## 2 Principle of Noise Reduction Based on Singular Value Decomposition (SVD)
### Improved Method Based on the Curvature Spectrum of Incremental Singular Entropy  
Curvature has better sensitivity to changes in curves and can be used to check for turning points. In addition, the degree of bend of a curve can be more accurately measured by curvature than difference. Thus, the curvature spectrum of incremental singular entropy was chosen for further analysis. The choice of an effective number of singular values is related to the integrity of the de-noised signal. An improper number will lead to the distortion of the de-noised signal. In the past, the effective number corresponded to the position of the maximum peak in the differential or curvature spectrum. In addition to the maximum peak, the other larger peaks also reveal the change of the spectrum as well as the difference of correlation between the useful signal and noise signal. Selecting a maximum peak while ignoring other larger peaks usually leads to loss of information about the useful signal. As a result, it is necessary to focus on other peaks and discuss the diverse changes about these peaks of noise and the pure signal. The pure signal can be extracted from noise due to a different variation of the spectrum of incremental singular entropy between the noise and pure signal; there is a more incisive peak in the curvature spectrum of a pure signal but none of noise. In other words, the difference between two adjacent curvature peaks at a turning point is significantly greater than the one at the other position (for details, see Section 3.1). In this paper, a new method is proposed to extract the effective singular values based on the difference spectrum of curvature. At first, all of the curvature peaks were extracted and the differences between two adjacent peaks were calculated. Then a difference sequence was obtained to reflect different variation trends of the curvature peaks. In order to reduce the influence of the forward and backward difference on the result due to the discontinuity of the curve, the absolute minimum difference was singled out for further analysis. According to the theoretical analysis, effective singular values are often concentrated in the front of the spectrum. This paper compared the difference between two adjacent curvature peaks from left to right, and selected the peak where the difference declines in an infinitely large degree for the first time to determine the number of effective singular values. It is worth noting that the concavity and convexity of curves needs to be considered in selecting the number of effective singular values based on curvature peaks. Supposing that the number of effective singular values is \(r\), if the spectrum curve is convex at the s point, we will select the first \(r\) effective singular value. Otherwise, we will select the first \(r-1\) effective singular value [20]. Similarly, the concavity and convexity of the curve at the curvature peak should be considered while determining the effective singular values based on the change of differential curvature peaks; the strategy is described above. In fact, the changes in the difference of the adjacent curvature peaks is equivalent to the fourth-order singular spectrum, and the difference spectrum and curvature spectrum amount to the first and second order of the singular sequence. For a smooth and continuous singular curve, we can use [PERSON]'s theorem to decompose curve function into the superposition of a different-order derivative of the singular point, and the larger bending curve could be described by a high-order spectrum. The major difference between pure signal and noise is that an obvious turning point exists in the spectrum of the pure signal but not in noise. When the pure signal is contaminated by the noise, the trend of the spectrum becomes flat, which makes it hard to precisely identify the turning point by the first order or second order. Consequently, the change level of curvature difference is proposed in this paper to orientate the turning point in the spectrum of incremental singular entropy.