aroundPCA

aroundPCA is a statistical technique that extends classical principal component analysis (PCA) by incorporating local neighborhood structures in high‑dimensional data sets. While traditional PCA seeks linear combinations of original variables that capture the greatest variance, aroundPCA introduces a weighting scheme based on proximity in the embedding space. This approach allows the method to preserve local geometry more effectively, especially in datasets where global variance explanations are insufficient.

The algorithm begins by computing a pairwise distance matrix for the data points. A neighborhood graph is constructed using a fixed radius or a k‑nearest neighbor rule. Edge weights are then assigned inversely proportional to distances, emphasizing relationships among nearby points. The weighted covariance matrix is formed and decomposed, yielding principal axes that are influenced by local interactions. The resulting components can be visualized in two or three dimensions, revealing clusters or trajectories that would be flattened or obscured by standard PCA.

Applications of aroundPCA include manifold learning, where it serves as a linear approximation to nonlinear structures, and exploratory data analysis in genomics and image processing. By retaining local variance, the method can enhance clustering accuracy and improve the interpretability of features that vary smoothly across a sample space. Limitations are similar to other variants: choice of neighborhood parameters critically affects results, and the computational cost grows with the size of the distance matrix. Nonetheless, aroundPCA presents a flexible alternative to classical PCA when preserving local data structure is essential.