Principal Component Analysis and Partial Least Squares

In analytical tools, Principal Component Analysis (PCA) and Partial Least Squares (PLS) are extensively utilized analytical methods. These approaches can be applied for data processing, feature extraction, and dimensionality reduction, thereby improving model accuracy and interpretability. In this code implementation, we employ the Nonlinear Iterative Partial Least Squares (NIPALS) algorithm to demonstrate the relationship between PCA and PLS. The NIPALS algorithm provides more accurate computation of variable correlations and can handle nonlinear relationships through iterative matrix decomposition and projection operations. Our implementation includes key functions for covariance matrix calculation, eigenvalue decomposition, and score/loading vector extraction. Additionally, we explore performance optimization techniques such as selecting optimal numbers of principal components through scree plot analysis and tuning regularization parameters using cross-validation methods. The code structure features modular functions for data preprocessing, iterative computation cycles, and result visualization to facilitate practical application.