## 2 Methodology and Principles
### CEEMDAN Principles and Improvements
This process is referred to as adaptive decomposition because the mean curve utilized in each iteration is the residual signal from the previous iteration. The principle of CEEMDAN is to decompose a complex signal into a finite number of IMFs that contain local features of the original signal at different time scales, which effectively improves the mode aliasing phenomenon of the EMD method and solves the problem of the large computational volume of the EEMD and CEEMD methods. However, there are also shortcomings, such as signal endpoint effect, envelope overshooting and undershooting, and so on. The construction of the mean value curve and the iterative sieving process play a decisive role in the computational process of EMD and its derivative methods. Improving these two core components must then be considered to solve the problems of CEEMDAN. To address the endpoint effect problem caused by the cubic spline curve interpolation and fitting process, the signal is considered for extension. Additionally, weighted averaging is performed on neighboring extreme points to even out their distribution and suppress envelope overshooting and undershooting problems. In the iterative sieving process, the mean curve is sieved as completely as possible from the current remaining signal before the mean curve is iteratively updated. Different weights are introduced in the sieving process, orthogonality is used as the basis for measuring whether the sieving is complete or not, and optimal results are selected from the sieving results under different weights to minimize the noise residue in the sieving results. Optimizing each sieving process after several iterations of updating can ensure that the IMF component reaches overall optimization to reduce the existence of false components and improve the accuracy of decomposition. The key to improving CEEMDAN lies in the construction of the mean value curve and the optimization of the iterative sieving process. In terms of mean value curve construction, the improved algorithm is different from CEEMDAN in directly fitting the envelope to the extreme value points, but the weighted average of the adjacent extreme value points is calculated first, and then the envelope is fitted to them. In addition, the improved method also performs end-point extension on the original signal and removes the extension part after decomposing the result to avoid the influence of the end-point effect. The specific steps of the method are as follows:  
Let the original signal be \(X(t)\) and the \(m\) times white noise sequence added to the signal be \(\omega^{n}(t)\); define the operator \(E_{p}(o)\) to be the \(p\)th IMF of the EMD decomposition, and write \(\overline{\text{Im}f_{i}}\) to be the \(I\) component of the improved CEEMDAN decomposition. 1. As in CEEMDAN, a noise-containing time-domain signal sequence \(x^{n}(t)\) is constructed first. Construct a white noise time-domain signal sequence \(\omega^{n}(t)\) with the standard deviation \(\varepsilon_{o}\). Combine the white noise signal with the original signal \(X(t)\) to combine the new signal as shown in the following equation: \[X^{n}(t)=X(t)+\frac{N_{std}}{\varepsilon_{0}}E_{i}(\omega^{n}(t))\qquad(n=1,2,\ldots,K;\;i=1,2,\ldots,I)\] (9)where \(N_{std}\) is the noise standard deviation, and the general case sets \(N_{std}=\epsilon_{o}\). 2. Perform end-point extension on the signal \(x^{n}(t)\) to obtain the extended signal \(x^{n}(t)\), discriminate all the extreme points of the extended signal, denoted as \((\tau_{k},X_{k})\), \(k=1,2,\ldots,K\), and introduce the weighting factor \(\alpha\) and define \(m_{k}\) as follows: \[m_{k}=\frac{X_{k-1}+\alpha X_{k}+X_{k+1}}{\alpha+2}\hskip 56.905512 pt(k=2,3, \ldots,K-1)\] (10) For the extreme points \(m_{1}\) and \(m_{k}\) at the two ends of the original signal, the neighboring points \(m_{2}\) and \(m_{k-1}\) can be further extended through mirror extension. Subsequently, they can be calculated according to Equation (10). 3.