PCA Dimensionality Reduction Code - MATLAB Implementation

This MATLAB program serves as a practical tool for dimensionality reduction of M*N large datasets, featuring a well-structured implementation that ensures ease of use. The input parameter mixedsig represents an n*T mixed data matrix where n indicates the number of signals and T corresponds to sampling points, while the output y is an m*T principal component matrix. The implementation follows these computational steps: 1. Mean Centering: This initial preprocessing step stabilizes the data by removing DC components, typically implemented using MATLAB's mean() function along appropriate dimensions. 2. Covariance Matrix and Eigendecomposition: Computes the covariance matrix using matrix multiplication operations, followed by eigenvalue decomposition through MATLAB's eig() function to obtain fundamental components for principal analysis. 3. Eigenvalue Thresholding: Determines the number of eigenvalues exceeding a predefined threshold to establish optimal dimensionality retention, often implemented via logical indexing operations. 4. Descending Eigenvalue Sorting: Employs sorting algorithms (like sort() with 'descend' option) to prioritize components with maximum variance contribution. 5. Small Eigenvalue Removal: Eliminates negligible eigenvalues associated with noise components using percentile-based or variance-based cutoff methods. 6. Large Eigenvalue Removal (typically omitted): Occasionally implemented to prevent overfitting by capping maximum variance contributions, though rarely used in standard PCA implementations. 7. Eigenvalue Merging: Combines selected eigenvalues through matrix concatenation operations to form the reduced feature set. 8. Eigenvector Selection: Retrieves eigenvectors corresponding to retained eigenvalues using index mapping techniques for component reconstruction. 9. Whitening Matrix Calculation: Applies whitening transformation to decorrelate components, often implemented via diagonal matrix operations with eigenvalue normalization. 10. Principal Component Extraction: Final projection step utilizing matrix multiplication between whitened data and selected eigenvectors to produce dimensionally-reduced output with preserved essential characteristics.