## 2 Materials and Methods  
### ICEEMDAN Decomposition Method  
EMD can handle the noise reduction in non-linear or non-stationary signals. EMD is based on two basic assumptions:  
1. For any signal to be analyzed, it can be decomposed into several different basic signal components; whether the component is linear or non-linear, there are equal extreme points and zero crossing points. The component also needs to satisfy the constraint conditions that the local symmetry is 0 and the mean value is 0. 2. For a certain point in time, the basic signal components decomposed in the signal to be analyzed can be reconstructed with certain rules to restore the signal to be analyzed. The instantaneous frequency of the signal to be analyzed can be obtained only when all the basic signal components are decomposed. Here, each of the underlying signal components is an intrinsic mode function. EMD is an adaptive method for the non-stationary signal analysis of non-linear systems but is prone to mode aliasing [24]. To reduce the mode-aliasing phenomenon in EMD, Gaussian noise is added to the entire time-domain space to decompose signals into more regular modes, and this is called EEMD [25]. However, a certain degree of residual noise exists in the EEMD-decomposed eigenmode component, trend component, and reconstructed signal. Complementary EEMD (CEEMD) was proposed to alleviate these problems by adding complementary noise [26]. However, deficiencies still exist in signal integrity and mean decomposition. Therefore, complete EMD with adaptive noise (CEEMDAN), which is based on CEEMD, was proposed and effectively prevents pseudo-modes and residual noise modes [27]; however, some residual noise remains in the decomposed IMF. ICEEMDAN, which is a complete-set EMD with improved adaptive noise, is an improved mode decomposition algorithm based on CEEMDAN that can effectively reduce the generation of pseudo-modal components and solve the residual noise problems existing in the intrinsic modal components, thus increasing the SNR. More precisely, the signal is decomposed into different IMF components, which improves the signal analysis capability [28]. ICEEMDAN's decomposition process is as follows: 1. Add noise \(\eta^{i}\) with a mean and variance of 0 and 1, respectively, to the original pipeline signal \(x\) to construct a new pipeline signal \(x^{i}\) and calculate the local mean of the signal using EMD to obtain a component \(r_{i}\) and several IMF components [29]. Equations (1)-(3) represent the denoted signals, first-order residuals, and first-order IMF components, respectively. The ICEEMDAN method can effectively remove the residual noise and mode aliasing that may exist in the _IMF_ components of the EMD and EEMD algorithms. This is expressed as follows: \[x^{i}(t)=x(t)+\beta_{0}E_{1}\Big{(}\eta^{i}(t)\Big{)}.\] (1) \[r_{1}=\Big{\langle}M\Big{(}x^{i}(t)\Big{)}\Big{\rangle}.\] (2) \[IMF_{1}(t)=x^{i}(t)-r_{1}(t).\] (3) where \(x(t)\) is a function that describes the value of the signal at different times, and \(t\) is the independent variable; \(\eta^{i}\) represents Gaussian white noise, where \(i=1,2\ldots n\); \(\beta_{0}=\frac{t_{\text{s}}t_{\text{s}}t_{\text{s}}(x(t))}{std\big{(}E_{1} \big{(}\eta^{i}(t)\big{)}\big{)}}\) represents the noise coefficient; and \(\varepsilon_{0}\) represents the reciprocal of the expected SNR between the first noise signal and the analyzed signal. Moreover, i represents the number of times noise is added and \(std(\bullet)\) is the standard deviation operator. Finally, \(E_{k}(\bullet)\) is the \(k\)-th mode obtained by EMD, \(M(\bullet)\) represents the local mean value used for the production signal, and \(\langle\bullet\rangle\) denotes averaging the whole. 2.