# Weak Fault Detection of Tapered Rolling Bearing Based on Penalty Regularization Approach
## 2 Augmented Huber Non-Convex Penalty Regularization
### Convexity Condition
For the selection of the regularization parameter \(\lambda\), we have  
\[\lambda=\gamma\times\beta_{0}\times\sigma \tag{22}\]  
where \(\sigma\) is the standard deviation (SD) of the additive noise, \(\beta_{0}\) is the constant so as to maximize the signal-to-noise ratio (SNR). Here, parameters \(\beta_{0}\) and \(\gamma\) are typically set to be constant values, i.e., \(\beta_{0}=[0.5,1]\), \(\gamma=[7.5,8]\). In practice, the standard deviation (SD) of the background noise in Equation  
Figure 3: The noise-free signal, simulated systematic signal and its frequency spectrum: (**a**) Noise-free signal; (**b**) simulated systematic signal; (**c**) frequency spectrum of simulated systematic signal. (22) can be computed using the fault signal and healthy data under the same operating environment. Moreover, when the healthy data are not available or are unknown, the standard deviation (SD) of background noise can still be estimated by the following formula:  
\[\hat{\sigma}=MAD(y)/0.6745 \tag{23}\]  
which is a traditional estimator of the noise level used for wavelet de-noising [33], where \(MAD(y)\) represents the median absolute deviation (MAD) of signal \(y\), i.e.,  
\[MAD(y)=\text{median}(|y_{i}-\text{median}(y)|),i=1,2,\ldots,N \tag{24}\]  
As an example, if we take the constant parameters to be \(\beta_{0}\) = 0.7 and \(\gamma\) = 7.5, with the regularization parameter \(\lambda\) = 7.5 \(\times\) 0.7 \(\times\) 1.5 = 7.875, the solutions of LFLO and proposed AHNPR method with parameter \(\tau\) = 0.8 are shown in Figure 5. Comparing the results of the LFLO and proposed AHNPR methods, the proposed AHNPR solution is sparser than the results of LFLO in frequency domain (see Figure 5c) and the LFLO underestimates significant coefficients of the underlying signal. Figure 4: The root-mean-square error (RMSE) values and optimized Fourier coefficients for the parameter \(\tau\), varying from 0.05 to 0.95. (**a**) The RMSE values; (**b**) the optimized Fourier coefficients. Figure 5: The de-noising signal generated by different methods with \(\tau\) = 0.8 and \(\lambda\) = 7.875. (**a**) De-noising signal generated by the L1-norm fused lasso optimization (LFLO) method; (**b**) de-noising signal generated by the proposed augmented Huber non-convex penalty regularization (AHNPR) method; (**c**) the Fourier coefficients of de-noising signals. Following this, the proposed AHNPR method is applied to process the simulated synthetic signal. Specifically, matrix \(\mathbf{A}\) is defined as \(\mathbf{A}_{m,n}=(1/\sqrt{N})\exp(j(2\pi/N)mn)\); here, \(m=0,1,\cdots,M-1;\ n=0,1,\cdots,N-1\). Assuming that matrix \(\mathbf{A}^{T}\) is the complex conjugate transpose of matrix \(\mathbf{A}\), it can be calculated that \(\mathbf{A}^{T}\mathbf{A}\) is a diagonal matrix, the parameter \(\tau\) is set to \(\tau=0.8\), and thus matrix \(\mathbf{B}\) is \(B=\sqrt{\tau/\lambda}A\). In addition, taking the constant parameters to be \(\beta_{0}=0.7\) and \(\gamma=7.5\), the parameter \(\lambda\) could be calculated by Equation (22), i.e., \(\lambda=7.5\times 0.7\times 0.4=2.1\). Since \(\mathbf{B}^{T}\mathbf{B}\) is a non-diagonal matrix, the augmented Huber penalty function \(\phi_{B}(x)\) is non-separable. The de-noising results of the LFLO method and proposed AHNPR method are shown in Figure 7a,c, respectively. Figure 7b,d are the corresponding Morlet wavelet time-frequency diagrams of Figure 7a,c, respectively. It can be noted that the periodic characteristics in Figure 7c are more obvious than in Figure 7a. The sparse Fourier coefficients of LFLO and the proposed AHNPR method are shown in Figure 8a,b. The non-convex penalty function method aims to induce sparsity in sparse coefficients more effectively than the LFLO method. Moreover, the LFLO method underestimates the significant (large-amplitude) sparse coefficients of the de-noised signal.