# Sparse Low-Rank Based Signal Analysis Method for Bearing Fault Feature Extraction
## 3 The Proposed Fault Information-Based Sparse Low-Rank Algorithm
### Correlated Kurtosis-Based Thresholding (CKT) Scheme Proposed in This Paper  
As mentioned above, the problem in (6) can be solved easily by applying the SVT. But as stated in Equation (11), SVT preserves the SVs greater than \(\lambda_{0}\) and the other SVs are set to zero [37]. This means that the SVT preserves components with large SVs in the matrix. However, it is worth noting that components with large SVs are not necessarily rich in information, especially when the repetitive impulses are weak. Therefore, when the bearing's fault features are weak or there are other large interferences such as HCs in the signal, SVT tend to ignore the fault features. Based on the above analysis, in the proposed FISLRA, the CKT scheme is used to instead SVT to save the components with a large amount of fault information. In the CKT method, the correlation kurtosis (CK) which is extensively applied to the fault diagnosis is used as an indicator for measuring the amount of fault information.  
\[\text{CK}_{M}(T)=\frac{\sum_{n=1}^{N}\bigl{(}\Pi_{m=0}^{M}y_{n-mT}\bigr{)}^{2} }{\left(\sum_{n=1}^{N}y_{n}^{2}\right)^{M+1}} \tag{12}\]  
where \(y\) is the signal with length \(N\). In this paper, shift number \(M=2\) is represented and period of interested is denoted by \(T\) which can be easily calculated according to fault characteristic frequency (FCF). A detailed description of CK can be found in [40]. CK measures both the kurtosis and periodicity of the signal, thus it is more conducive for the extraction of repetitive impulses from noisy signal than indicators that only emphasize impact or periodicity.  
\[Z\gets U\bullet CKT(\Sigma,\text{CK},k)\bullet V^{T} \tag{13}\]  
where \(CKT\) (\(\Sigma\), CK, \(k\)) indicates that the SVs with a large amount of fault information are retained, and others are discarded; \(k\) is the number of the components to be saved. According to simulation analysis and experimental verification, it is recommended to set it to 10%-50% of the number of SVs. In the experimental verification and simulation analysis section of this article, \(k\) is set it to 20% of the number of SVs. In addition, based on the analysis above, the scalar-augmented Lagrangian parameter \(\mu=1.1\) is used in this paper. Non-convex penalty parameter \(a\) is set to \(1/\lambda_{1}\) to maximize the sparsity of the solution while ensuring convexity. Furthermore, the regularization parameter \(\lambda_{1}\) which has a great influence on the processing result of FISLRA must be appropriately selected according to the signal itself. As mentioned above, the purpose of this article is to extract the fault information caused by bearing defects, and CK is an excellent criterion to indicate the amount of fault information, then the parameter \(\lambda_{1}\) that corresponds to the maximum CK of the denoised signal is regarded as the optimal value in this article.  
The algorithm of the proposed FISLRA for extracting fault features is summarized in Algorithm 1.  
```
Require: Raw signal \(y\) needs to be converted into a matrix \(Y\) through the STFT operator \(A^{T}\) and maximum iteration number \(I\) is set. Input: \(Y\), \(\lambda_{1}\), \(\mu\), \(a\), \(k\) Initialization:\(X\), \(Z\), \(D\) for\(i\leq 1\)do
4\(X\leftarrow\arg\!\!\min\limits_{X}\left\{\frac{1}{2}\|\!\!\|_{1+\mu}(Y+\mu(Z+D))- X\|_{F}^{2}+\frac{\lambda_{1}}{1+\mu}\sum\limits_{m_{1}=1}^{M_{1}}\sum\limits_{m_{2}=1}^{M_ {2}}\phi_{\varepsilon}(X_{m_{1},m_{2}};a)\right\}\)
5\([U,\sum,V]\gets SVD(X-D)\)
6 Calculate CK value of each component of \(X\)
7\(Z\gets U\bullet CKT(\Sigma,\text{CK},k)\bullet V^{T}\)
8\(D\gets D-(X-Z)\)
9 End
10 Output:\(X\)
```  
**Algorithm 1** Algorithm of fault information-based sparse low-rank algorithm (FISLRA) for bearing fault feature extraction