## 2 Methodology
### Multiscale Local Projection Algorithm  
According to the steps of the local projection noise reduction algorithm (3), [PERSON] et al. (2014) used the mean value of the neighborhood point as the neighborhood centroid as follows:  
\[\overline{\mathbf{X}}_{n}=\frac{1}{\mu_{n}}\sum_{k\in\mu_{n}}\mathbf{X}_{k}. \tag{3}\]  
As shown in Figure 1, it is considered that points gather around a curve, and the local linear approximations (straight lines) are not tangents but secants. All the centroids (circles) of different neighborhoods are shifted inward with respect to the curvature, which has a great effect on the performance of noise reduction. To further suppress the noise, [PERSON] et al. (2015) estimated the neighborhood centroid using the second order polynomial. \[\overline{\mathbf{X}}_{n}^{\prime}=2\overline{\mathbf{X}}_{n}-\frac{1}{\mu_ {n}}\sum_{k\in\mu_{n}}\overline{\mathbf{X}}_{k}. \tag{4}\]  
As shown in Figure 2, a tangent approximation is obtained by shifting the centroid outward with respect to the curvature. The filled circle denotes the average of the centroid of adjacent neighborhoods, where the square is the corrected centroid \(\overline{\mathbf{X}}_{n}^{\prime}\). In this article, we call it the standard local projection algorithm. [PERSON] et al. (2017) considered both the geometrical shape and statistical error of the signal in the high dimensional phase space, and proposed a multiscale selection method of the centroid, which cannot only effectively eliminate the noise, but can also better preserve the complete geometric structure of the attractors. It was confirmed that the method had a good effect on medical signals. The basic principles are as follows:  
Figure 1: Schematic diagram of local linear approximations. Figure 2: Improved method by [PERSON] (2015). For real numbers \(\delta>0\), the continuous moving average operator can be described as \(I_{\delta}\) by:  
\[(I_{\delta}f)(x)=\frac{1}{2\delta}\int_{x-\delta}^{x+\delta}f(t)dt. \tag{5}\]  
[PERSON] et al. [15] considered a linear combination of the centroid of the first and second order, to achieve a more accurate estimate of \(m\)-dimensional of the original signal. In analogy, it requires the equation below to hold for all monomials \(f_{n}(x)=1\) (\(n=0\)) and \(f_{n}(x)=x^{n}\) (\(x\
eq 0\)) with \(0\leq n\leq n_{max}\), and with \(n_{max}\) as large as possible. \[f_{n}(0)=a_{1}(I_{\delta}f_{n})(0)+a_{2}({I_{\delta}}^{2}f_{n})(0). \tag{6}\]  
For odd \(n\), both the sides of (6) are zero. Letting \(n=0\) and \(n=2\) in (6) yields a system of two linear equations with the unique solution: \(a_{1}=2\), \(a_{2}=-1\), and then identifies the \(m\)-dimensional analogue of the operator \(I_{\delta}\) with the centroid of points within a ball of radius \(\delta\) in Euclidean \(m\)-space, resulting in the second-order local projection algorithm. As for Equation (6), the geometric structure of the attractor is taken into consideration, so the scale factor \(\beta\) is introduced. It is eligible to represent \(f_{n}(0)\) by using a linear combination of iterated moving averages of \(f_{n}\) evaluated at zero for integers \(n\), such that \(0\leq n\leq 5\) without using moving averages of an order greater than two. It requires the following equation:  
\[f_{n}(0)=a_{2}({I_{\delta}}^{2}f_{n})(0)+b_{1}(I_{\beta\delta}f_{n})(0), \tag{7}\]  
which is suitable for the situation with integers \(n\) such that \(0\leq n\leq 5\) (\(\beta\geq 0\)). It gives two solutions: \(a_{2}=0\), \(b_{1}=1\), \(\beta=0\), and \(a_{2}=4\), \(b_{1}=-3\), \(\beta=\sqrt{8/3}\). Then, the corresponding coefficient of the standard local projection algorithm is: \(a_{2}=0\), \(b_{1}=1\), \(\beta=1\). Then, the aspects of statistical error are analyzed. For non-negative integer \(d\), define the discrete moving average operator \(J_{d}\) by the following equation:  
\[(J_{d}f)(j)=\frac{1}{2d+1}\sum_{i=-d}^{d}f(j+i). \tag{8}\]  
Let \(e(-2)\), \(e(-1)\), \(e(0)\), \(e(1)\), \(e(2)\),... \(\in R\) be independent random variables with mean zero and unit variance.