## 4 Fault Experiment Signal Analysis
### Paderborn University (PU) Dataset
#### 4.2.2 Experimental Validation in Noisy Environments  
To validate the proposed method's efficacy against noise, Gaussian white noise was introduced into the bearing vibration signal, creating noisy datasets at \(-16\) dB, \(-12\) dB, \(-8\) dB, \(-4\) dB, and \(0\) dB. The performance of the traditional CNN and MSCNN and the proposed MRCNN were compared, both with and without optimized VME parameters. The results are shown in Figure 23. At an SNR of \(0\) dB, the accuracy levels of the six methods are comparable, each surpassing \(95\%\) accuracy. However, with a further reduction in the SNR, the benefits of parameter-optimized VME preprocessing become increasingly evident. Additionally, the proposed MRCNN demonstrates higher recognition accuracy and greater robustness compared to the traditional CNN and MSCNN.  
The impacts of parameter-optimized VME and empirical signal preprocessing on MRCNN training were compared using a fixed value of \(1000\) and a frequency corresponding to the highest spectral peak. ORF2 is used as an example, as shown in Figure 24. Figure 24 offers a graphical representation of the initial data. These points, indicative of different bearing faults and distinguished by color, are dispersed and difficult to identify. In Figure 24b-d, the outcomes for the three separate branches are visually represented, showing a greater clustering of points within the same category. In Figure 24e, the outcomes post-global average pooling are displayed, highlighting the clustering of points within the same category with minimal outliers, showcasing the MRCNN's proficiency in extracting features. Table 10 displays the center frequencies associated with each fault type.  
Figure 23: Comparison of IVME-MRCNN with other methods.  
Figure 22: The confusion matrix without noise.  
Table 11 presents the comparison results for parameter-optimized VME and empirical VME within the MRCNN model. The choice of \(\alpha\) for Empirical VME is the same as by [PERSON] et al. [26], which is 1000. The frequency value corresponding to the highest spectral peak takes ORF2 in Figure 25 as an example. Under noise-free conditions and at 0 dB and \(-4\) dB SNR levels, the accuracy of empirical VME slightly surpasses that of parameter-optimized VME. However, at lower SNR levels, parameter-optimized VME demonstrates greater robustness than empirical VME, maintaining an accuracy of 94.4 \(\pm\) 0.4 (%) at \(-16\) dB. In contrast, the accuracy of empirical VME drops to 77.3 \(\pm\) 0.9 (%) at the same SNR level. Figure 26 illustrates the confusion matrixes for both methods at \(-16\) dB, and it is demonstrated that parameter-optimized VME is significantly more effective in high-noise environments.  
To further assess the impact of various signal preprocessing methods on the outcomes, the approach depicted in Figure 17 was applied. The results are shown in Figure 27. At no noise, \(0\) dB, and \(-4\) dB, parameter-optimized VMD shows higher accuracy than parameter-optimized VME; however, VME is less time consuming and better suited to preprocessing  
Figure 26: Confusion matrixes at \(-16\) dB: (**a**) parameter-optimized VME; (**b**) empirical VME.  
Figure 25: The frequency values corresponding to the highest spectral peak of ORF2.  
multiple signal samples. At \(-8\) dB and below, VME outperforms the other algorithms in terms of accuracy. Thus, parameter-optimized VME emerges as a more advantageous method for signal preprocessing.