## 2 Materials and Methods  
### CEEMDAN Algorithm  
EMD [16] can adaptively decompose the original signal into an intrinsic mode function (IMF) according to its own scale. However, due to the problem of EMD's own computing theory, mode mixing often occurs in its decomposition. EEMD [21] is an improvement on EMD; its calculation principle is to add the corresponding white noise to the original signal and eliminate the mode-mixing phenomenon in EMD decomposition at one time by suppressing and canceling the influence of decomposition noise through multiple integration. After multiple integration averaging, the influence of white noise on the decomposition results is offset, but a reconstruction error appears. The reconstruction error relies on the integration number, and increasing the integration number can diminish the reconstruction error, but it increases the computational volume to some extent and seriously affects the computational efficiency.  
In order to solve the problems of mode mixing, calculation accuracy and computational efficiency, the CEEMDAN algorithm is proposed. Compared with the EEMD algorithm, the CEEMDAN algorithm adds a finite amount of adaptive white noise at each stage of EMD decomposition. When the number of integrations is small, its reconstruction error is almost zero, and the reconstructed signal is almost identical to the original signal. Therefore, the CEEMDAN algorithm can solve the mode-mixing phenomenon existing in the EMD algorithm and overcome the incompleteness as well as low computational efficiency of the EEMD algorithm.  
Define an operator \(E_{k}(\cdot)\) that represents the process of EMD obtaining the \(k\)-th mode component \(IMF_{k}\); let \(\omega^{i}(t)\) be the white-noise-satisfying distribution of \(N(0,1)\) and \(\varepsilon_{k}\) is the amplitude coefficient of white noise added for the \(k\)-th time. The decomposition process for the CEEMDAN algorithm is shown below [25]:  
(1) The white noise \(X(t)+\varepsilon_{0}\omega^{i}(t)\) is added to the original signal, and \(I\)-th EMD decomposition is performed. An average operation is then performed on the result to obtain IMF\({}_{1}\).  
\[\text{IMF}_{1}=\frac{1}{I}{\sum\limits_{i=1}^{1}E_{1}(X(t)+\varepsilon_{0} \omega^{i}(t))} \tag{1}\]  
(2) The first stage residual component can be calculated.  
\[r_{1}(t)=X(t)-\text{IMF}_{1} \tag{2}\]  
The white noise \(r_{1}(t)+\varepsilon_{1}E_{1}\big{(}\omega^{i}(t)\big{)},i=1,2,\ldots,I\) is added to the first-stage residual component, and the EMD is performed.  
IMF\({}_{2}\) can then be calculated with the mean value of the first IMF.  
\[\text{IMF}_{2}=\frac{1}{I}{\sum\limits_{i=1}^{1}E_{1}\Big{(}r_{1}(t)+ \varepsilon_{1}E_{1}\Big{(}\omega^{i}(t)\Big{)}\Big{)}} \tag{3}\]  
For \(k=1,\ 2,\ \ldots,K\), the \(k\)-th residual component can be calculated.  
\[r_{k}(t)=r_{k-1}(t)-\text{IMF}_{k} \tag{4}\]  
(3) White noise \(r_{1}(t)+\varepsilon_{1}E_{1}\big{(}\omega^{i}(t)\big{)},i=1,2,\ldots,I\) is added to the \(k\)-th residual component and EMD decomposition performed. IMF\({}_{k+1}\) can then be calculated with the mean value of the first IMF.  
\[\text{IMF}_{k+1}=\frac{1}{I}{\sum\limits_{i=1}^{1}E_{1}\Big{(}r_{k}(t)+ \varepsilon_{k}E_{k}\Big{(}\omega^{i}(t)\Big{)}\Big{)}} \tag{5}\](4) Step (4) and Step (5) are repeated until the value of the residual component is less than two extremes, then the decomposition stops. Eventually the residual variable is obtained.  
\[r(t)=X(t)-\sum_{k=1}^{K}\text{IMF}_{k} \tag{6}\]  
where \(K\) is the total number of modes in the decomposition process.  
The reconstructed signal can be expressed as follows:  
\[X(t)=r(t)+\sum_{k=1}^{K}\text{IMF}_{k} \tag{7}\]