## 8 Principal Component Analysis (PCA): Unpacking the Fundamentals  
It is a statistical technique used to reduce the dimensionality of data while preserving most of the original variance. Applied to the removal of artifacts in EEG signals, PCA can isolate and eliminate major components that represent noise or artifacts, thereby improving the quality of the EEG signal for further analysis. Let us consider a matrix of EEG signals \(X\) of size \(m\times n\), where \(m\) represents the number of channels (electrodes) and \(n\) the number of time points (samples). The goal is to transform \(X\) into a new coordinate space that better represents the variance in the data. Initially, you average each channel and subtract this average from each channel to center the data around zero. This step is critical to ensure that the PCA correctly identifies the directions of maximum variance without being influenced by the original location of the data. The centered matrix [104; 105]  
\[\mathbf{C}=\frac{1}{n-1}\mathbf{X}_{c}\mathbf{X}_{C}^{T} \tag{13}\]  
which reflects how variances and covariances are distributed across channels. Subsequently, we proceed with the eigenvalue decomposition of \(\mathbf{C}\), finding the eigenvalues, \(\lambda_{i}\), and the eigenvectors \(\mathbf{v}_{i}\). Eigenvalues reflect the amount of variance captured by each principal component (PC), while eigenvectors indicate the directions (or principal components) in the original data space that maximize variance. Mathematically, this step is expressed as  
\[\mathbf{C}\mathbf{v}_{i}=\lambda_{i}\mathbf{v}_{i},\ \ i=1,2,\ldots,m. \tag{14}\]  
where the eigenvalues are typically sorted as  
\[\mathbf{Y}=\mathbf{V}^{T}\mathbf{X}_{c} \tag{15}\]  
where \(\mathbf{V}\) is the matrix containing the eigenvectors of \(\mathbf{C}\) as columns. \(\mathbf{Y}\) represents the data projected onto the principal components, where each row corresponds to a PC.  
To remove artifacts, the main components that correspond to the artifacts are identified by analyzing the characteristics of the PCs, such as the waveform or spectral content. These PCs are therefore excluded from data reconstruction.  
Finally, the clean EEG signals are reconstructed using only the PCs deemed free of artifacts. If \(\mathbf{V}_{\text{clean}}\) is the reduced eigenvector matrix that excludes the artifactual PCs, the reconstruction of the clean data \(\mathbf{V}_{\text{clean}}\) is obtained as  
\[\mathbf{X}_{\text{clean}}=\mathbf{V}_{\text{clean}}\mathbf{Y}_{\text{clean}} +\text{media\ originale}, \tag{16}\]  
where \(\mathbf{Y}_{\text{clean}}\) represents the selected principal components (excluding those of the artifacts).  
PCA minimizes the impact of artifacts in EEG signals by preserving essential neural information, using dimensionality reduction to isolate and eliminate unwanted influences, and improving signal quality for future analysis.  
\begin{table}
\begin{tabular}{l p{284.5 pt}} \hline \hline  & **EMD Approaches** \\ \hline
**Strengths** & Ability to adapt to data, flexible decomposition. Effective artifact isolation. No predefined windows are required. Identification of complex and nonlinear dynamics. \\ \hline
**Weaknesses** & Computationally expensive. Significant intuition and specific skills are required. Change in noise and artifact management. Difficulty in processing data affected by uncertainties and/or inaccuracies. \\ \hline
**Opportunities** & Development of more computationally efficient variants. Integration with other signal analysis techniques. Adoption of hybrid approaches with machine learning. \\ \hline
**Threats** & Real-time EEG processing for advanced applications. \\ \hline \hline \end{tabular}
\end{table}
Table 4: SWOT Analysis for EMD.