## 3 Neighborhood Statistical De-Noising Method  
Although the high-frequency modes extracted by the optimal VMD can well eliminate the deterministic interference components, the random and uncertain noises still exist and are generally sparse, oscillatory and smooth. Therefore, the subsequent de-noising strategies must be thoroughly studied to enhance the fault characteristics.  
Different from the classical wavelet transform, the SGWT is implemented based on the lifting scheme according to data content [29; 30], which is very suitable for mechanical fault diagnosis. Ref. [31] proposed the following threshold scheme for wavelet de-noising. If \(D_{j,n}^{2}=d_{j,n-1}^{2}+d_{j,n}^{2}+d_{j,n+1}^{2}\) is less than or equal to \(\lambda^{2}\), then the wavelet coefficient \(d_{j,n}\) is set to zero. Otherwise, it is set to  
\[d_{j,n}=d_{j,n}\left(1-\frac{\lambda^{2}}{D_{j,n}^{2}}\right) \tag{19}\]  
Since the reconstruction of thresholded wavelet coefficients will be based on the true nature of the impulses and local fluctuation details for signals with low SNR, it is necessary to carefully determine \(\lambda\) from the perspective of statistical data modeling. In reality, many problems in engineering diagnosis can be successfully solved by mathematical statistics. The actual test signal inevitably contains noise, and we hope to recover the useful signal as accurately as possible according to some criteria. The problem can be formulated as:  
\[y=\omega+n \tag{20}\]  
where \(y\), \(\omega\) and \(n\) represent noisy data, true data and noise, respectively.  
In Bayesian statistical theory, maximum a posteriori (MAP) estimation is a classical method. MAP estimation is to find the \(\omega\) that maximizes the posterior probability density under the condition of given observation data \(y\):  
\[\begin{array}{rl}\widetilde{\omega}(y)&=\underset{\omega}{\text{argmax}} \Big{[}p_{y|\omega}(y|\omega)\cdot p_{\omega}(\omega)\Big{]}\\ &=\underset{\omega}{\text{argmax}}[p_{n}(y-\omega)\cdot p_{\omega}(\omega)] \end{array} \tag{21}\]  
When the noise is additive and independent, as in (20), the MAP estimator is given by:  
\[\widetilde{\omega}(y)=\underset{\omega}{\text{argmax}}[\log(p_{n}(y-\omega) )+\log(p_{\omega}(\omega))] \tag{22}\]  
Let \(f(\omega)=\log(p_{\omega}(\omega))\), when the noise is Gaussian distribution, the above formula can be expressed as:  
\[\frac{y-\widehat{\omega}}{\sigma_{n}^{2}}+f^{{}^{\prime}}\Big{(}\widetilde{ \omega}\Big{)}=0 \tag{23}\]  
where \(\sigma_{n}\) denotes the standard deviation of the noise, which can be approximated using the mean absolute deviation of the values in \(y\). It is worth stating that \(p_{\omega}(\omega)\) should meet the joint characteristics of sparsity, oscillation and spikes so as to effectively serve the purpose of noise removal and fault-signal preservation. Next, we use the FE to write a new adaptive Laplacian PDF to describe the structure of the fault signal, which can concurrently quantifythe noise morphology and extract the periodic sparse components. The PDF of the best failure mode is given by:  
\[p_{\omega}(\omega)=\frac{1}{FE\cdot\sigma}\exp\biggl{(}-\frac{FE\cdot|\omega|}{ \sigma}\biggr{)} \tag{24}\]  
Here, \(FE\) stand for the fluctuation entropy of the input signal and \(\sigma\) denotes the standard deviation of true data. Since the data is independent of noise, the estimate of the noiseless data standard deviation is \(\sigma=\sqrt{\sigma_{y}^{2}-\sigma_{n}^{2}}\). In addition, when the square root is imaginary, \(\sigma\) is set to zero.  
Consequently, \(f(\omega)=-\log(FE\cdot\sigma)-FE\cdot|\omega|/\sigma\) and the threshold \(\lambda\) will be:  
\[\lambda=FE\cdot\sigma_{n}^{2}/\sigma \tag{25}\]