Scaled torus principal component analysis

TL;DR

This paper introduces ST-PCA, a new method for dimensionality reduction of multivariate circular data on a torus, using a data-driven map to a sphere and nested spheres to better analyze complex datasets.

Contribution

ST-PCA is a novel approach that combines multidimensional scaling and principal nested spheres to effectively reduce dimensions of toroidal data.

Findings

ST-PCA outperforms existing methods in clustering tasks.

Effective dimensionality reduction demonstrated on astronomy and molecular biology datasets.

Method preserves meaningful structures in low-dimensional torii.

Abstract

A particularly challenging context for dimensionality reduction is multivariate circular data, i.e., data supported on a torus. Such kind of data appears, e.g., in the analysis of various phenomena in ecology and astronomy, as well as in molecular structures. This paper introduces Scaled Torus Principal Component Analysis (ST-PCA), a novel approach to perform dimensionality reduction with toroidal data. ST-PCA finds a data-driven map from a torus to a sphere of the same dimension and a certain radius. The map is constructed with multidimensional scaling to minimize the discrepancy between pairwise geodesic distances in both spaces. ST-PCA then resorts to principal nested spheres to obtain a nested sequence of subspheres that best fits the data, which can afterwards be inverted back to the torus. Numerical experiments illustrate how ST-PCA can be used to achieve meaningful dimensionality reduction on low-dimensional torii, particularly with the purpose of clusters separation, while two data applications in astronomy (three-dimensional torus) and molecular biology (on a seven-dimensional torus) show that ST-PCA outperforms existing methods for the investigated datasets.