.70 & 94.79 & 1.0932 & (95.65, 97.21) \\
7-IGTO-VMD-PE-GTO-LSTM & 97.11 & 97.92 & 96.09 & 0.5956 & (96.73, 97.49) \\
8-IGTO-VMD-PE-CDO-LSTM & 97.89 & 98.70 & 96.61 & 0.7002 & (97.39, 98.39) \\
9-GWO-VMD-PE-IGTO-LSTM & 93.98 & 97.92 & 92.45 & 1.6536 & (94.79, 97.08) \\
10-PSO-VMD-PE-IGTO-LSTM & 94.74 & 95.57 & 94.01 & 0.5463 & (95.65, 97.21) \\
11-GTO-VMD-PE-IGTO-LSTM & 96.12 & 96.88 & 95.05 & 0.6548 & (96.73, 97.49) \\
12-CDO-VMD-PE-IGTO-LSTM & 96.48 & 97.92 & 95.05 & 0.9916 & (97.39, 98.39) \\
13-IGTO-VMD-PE-IGTO-LSTM & 98.85 & 99.48 & 97.92 & 0.5371 & (98.47, 99.24) \\ \hline \end{tabular}
\end{table} 
Table 10: Results of the testing phase for JNU dataset.