## 2 Materials and Methods
### Principal Component Analysis (PCA) for Noise Filtering  
[PERSON] posited principal component analysis (PCA) for the first time in 1901. It has gained specific applications in many areas, such as chemometrics, image processing, sociology, and economics. PCA is commonly used for data clustering, filtering, extraction and classification, outlier detection, data compression, and for minimizing the correlation between variables [34; 35; 36]. PCA is a data compression approach that reduces the data dimensionality. PCA collects the covariance matrix's eigenvectors and eigenvalues to construct the data's uncorrelated principal components. Principal components are conceived of as new axes, the orientations of which reveal the most significant variance in the source data. The eigenvectors of the covariance matrix define the directions of the principal components. The eigenvalues, on the other hand, are the coefficients or weights of the principal components that reflect the amount of variation carried by each component [37; 38]. [PERSON] et al. [39] investigated how data mining can be employed to estimate the aggregated solar power generation from a large set of solar power generation plants without continuously measuring the output of every single site by using only the measured values from a small number of representative sites. The obtained results showed that the proposed framework is capable of estimating solar generation with good accuracy; the combination of linear regression and the proposed hybrid k-means + PCA dimension reduction method gave the best results, demonstrating that PCA is very fast and computationally efficient. Another characteristic of PCA is that the loading factors are sorted by their contribution to the variance of the original data, which means that the first loading factor captures most of the total variance of the original data among the other elements. The second factor then accounts for the second-largest portion of the total variance, and so on. The contribution of the final loading factors will be minimal, and they are typically used to model noise. As a result, these factors may be overlooked to suppress most of the noise and eliminate the variables' redundancy. Once the loading factors are determined, they are used in connection with the original data to compute the raw data scores. Score vectors are the projection amounts of the nth feature in the original data on the entire loading vectors. Because the scores in each data set are orthogonal, uncorrelated, and span the entire data range, the pseudoinverse of the score matrix can be computed. Thus, the raw data can be represented based on the definition of the loading vectors and scores as:  
\[\text{A}=\text{TP}^{\prime} \tag{2}\]  
The T rows are the score vectors, and P columns are the loading vectors. Singular value decomposition (SVD) [40], Eigenvector decomposition [41], nonlinear iterative partial least squares (NIPALS) [42], the covariance method [12], expectation-maximization method [13], and successive average orthogonalization (SAO) are all used to generate the PCA model [35]. Figure 2 shows the effect of applying the PCA-based noise filter on a set of observations representing the temperature data; from Figure 2, it is clear that the noise variance of the signal is reduced. For comparison, we added another denoising tool, wavelet signal denoising, and we compared the results of the two techniques. In Figure 3, wavelet signal denoising is used; it decomposes the signal into different scales, significantly improving the denoising step in the prediction process. The denoising performance of each method (PCA and wavelet signal denoising) was assessed using Signal-to-Noise Ratio (SNR) parameter.