# Bearing Fault Diagnosis Method Based on Improved VMD and Parallel Hybrid Neural Network
## 5 Discussion  
### Comparison of Denoising Experiments  
To prove the advantages of the improved VMD method, Gaussian noise with signal-to-noise ratios (SNRs) of 5 dB, 10 dB, and 15 dB are added to the fault signal, respectively, and fast Fourier transform (FFT), WT, WOA-VMD, Kurtosis-VMD, EWT, and the method in  
Figure 11: Fault diagnosis results. Figure 10: Training error curve. this paper were employed for denoising, respectively. The SNR and the Pearson correlation coefficient between the denoised signal and the fault signal were employed as evaluation indexes. The average values obtained after ten experiments are shown in Table 5 below:  
SNR is the ratio of relative strength or energy between signal and noise. It is a common indicator for evaluating signal quality and noise level, and the formula is  
\[\text{SNR}=10 log_{10}(signal/noise) \tag{14}\]  
where signal and noise represent their power, respectively. Pearson correlation coefficient (_use r instead_) can express the linear correlation degree of two variables, that is, the intensity and direction of their linear relationship, and the expression is  
\[r=\frac{\sum\limits_{i=1}^{i=1}\left(X_{i}-\overline{X}\right)\left(Y_{i}- \overline{Y}\right)}{\sqrt{\sum\limits_{i=1}^{i=1}\left(X_{i}-\overline{X} \right)^{2}\sqrt{\sum\limits_{i=1}^{i=1}\left(Y_{i}-\overline{Y}\right)^{2}}}} \tag{15}\]  
where \(r\) is the Pearson correlation coefficient, \(X\) and \(Y\) are two variables, and \(\overline{X}\) and \(\overline{Y}\) represent the average of \(X\) and \(Y\), respectively. In Table 5, the denoising results of various methods are compared using signal-to-noise ratios and the Pearson correlation coefficient. A large SNR value means that the strength of the signal is higher than that of the noise. Therefore, a larger SNR usually indicates a better denoising effect. Also, \(r\) is used to measure the linear correlation between the two variables, so a larger \(r\) value means that the higher the similarity between the denoised signal and the original signal, the better the denoising effect. The denoising effect of FFT is not ideal, and SNR and \(r\) are basically not improved compared with those before denoising. The denoising effect of WT and WOA-VMD is obvious, and their SNR and R have been improved in 5 dB, 10 dB, and 15 dB Gaussian noise environments. At the same time, compared with WT, WOA-VMD is better in a 15 dB noise environment, indicating that VMD has certain advantages over other denoising methods in a low noise environment. Compared with WOA-VMD, the kurtosis value VMD method has little difference in the denoising effect, but the SNR and Pearson correlation coefficient are lower than those before denoising under 15 dB noise, and there is a negative increase, which may be due to the impact of some strong impact in the noise signal, resulting in improper decomposition. The EWT algorithm has achieved a good denoising effect in three kinds of noise cases, which is better than the WT algorithm and has little difference with the method in this paper. Its calculation is faster and suitable for real-time processing. The RK-VMD proposed in this paper has the highest signal-to-noise ratio and Pearson correlation coefficient under three kinds of noise conditions, which represents the best denoising effect. It can remove \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & **SNR (dB)** & \(r\) **(\%)** & **SNR (dB)** & \(r\) **(\%)** & **SNR (dB)** & \(r\) **(\%)** \\ \hline Noise Signal & 5 & 86.84 & 10 & 95.21 & 15 & 98.43 \\ FFT & 6.21 & 87.85 & 10.41 & 95.24 & 15.25 & 98.45 \\ WT & 8.93 & 93.26 & 12.65 & 97.17 & 16.65 & 98.88 \\ WOA-VMD & 8.58 & 92.93 & 13.74 & 97.86 & 18.20 & 98.96 \\ Kurtosis–VMD & 8.6 & 92.80 & 13.23 & 97.70 & 13.59 & 98.06 \\ EWT & 9.04 & 93.58 & 13.71 & 97.88 & 17.57 & 98.92 \\ Proposed method & 9.13 & 93.64 & 14.5 & 98.20 & 18.89 & 99.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Denoising results of various methods.