## 3 Methodology
### Mc-Tcgrul  
#### 3.2.1 CreamDAN Decomposition Algorithm  
The empirical-mode decomposition (EMD) algorithm has significant advantages in handling non-stationary signals, but it still suffers from mode-mixing issues. EMD decomposes a signal into a series of intrinsic mode functions (IMFs) by interpolating local extrema, which can lead to mode mixing and unstable results. Mode mixing occurs when a single IMF contains widely different characteristic time scales, or similar time scales are distributed across different IMFs, leading to waveform mixing between adjacent components. [PERSON] et al. proposed ensemble empirical-mode decomposition (EEMD) based on EMD, which mitigates mode mixing by adding white noise to the signal before decomposition [33]. However, the reconstructed components still retain some residual white noise,affecting the subsequent analysis. To address this, [PERSON] et al. introduced the CEEMDAN, which further improves the completeness of EEMD and effectively reduces reconstruction errors by adding specific adaptive white noise at each stage of the decomposition [29]. In summary, while EMD serves as the foundational method for signal decomposition, EEMD improves upon it by averaging multiple noisy realizations, and CEEMDAN provides an even more refined technique by utilizing adaptive noise to enhance the stability and accuracy of the decomposition.  
Let \(\varepsilon_{j}(\cdot)\) denote the operator that generates the \(j\)-th component using EMD and let \(\omega_{i}(x)\) represent Gaussian white noise with a normal distribution \(N(0,1)\). The basic procedure of the CEEMDAN algorithm is as follows:  
1. Add Gaussian white noise with an amplitude of \(\varepsilon_{0}\) to the signal \(H(x)\), resulting in the following: \[H_{i}(x)=H(x)+\varepsilon_{0}\omega_{i}(x)\] (3) where \(i\) denotes the number of added white noise instances, \(I=1\), \(2\), \(\ldots\), \(I\).
2. Decompose each \(H_{i}(x)\) using the EMD algorithm. The first component of each sample is averaged to obtain \(IMF_{1}\): \[IMF_{1}(x)=\frac{1}{I}{\sum_{i=1}^{l}imf_{1}^{i}(x)}\] (4)
3. Subtract the \(IMF_{1}(x)\) obtained above from the original signal to obtain the residual component \(R_{1}(x)\): \[R_{1}(x)=H(x)-IMF_{1}(x)\] (5)
4. Add white noise to the residual component to obtain \(R_{1}(x)+\varepsilon_{1}E_{1}(\omega_{i}(x))\). This new sequence is decomposed using EMD to calculate the \(IMF_{2}(x)\) and its residual \(R_{2}(x)\): \[IMF_{2}(x)=\frac{1}{I}{\sum_{i=1}^{I}E_{1}(R_{1}(x)+\varepsilon_{1}E_{1}( \omega_{i}(x)))}\] (6) \[R_{2}(x)=R_{1}(x)-IMF_{2}(x)\] (7)
5. Repeat the above steps for other stages to compute \(IMF_{j}\) and its residual: \[IMF_{j}(x)=\frac{1}{I}{\sum_{i=1}^{I}E_{1}(R_{j-1}(x)+\varepsilon_{j-1}E_{j-1} (\omega_{i}(x)))}\] (8) \[R_{j}(x)=R_{j-1}(x)-IMF_{j}(x)\] (9) where \(j=2,3,\cdots,M\), and \(M\) is the total number of IMFs.
6. The algorithm terminates when the residual component can no longer be decomposed using EMD. The final residual component can be expressed as follows: \[R(x)=H(x)-{\sum_{j=1}^{M}IMF_{j}(x)}\] (10)  
The IMF components collectively represent the characteristics of the original sequence at different time scales, while the residual represents the overall trend of the original sequence. The components decomposed by CEEMDAN are purer, with lower reconstruction errors, making this method highly suitable for decomposing and processing time-series data with significant volatility.