New Interpretation of Principal Components Analysis

TL;DR

This paper offers a novel geometric interpretation of PCA, introduces an enhanced selection criterion for principal components, and proposes a new clustering algorithm for variables based on their similarity to principal components.

Contribution

It presents a new geometric perspective on PCA, enriches the classical method with a novel component selection criterion, and introduces a variable clustering algorithm based on similarity measures.

Findings

Geometric interpretation of the determination coefficient.

A new criterion for selecting important principal components.

A clustering algorithm for primary variables based on similarity to principal components.

Abstract

A new look on the principal component analysis has been presented. Firstly, a geometric interpretation of determination coefficient was shown. In turn, the ability to represent the analyzed data and their interdependencies in the form of easy-to-understand basic geometric structures was shown. As a result of the analysis of these structures it was proposed to enrich the classical PCA. In particular, it was proposed a new criterion for the selection of important principal components and a new algorithm for clustering primary variables by their level of similarity to the principal components. Virtual and real data spaces, as well as tensor operations on data, have also been identified.The anisotropy of the data was identified too.