## 2 Materials and Methods  
### CEEMDAN  
EMD algorithm is an adaptive method used for analyzing nonlinear signals. This method can decompose complex time series into multiple Intrinsic Mode Functions (IMFs) and residue signals (R) with different frequencies and scales. However, the biggest drawback of EMD is the occurrence of endpoint effects and mode-mixing phenomena when used. To address the mode mixing issue in EMD, the EEMD algorithm proposes a noise-assisted data analysis method to overcome the deficiencies of the EMD method. At each step of signal decomposition, white noise with zero means, and fixed variance is added to the signal being decomposed, effectively resolving the endpoint effects and mode-mixing phenomena encountered by the EMD algorithm. However, this method still has a drawback: it cannot eliminate the added white noise, thereby affecting the extraction of feature information.  
Therefore, based on EEMD, the CEEMDAN algorithm is proposed by adaptively introducing white noise and performing cumulative averaging after obtaining the IMF components to improve EEMD. This method effectively addresses the modal aliasing phenomenon in EMD and overcomes the issues of incompleteness and large reconstruction errors in EEMD [43]. Figure 1 briefly describes the CEEMDAN process.  
The steps of this decomposition method can be represented by Equations (1)-(9):  
Step 1: In the original displacement signal \(X(t)\), white noise \(\omega^{e}(t)\) is added, represented mathematically as:  
\[X^{e}(t)=X(t)+\xi_{0}\omega^{e}(t),e=1,2,3,\ldots,E \tag{1}\]  
where \(\xi_{0}\) is the noise factor; \(\omega^{e}(t)\) is the white noise added for the \(e\)th time; \(E\) is the number of integration times, which is usually set to 15-25; \(X^{e}(t)\) is the signal after adding white noise for the \(e\)th time.  
Step 2: The first IMF component \(\overline{IMF}_{1}(t)\) is obtained by performing EMD decomposition on \(X^{e}(t)\) and then taking the average value, that is:  
\[\overline{IMF}_{1}(t)=\frac{1}{E}\sum_{i=1}^{M}IMF_{1}^{e}(t) \tag{2}\]  
\[r_{1}(t)=X(t)-\overline{IMF}_{1}(t) \tag{3}\]  
where \(\overline{IMF}_{1}(t)\) is the first IMF component after EMD decomposition of the signal and \(r_{1}(t)\) is the first residual signal.  
Step 3: The signal \(r_{1}(t)+\xi_{1}EMD_{1}(\omega^{e}(t))\) is decomposed using EMD to obtain the second IMF component and the residue signal as shown below:  
\[\overline{IMF}_{2}(t)=\frac{1}{E}\sum_{e=1}^{E}EMD_{1}(r_{1}(t)+\xi_{1}EMD_ {1}(\omega^{e}(t))) \tag{4}\]  
\[r_{2}(t)=r_{1}(t)-\overline{IMF}_{2}(t) \tag{5}\]  
where \(\xi_{1}\) is the first noise factor, \(\omega^{e}(t)\) is the white noise added for the \(e\)th time,\(EMD_{1}(\cdot)\) represents the first IMF obtained by EMD, \(\overline{IMF}_{2}(t)\) is the second IMF component after EMD decomposition of the signal and \(r_{2}(t)\) is the second residual signal.  
Step 4: Repeat steps (2) and (3) to obtain residuals and IMF components, that is:  
\[r_{k}(t)=r_{k-1}(t)-\overline{IMF}_{k}(t),k=2,3,\ldots,K \tag{6}\]  
\[\overline{IMF}_{k+1}(t)=\frac{1}{E}\sum_{i=1}^{E}EMD_{1}\Big{(}r_{k}(t)+\xi_ {k}EMD_{k}\Big{(}\omega^{i}(t)\Big{)}\Big{)} \tag{7}\]  
where \(EMD_{k}(\cdot)\) denotes the \(k\)th IMF component obtained by EMD and \(K\)is the total number of modes.  
Figure 1: CEEMDAN flowchart.  
Step 5: Repeat step (4), and when the signal no longer exceeds two extreme points during decomposition, indicating that the residual signal cannot be further decomposed, the algorithm stops. The residual signal can be represented as:  
\[R(t)=X(t)-\sum_{k=1}^{K}IMF_{k}(t) \tag{8}\]  
After undergoing CEEMDAN decomposition, the original displacement sequence is iteratively decomposed into \(n\)IMF components, denoted as \(\overline{IMF_{i}}(t)\), \(i=1,2,3,\dots,n\). Finally, the displacement sequence \(X(t)\) after CEEMDAN decomposition can be represented as:  
\[X(t)=\sum_{k=1}^{K}\overline{IMF_{k}}(t)+R(t) \tag{9}\]