## 4 Analysis and Verification of Rolling Bearing Fault Diagnosis Tests
### Comparison Test of Diagnostic Models
In contrast, model 8, which exhibited the second  
Figure 11: (**a**) Confusion matrix for VMD-PE-LSTM models; (**b**) confusion matrix for VMD-PE-IGTO-LSTM models; (**c**) confusion matrix for GTO-VMD-PE-LSTM models; (**d**) confusion matrix for IGTO-VMD-PE-LSTM models; (**e**) confusion matrix for IGTO-VMD-PE-GWO-LSTM models; (**f**) confusion matrix for IGTO-VMD-PE-PSO-LSTM models; (**g**) confusion matrix for IGTO-VMD-PE-GTO-LSTM models; (**h**) confusion matrix for IGTO-VMD-PE-CDO-LSTM models; (**i**) confusion matrix for IGTO-VMD-PE-IGTO-LSTM models. highest accuracy with 97.03% in training and 95.00% in testing, underscores the advantages of employing IGTO twice for parameter optimization. This approach allowed model 9 to exceed the performance of both GTO and other algorithms. In order to better evaluate the performance of the models, the standard deviation and 95% confidence intervals of their diagnostic accuracies for each model in the testing phase were calculated and the key parameters are listed in Table 6 below. Based on the experimental results in Table 6 above, the IGTO-VMD-PE-IGTO-LSTM model performs well in the fault diagnosis task, with an average accuracy of 96.67%, a maximum accuracy of 98.96%, and a standard deviation of 1.2148, which shows high stability and small fluctuations. In contrast, the other models have obvious gaps in accuracy and stability with large standard deviations, indicating that they do not perform as well as IGTO-VMD-PE-IGTO-LSTM. In addition, the 95% confidence intervals of IGTO-VMD-PE-IGTO-LSTM are [95.75%, 97.58%], which further verifies its reliability and accuracy. Taken together, the model not only provides higher accuracy and lower volatility but also has strong stability in practical applications. As shown in Figure 10, it can be found that Model 9 exhibits the highest median accuracy of 98.44% in the training phase, with a narrow fluctuation range concentrated around 98%. This observation indicates that Model 9 not only achieves the highest accuracy during the training process but also demonstrates commendable stability. Furthermore, in the boxplot of the testing phase, Model 9 continues to uphold its superior position in terms of both accuracy and stability, achieving a median accuracy of 96.35% and a fluctuation range between 95% and 97%. These results suggest that Model 9 possesses exceptional stability while maintaining high accuracy, rendering it the model likely to excel in practical applications, particularly in fault diagnosis environments that necessitate both high accuracy and stability. The confusion matrix presented above in Figure 11, which represents one of ten results, indicates that the proposed Model 9 demonstrates exceptional performance in the fault diagnosis of rolling bearings. Also, it achieves high accuracy in all fault categories, especially in the L1 category, reaching a 29 times high accuracy diagnosis, which is the highest value in the comparison model. Furthermore, the classification error rate for Model 9 remains consistently low across all fault types, reflecting its high stability and superior classification efficacy. In comparison to other models, Model 9 not only excels in diagnosing individual fault types but also effectively minimizes misclassification while sustaining an overall high level of accuracy. In order to evaluate the performance of the IGTO-VMD-MSE-IGTO-LSTM models, Precision, Recall, and F1 Score are used, and the resulting fault diagnosis evaluation metrics for each model under different operating conditions are shown in Table 7.