## 2 Principle of Noise Reduction Based on Singular Value Decomposition (SVD)
### Curvature Spectrum of Incremental Singular Entropy  
Incremental singular entropy usually indicates the amount of information about the singular value that is contained by the corresponding component in the signal. The pure signal information focuses on the previous singular values whose incremental singular entropy is much larger than others. With the increase in the number of singular values, the incremental singular entropy gradually decreases to a stable one. In contrast, the amount of information is less in noise, and no such distinct variation appears in the spectrum of incremental singular entropy. As a result, a turning point indicates that the state of the spectrum of incremental singular entropy has experienced a drastic change that can be used to distinguish the useful signal from the noise. Incremental values of singular entropy are defined as follows [23]:  
\[\Delta E_{i}=-(\alpha_{i}/\sum_{j=1}^{q}\alpha_{j})\log(\alpha_{i}/\sum_{j=1} ^{q}\alpha_{j})(i=1,2,3\ldots p) \tag{8}\]  
where \(\Delta E_{i}\) is the \(i_{\text{th}}\) incremental singular entropy, and the sequence composed of \(\Delta E_{i}\) is the spectrum of incremental singular entropy. The curve of incremental singular entropy is composed of a series of discrete incremental singular entropy points. The incremental singular entropy constructs a sequence in order from large to small, which is defined in Equation (9).  
\[\Delta E=\left\{\Delta E_{1},\Delta E_{2},\Delta E_{3}\cdots\Delta E_{p}\right\} \tag{9}\]  
Curvature is the division of tangent angle to rotation about the arc length on a certain point of the curve. Curvature shows the degree to which the curve deviates from a straight line. The definition of curvature type is as follows:  
\[C=\frac{|y^{\prime\prime}|}{\left(1+y^{\prime}\right)^{2}} \tag{10}\]  
Curvature expresses the sensitivity to changes of state; for the pure signal, there is a large curvature peak where the state of curve undergoes an evident change. This peak can be used to reflect the changing states of the incremental singular entropy sequence. Because the incremental singular entropy sequence is discrete, the difference was used to approximate the derivative, which is defined in Equations (11) and (12) [20].  
\[y^{\prime\prime} =\Delta E_{i+1}-2\Delta E_{i}+\Delta E_{i-1}(i=1,2,3\ldots p-1) \tag{11}\] \[y^{\prime} =\left\{\begin{array}{c}\Delta E_{i+1}-\Delta E_{i}\\ \Delta E_{i}-\Delta E_{i-1}\end{array}\right.\quad(i=1,2,3\ldots p-1) \tag{12}\]  
For a pure signal, the variation of the curve is continuous, and the forward difference operator has the same effect as the backward difference operator. However, the variation of the curve becomes discontinuous after adding the noise to the pure signal, which influences the results, while choosing an improper type of difference operator [20]. Thus, the fraction value is inversely proportional to that of the denominator, which is composed of the square of first-order difference values. The deviation of incremental singular entropy resulting from improper first-order difference values may cause negative effects for de-noising. As a result, this paper calculates the forward and backward difference of every singular value and chooses the absolute minimum of difference to substitute the first derivative in Equation (10). The curvature spectrum can be described as follows:  
\[\text{CE}_{i}=\frac{|y^{\prime\prime}|}{\left(1+y_{i}^{\prime 2}\right)^{1.5}}(y^{ \prime}_{i}=\min(|\Delta E_{i+1}-\Delta E_{i}|,|\Delta E_{i}-\Delta E_{i-1}|),i=1,2,3\ldots p-1) \tag{13}\]