# Weak Fault Feature Extraction and Enhancement of Wind Turbine Bearing Based on OCYCBD and SVDD
## 3 Singular Value Decomposition Denoising
### Singular Value Order Determination  
The order of singular value needs to be predefined in denoising process. And the result is closely depended on the selected singular value order. However, this problem is a difficult barrier for lack of priori knowledge about the original signal, which is very complex in actual engineering application. The useful feature will be lost if the selected singular value order is too small using the traditional difference spectrum method, while the excessive redundant noises will be remained if the selected order is too large using the median value or the mean value method [30; 31]. Usually, after performing SVD on the original signal, there exists an elbow in the obtained singular value curve. In the left side of the elbow, the singular values decrease rapidly, while the singular values in the right side of the elbow decrease slowly. Thus, the elbow of singular value curve should be given more attention. The trend lines of singular value distribution can be drawn using fitting method. And the diacritical location of the available and the useless singular values in the elbow can be determined by the degree of closeness with these two trend lines. Then, in order to select the reasonable order of reconstructed matrix during denoising process, a principle based on fitting error minimum is utilized in this paper. The realization processes of singular value order determination are as follows:  
1. The singular value sequence \(Q=[\sigma_{1},\sigma_{2},\cdots,\sigma_{p}]\) and the order sequence \(D=[1,2,\cdots,p]\) are obtained by applying SVD to the given signal using Equations (22) and (23). The first singular value, which corresponding the trend component of the signal, is obvious larger than the others in the singular value sequence. In order to avoid causing large deviation, then the first singular value is removed in the whole process.
2. The initial order \(r\) is confirmed.
3. According to the order \(r\), the original singular value sequence \(Q\) is split into two set of sequences \(Q_{1}=[\sigma_{2},\sigma_{3},\cdots,\sigma_{r}]\) and \(Q_{2}=[\sigma_{r+1},\sigma_{r+2},\cdots,\sigma_{p}]\), and the original order sequence \(D\) is also split into two set of sequences \(D_{1}=[2,3,\cdots,r]\) and \(D_{2}=[r+1,r+2,\cdots,p]\).
4. The order sequences \(D_{1}=[2,3,\cdots,r]\) and \(Q_{2}=[\sigma_{r+1},\sigma_{r+2},\cdots,\sigma_{p}]\) are respectively considered as the independent variables and the singular value sequences \(Q_{1}=[\sigma_{2},\sigma_{3},\cdots,\sigma_{r}]\) and \(Q_{2}=[\sigma_{r+1},\sigma_{r+2},\cdots,\sigma_{p}]\) are respectively regarded as the dependent variables. Then the least squares quadratic polynomial fitting method is utilized to construct the fitting function and. Where \(a_{1}\), \(a_{2}\), \(a_{3}\), \(b_{1}\), \(b_{2}\) and \(b_{3}\) are the polynomial coefficients, \(k=2,3,\cdots,r\) and \(g=r+1,r+2,\cdots,p\) are the independent variables.
5. The fitting error \(\mathrm{FE}_{r}\) corresponding to the order \(r\) is calculated by the following expression: \(\mathrm{FE}_{r}=\sqrt{\sum\limits_{k=2}^{r}{(\sigma_{k}-\varsigma_{1}(k))^{2} }}+\sqrt{\sum\limits_{g=r+1}^{p}{(\sigma_{g}-\varsigma_{2}(g))^{2}}}\).
6. The order \(r\) is reassigned as \(r=r+1\).
7. The steps (3)-(6) are repetitive executed until \(r=p\).
8. The minimum value of fitting error \(\mathrm{FE}_{i}\) is confirmed, then the corresponding order \(i\) is regarded as the diacritical location of the available and the useless singular values. And the former \(i\) singular values except for the first one are applied for purified signal reconstruction.