## 4 Analysis and Verification of Rolling Bearing Fault Diagnosis Tests
### Additional Validation Experiments  
The previous experiment was validated using only a single dataset, which is insufficient to demonstrate its effectiveness. To further validate the proposed model's applicability in fault diagnosis, it was applied to another dataset for supplementary validation. The chosen dataset is the Jiangnan University bearing dataset, which includes roller bearings with faults such as outer race faults, inner race faults, and rolling element faults. The test bearings used are of two types: N205 and NU205. The N205 bearings were used for normal, outer race fault, and rolling element fault conditions, while the NU205 bearings, which have a separable outer race, were used for inner race fault conditions. Each fault type is tested under three different operating conditions, corresponding to three different rotational speeds: 600 rpm, 800 rpm, and 1000 rpm. The data are collected using a single accelerometer with a sampling frequency of 50 kHz and a sampling duration of 10 s. The structure diagram and schematic of the test rig are shown in Figure 12 below. \begin{table}
\begin{tabular}{c c c c c} \hline
**Model** & **Operating States** & **Precision** & **Recall** & **F1 Score** \\ \hline \multirow{3}{*}{1-VMD-PE-LSTM} & L0 & 0.7714 & 0.9000 & 0.8308 \\  & L1 & 0.8333 & 0.7895 & 0.8108 \\  & L2 & 0.8462 & 0.8462 & 0.8462 \\  & L3 & 0.8095 & 0.6800 & 0.7391 \\ \hline \multirow{3}{*}{2-VMD-PE-IGTO-LSTM} & L0 & 0.9333 & 0.9655 & 0.9492 \\  & L1 & 0.9545 & 0.9130 & 0.9333 \\  & L2 & 0.8846 & 0.8519 & 0.8679 \\  & L3 & 0.7778 & 0.8235 & 0.8000 \\ \hline \multirow{3}{*}{3-GTO-VMD-PE-LSTM} & L0 & 0.8696 & 0.9524 & 0.9091 \\  & L1 & 0.9615 & 0.8929 & 0.9259 \\  & L2 & 0.9600 & 0.8889 & 0.9231 \\  & L3 & 0.8636 & 0.9500 & 0.9048 \\ \hline \multirow{3}{*}{4-IGTO-VMD-PE-LSTM} & L0 & 0.8333 & 0.9091 & 0.8696 \\  & L1 & 1.0000 & 1.0000 & 1.0000 \\  & L2 & 1.0000 & 0.8621 & 0.9259 \\  & L3 & 0.7692 & 0.8333 & 0.8000 \\ \hline \multirow{3}{*}{5-IGTO-VMD-PE-GWO-LSTM} & L0 & 0.8500 & 0.8095 & 0.8293 \\  & L1 & 1.0000 & 1.0000 & 1.0000 \\  & L1 & 1.0000 & 1.0000 & 1.0000 \\  & L3 & 0.8750 & 0.9032 & 0.8889 \\ \hline \multirow{3}{*}{6-IGTO-VMD-PE-PSO-LSTM} & L0 & 0.8966 & 0.8966 & 0.8966 \\  & L1 & 1.0000 & 1.0000 & 1.0000 \\  & L2 & 1.0000 & 1.0000 & 1.0000 \\  & L3 & 0.8235 & 0.8235 & 0.8235 \\ \hline \multirow{3}{*}{7-IGTO-VMD-PE-GTO-LSTM} & L0 & 1.0000 & 0.8519 & 0.9200 \\  & L1 & 1.0000 & 1.0000 & 1.0000 \\  & L2 & 1.0000 & 1.0000 & 1.0000 \\  & L3 & 0.8621 & 1.0000 & 0.9259 \\ \hline \multirow{3}{*}{8-IGTO-VMD-PE-CDO-LSTM} & L0 & 0.9048 & 0.9048 & 0.9048 \\  & L1 & 1.0000 & 1.0000 & 1.0000 \\  & L2 & 1.0000 & 1.0000 & 1.0000 \\  & L3 & 0.9000 & 0.9000 & 0.9000 \\ \hline \multirow{3}{*}{9-IGTO-VMD-PE-IGTO-LSTM} & L0 & 0.9600 & 0.9600 & 0.9600 \\  & L1 & 1.0000 & 1.0000 & 1.0000 \\ \cline{1-1}  & L2 & 1.0000 & 1.0000 & 1.0000 \\ \cline{1-1}  & L3 & 0.9615 & 0.9615 & 0.9615 \\ \hline \end{tabular}
\end{table} 
Table 7: Evaluation metrics for fault diagnosis in different operating states for each model. Taking the data at 600 rpm as an example, the original signal is segmented into 1024-length windows, and 480 samples are generated for each fault category. With four categories in total, there are 1920 samples. The data are split into a training set and a test set with an 8:2 ratio, resulting in 1536 training samples and 384 test samples. The bearing condition labels and corresponding information are shown in Table 8 below. To verify the applicability of the IGTO-VMD-PE-IGTO-LSTM model on this dataset, it was compared with a series of benchmark models, including VMD-PE-LSTM, VMD-PE-IGTO-LSTM, GTO-VMD-PE-LSTM, IGTO-VMD-PE-LSTM, IGTO-VMD-PE-PSO-LSTM, IGTO-VMD-PE-GWO-LSTM, IGTO-VMD-PE-CDO-LSTM, IGTO-VMD-PE-GTO-LSTM, GWO-VMD-PE-IGTO-LSTM, PSO-VMD-PE-IGTO-LSTM, GTO-VMD-PE-IGTO-LSTM, and CDO-VMD-PE-IGTO-LSTM. The parameter settings used were consistent with those described in Section 4.2.