# Rolling Bearing Incipient Fault Detection Based on a Multi-Resolution Singular Value Decomposition
## 2 Overview of the Method of Multi-Resolution Singular Value Decomposition
### Principle of the SVD Matrix's Dichotomy Recursive Algorithm  
Presuming that the (\(j\)-1)th MRSVD decomposition of the given signal \(\mathrm{A}_{0}=(x_{1},x_{2},\ldots\ x_{N})\) has been carried out, the (\(j\)-1)th SVD approximate signal will have also been obtained and denoted as \(\mathrm{A}_{j-1}=(a_{j-1,1},a_{j-1,2},\ldots\ n\ a_{j-1,N})\). A 2-row Hankel matrix \(\mathrm{H}_{j}\) can be constructed, i.e., \(\mathrm{H}_{j}=\left(\begin{array}{c}a_{j-1,1},a_{j-1,2},\cdots,a_{j-1,N-1} \\ x_{j-1,2},x_{j-1,3},\cdots,x_{j-1,N}\end{array}\right)\).  
The SVD result of the matrix \(\mathrm{H}_{j}\) can be expressed as  
\[\mathrm{H}_{j}=\mathrm{U}_{j}\ S_{j}V_{j} \tag{1}\]  
where \(\mathrm{U}_{j}=(u_{j1},u_{j2})\), \(\mathrm{U}_{j}\in\mathrm{R}^{2\times 2}\), \(\mathrm{V}_{j}=(v_{j1},v_{j2},\ldots,v_{j(N-1)})\), \(\mathrm{V}_{j}\in\mathrm{R}^{(N-1)\times(N-1)}\), and \(\mathrm{U}_{j}\) and \(\mathrm{V}_{j}\) are the left and right orthogonal matrices obtained by the \(j\)th decomposition. \(\mathrm{S}_{j}=(\mathrm{diag}(\sigma_{aj},\ \sigma_{dj}),0)\), \(\mathrm{S}_{j}\in\mathrm{R}^{2\times(N-1)}\), and \(\sigma_{aj}\) and \(\sigma_{dj}\) are the approximate singular value and the detail singular value obtained using the \(j\)th decomposition.  
In order to obtain the SVD approximate signal \(\mathrm{A}_{j}\) and the detail signal \(\mathrm{D}_{j}\) for the \(j\)th decomposition, \(\mathrm{H}_{j}\) in the formula (1) can be expressed as the following form.  
\[\mathrm{H}_{j}=\sigma_{aj}u_{j1}v_{j1}^{T}+\sigma_{aj}u_{j2}v_{j2}^{T} \tag{2}\]  
where \(u_{j1}\in\mathrm{R}^{2\times 1}\), \(v_{j1}\in\mathrm{R}^{(N-1)\times 1}\), and \(i=1,2\). Denoting \(\mathrm{H}_{aj}=\sigma_{aj}u_{j1}v_{j1}^{T}\), then \(\mathrm{H}_{aj}\in\mathrm{R}^{2\times(N-1)}\) and \(\mathrm{H}_{aj}\) corresponds to the larger singular value \(\sigma_{aj}\), which reflects the body of the original signal, so it is called the approximation matrix. In addition, \(\mathrm{H}_{dj}=\sigma_{dj}u_{j2}v_{j2}^{T}\), \(\mathrm{H}_{dj}\in\mathrm{R}^{2\times(N-1)}\), and \(\mathrm{H}_{dj}\) corresponds to the smaller singular value \(\sigma_{d}\), which reflects the detail of the original signal, so it is called the detail matrix.  
Figure 1: The multi-resolution singular value decomposition (MRSVD) process of a given signal.  
The SVD approximate signal A\({}_{j}\) and the detail signal D\({}_{j}\) are obtained from H\({}_{aj}\)_and_ H\({}_{dj}\), respectively, and the specific process is described as follows.  
Both of the matrixes, H\({}_{aj}\) and H\({}_{dj}\), can be expressed in the form of row vectors, i.e., H\({}_{aj}\)\(=\)\(\left(\begin{array}{c}\sigma_{a}\mu_{j,1},v_{j1}^{T}\\ \sigma_{a}\mu_{j,2}v_{j1}^{T}\end{array}\right)\) and H\({}_{dj}\)\(=\)\(\left(\begin{array}{c}\sigma_{dj}\mu_{j,2}v_{j2}^{T}\\ \sigma_{dj}\mu_{j,2}v_{j2}^{T}\end{array}\right)\). The specific row vector forms of the H\({}_{aj}\) and H\({}_{dj}\) matrixes are displayed in Figure 2a,b.  
Observably, both L\({}_{a,1}\) and L\({}_{a,2}\), as shown in Figure 2a, represent the data points \(a_{j,1}\), \(a_{j,2}\),..., \(a_{j,N-1}\) of the SVD approximate signal A\({}_{j}\). However, L\({}_{a,1}\) is not equal to L\({}_{a,2}\). In this study, all of the elements in matrix H\({}_{aj}\) that represent the same data in signal A\({}_{j}\) are averaged and the acquired mean value of the corresponding data of A\({}_{j}\) is used. Therefore, the SVD approximate signal A\({}_{j}\) is obtained using Formula (3).  
\[\text{A}_{j}=(a_{j,1},(\text{L}_{a,1}+\text{L}_{a,2})/2,\,a_{j,N}) \tag{3}\]  
In a similar manner to the above description, the SVD detail signal D\({}_{j}\) is obtained using Formula (4).  
\[\text{D}_{j}=(d_{j,1},(\text{L}_{d,1}+\text{L}_{d,2})/2,\,d_{j,N}) \tag{4}\]