PCA (Principal Component Analysis) Algorithm

In engineering and scientific research, the PCA (Principal Component Analysis) algorithm finds extensive application. This report focuses on studying PCA's fundamental structure and underlying principles. While linear PCA algorithms are commonly implemented using covariance matrix decomposition through eigenvalue decomposition or SVD (Singular Value Decomposition) in code, research demonstrates their limitations. For instance, linear PCA cannot separate independent signal components from linear mixtures since principal components are determined exclusively by second-order statistics—autocorrelation matrices—which only characterize stationary Gaussian distributions. Consequently, improved versions have been developed, including nonlinear PCA algorithms employing neural networks or kernel methods, and robust algorithms using iterative reweighting techniques. The report illustrates PCA's engineering applications through a fitting example on lines/planes, implemented by extracting minor components (variance-minimizing components) via secondary eigenvalue analysis. Result analysis confirms that PCA achieves satisfactory fitting outcomes. Notably, PCA applies not only to linear data but also to nonlinear datasets through appropriate transformations. Furthermore, PCA finds broad utilization in signal processing (dimensionality reduction), image processing (feature extraction), pattern recognition (data preprocessing), and related domains.