## 2 Methodology and Principles
### The Principle of Singular Value Decomposition
#### 2.2.2 SVD Noise Reduction Order Determination  
In the process of singular value decomposition and the reconstruction of each order IMF after decomposing the vibration signal, it is necessary to select the first few singular values that can represent the characteristics of the effective signal in a reasonable manner. If too few orders are selected, some of the feature information of the effective signal will be lost; if too many orders are selected, it will lead to residual noise components and fail to achieve the purpose of noise reduction [33; 34]. Therefore, the key to noise reduction by the SVD method is to select the optimal noise reduction order. To reasonably select the noise reduction order for SVD noise reduction, a judgment basis needs to be defined. The selection of the noise reduction order often adopts the trial-and-error method and the threshold method, which relies heavily on the user's experience and lacks a theoretical basis for the selection of the reconstruction order; the adaptability of the noise reduction method is poor, although it has achieved some results on some feature sets. [PERSON] et al. [20] proposed an SVD noise reduction method using the singular value difference spectrum as a criterion and achieved better results. [PERSON] [21] and [PERSON] et al. [22]separately made some improvements on its judgement basis, proposed the concept of energy difference spectrum, and normalized it with the second-order moments of singular values; its mathematical expression can be defined as:  
\[Q_{0}(\text{i})= \lambda_{i}-\lambda_{i+1}\qquad\qquad\qquad(i=1,2,\ldots,p) \tag{22}\] \[Q_{1}(i)=\frac{\sqrt{(\lambda_{i}^{4}-\lambda_{i+1}^{4})}}{\frac {1}{p}\frac{p}{i}\lambda_{i}^{2}} \tag{23}\]  
where \(\lambda_{i}\) is the ith singular value obtained after SVD decomposition; all the values of \(Q_{o}(i)\) are arranged in the order of \(i\) as the singular value difference spectrum; all the values of \(Q_{1}(i)\) are arranged in the order of \(i\) as the energy difference spectrum; \(p\) is the number of singular values after SVD decomposition. When the difference between two neighboring singular values is large, a distinct spectral peak is formed in the energy difference spectrum. When \(x\geq y\geq 0\), there is \(\sqrt{x^{4}-y^{4}}\geq x^{2}-y^{2}\); it can be concluded that the square root of the fourth-order energy difference spectrum provides a better reflection of the characteristics of neighboring singular value variations compared to the second-order energy difference spectrum, and the fluctuation of its spectral value is larger than that of the difference spectrum. According to Equations (19)-(21), the Hankel matrix of SVD can be expressed as:  
\[H=U\Lambda V^{\text{T}}=\sum_{i=1}^{p}\lambda_{i}u_{i}V_{i}^{\text{T}}=\sum_{ i=1}^{p}\lambda_{i}A_{i} \tag{24}\]  
According to Equation (24), the Hankel matrix is the sum of the products of the singular values of each order and the corresponding eigenvectors. The larger the singular values, the larger the proportion of the corresponding eigenvectors in the reconstructed signal. From Equation (23), it can be seen that there is a maximum spectral peak in the computed energy difference spectrum, and since it is arranged from the largest to the smallest, it indicates that the variation of the singular value is the largest at the singular value corresponding to the spectral peak. Due to the uniform characteristics of noise in the time-frequency domain, its contribution to the singular values after SVD decomposition is almost equal. Therefore, the singular values before the spectral peak in the energy spectrum are considered as corresponding to the contribution of the effective signal, while the smaller spectral values with flat and near-zero changes are considered as corresponding to the contribution of noise to the singular values [35].