## 2 Materials and Methods
### Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN)  
CEEMDAN was created as an improved version of the EEMD method. EEMD is a noise-assisted signal decomposition technique that solves the problem of modal mixing in EMD by introducing white Gaussian noise to the original signal, allowing the complex signals to be decomposed into high-frequency intrinsic modal functions (IMF) and a low-frequency part called residual (\(R\)) over several iterations [46]. However, the white Gaussian noise added to the EEMD must be repeatedly averaged to be removed, which consumes considerable time. In addition, the reconstructed signal, which includes residual noise, can produce a variable number of IMFs. Therefore, we use the CEEMDAN method to overcome these problems. The CEEMDAN process is described as follows:  
1. White noise \(w\) with a standard deviation \(\epsilon\) is added to the original signal \(X(t)\), which can be expressed by Equation (8). \[X_{i}(t)=X(t)+\epsilon w_{i}(t)\] (8) where \(i\) represents the number of realizations. 2. The EMD decomposition step is performed on the signal after each realization, and then the first IMF \(\overline{\mathrm{C}}_{1}(t)\) is calculated by averaging the decomposition components (see Equation (9)). \[\overline{\mathrm{C}}_{1}(t)=\frac{1}{N}\sum_{i=1}^{N}\mathrm{C}_{1}^{i}(t)\] (9)
3. The residual of the first stage is given as \(R_{1}(t)=X(t)-\overline{\mathrm{C}}_{1}(t)\). Then, the signal \(R_{1}(t)\) can be further decomposed by EMD to calculate the second IMF mode, which can be formulated as in Equation (10). \[R_{2}(t)=R_{1}(t)-\overline{\mathrm{C}}_{2}(t)\] (10) where \(\overline{\mathrm{C}}_{2}(t)\) represents the second IMF mode obtained by the EMD algorithm. 4. In the next stage, the residual \(R_{k}(t)\) and the component \(C_{k}(t)\) can be calculated using Equation (11). \[R_{k+1}(t)=R_{k}(t)-\overline{\mathrm{C}}_{k+1}(t)\] (11) where \(\overline{\mathrm{C}}_{k+1}(t)\) represents the \((k+1)\)-th IMF mode obtained by the CEEMDAN algorithm. Repeat steps 3 and 4 until the residual \(R_{n}(t)\) meets the stopping criterion of Equation (12). \[\frac{\|R_{n}(t)\|^{2}}{\overline{T}}<\delta \tag{12}\]where \(T\) represents the length of the sequence \(X(t)\), \(R_{n}(t)\) is the sequence after the \(n\)-th decomposition, and the empirical value of \(\delta\) is set to 0.2. Finally, the original signal \(X(t)\) can be decomposed as Equation (13). \[X(t)=\sum_{k=1}^{n}\overline{C}_{k}(t)+R_{n}(t) \tag{13}\]  
where \(R_{n}(t)\) is the final residual. In selecting the parameters in Table 4, consideration was given between decomposition accuracy and computational efficiency. First, 100 realizations (trials) were selected so that the intrinsic mode functions (IMFs) would capture the original signal features without adding excess noise. Then, \(\epsilon=0.1\) was used to control the magnitude of the added noise, with the noise_scale set to 0.5 to maintain an appropriate signal-to-noise ratio. The number of siftings (num_siftings) was set to 100 to ensure that each IMF was extracted with maximum accuracy. CEEMDAN helps to analyze solar irradiance data due to the complex and variable nature of these signals, especially in regions with high climatic variability. Under these conditions, solar irradiance signals exhibit rapid fluctuations and seasonal variations, which can be difficult for the model to learn directly. CEEMDAN can effectively separate high-frequency components, such as noise and rapid fluctuations, from low-frequency trends, such as seasonal and long-term variations, by decomposing the data into different IMFs, allowing the model to learn these patterns more efficiently, increasing its robustness and accuracy. In Figure 4, the original solar irradiance (10 min resampled) time series is presented at the top (labeled \"True\"), followed by the different IMFs labeled 1 through 14.