## 4 Fault Experiment Signal Analysis
### Case Western Reserve University (CWRU) Dataset
#### 4.1.3 Experiment Validation in Noisy Environments  
To further validate the effectiveness of the method in noisy environments, Gaussian white noise was introduced into the bearing vibration signals. Noisy datasets with signal-to-noise ratios of \(-16\) dB, \(-12\) dB, \(-8\) dB, \(-4\) dB, and \(0\) dB were generated. Figure 14 illustrates the mixed signal resulting from the addition of noise. In order to demonstrate the effectiveness and superiority of the IVME-MRCNN method, we processed the same experimental data with the MRCNN, akin to a traditional CNN and MSCNN, and compared it with the non-parameter-optimized VME method. The  
Figure 14: Mixed signal after adding noise. Figure 12: Training curves: (a) accuracy curve; (b) loss curve. Figure 13: The confusion matrix without noise. outcomes are displayed Figure 15. Parameter-optimized VME significantly enhances the overall accuracy when analyzing noisy bearing signal data, particularly at noise levels of \(-16\) dB and \(-12\) dB. As the SNR decreases, all six methods experience a drop in accuracy. However, the IVME-MRCNN demonstrates superior robustness, maintaining an accuracy of 97.7 \(\pm\) 0.93 (%), even in the \(-16\) dB environment. To enhance the intuitiveness of the feature learning process in the proposed method, t-distributed Stochastic Neighbor Embedding (t-SNE) was employed to transform high-dimensional features into two-dimensional ones, facilitating the visualization of different network layers. The results are shown in Figure 16. Figure 16a provides a visualization of the original data. The points, representing various bearing faults and differentiated by color, are scattered and challenging to distinguish. Figure 16b-d provides a visualization of the results for the three distinct branches, where points within the same category appear more clustered. Figure 16e presents the results after global average pooling, where points in the same category are clustered, with only a few outliers, demonstrating the MRCNN's feature extraction capability. Figure 16: _Cont._  
Figure 15: Comparison of IVME-MRCNN with other methods. To demonstrate the benefits of parameter-optimized VME, we compared it to the approach by [PERSON] et al. [26], who employed a fixed \(\alpha\) value of 1000. The selection was based on the frequency value corresponding to the highest spectral peak in the bearing vibration signal. The selected values are shown in Table 5.  
Table 6 presents the results of the comparison between the parameter-optimized VME and empirical VME, within the MRCNN model. The signal features extracted using parameter-optimized VME are more obvious than those extracted using empirical VME. In the case of no noise, 0 dB and \(-4\) dB, the accuracy of VME using the empirical method is also close to 100%. As the SNR decreases further, the advantages of signal extraction using parameter-optimized VME become more evident. Figure 17 displays confusion matrixes for both methods at \(-16\) dB. The diagnostic accuracy, as shown in the confusion matrixes, is significantly enhanced with parameter-optimized VME. The impact of the signal preprocessing method (VMD, EEMD) on the results were analyzed. VMD employs WCK as the objective function for signal decomposition, selecting the IMF's mode component with the maximum WCK value as the optimal component. In EEMD decomposition, the noise amplitude ratio is set to \(\varepsilon=0.2\) relative to the signal amplitude's standard deviation, with N = 100 trials conducted, and mode component 1 is selected as the principal component. Hence, the results indicate that preprocessing the noisy original vibration signal with parameter-optimized VME can enhance accuracy and save time. Table 7 presents the average decomposition time for each signal preprocessing method. Figure 18 illustrates the results from employing different signal preprocessing methods within the MRCNN. With parameter-optimized VME, the MRCNN demonstrates robustness as the SNR decreases.