Toroidal PCA via density ridges

TL;DR

This paper introduces Toroidal Ridge PCA (TR-PCA), a new method for dimension reduction of bivariate circular data that uses density ridges, with algorithms and an R package, demonstrated on ocean current data.

Contribution

It proposes TR-PCA, a novel PCA extension for toroidal data based on density ridges, including algorithms and an R package for practical implementation.

Findings

Effective algorithms for density ridge computation for specific circular distributions

Complete PCA methodology for toroidal data including scores and variance

Application to ocean currents demonstrating practical utility

Abstract

Principal Component Analysis (PCA) is a well-known linear dimension-reduction technique designed for Euclidean data. In a wide spectrum of applied fields, however, it is common to observe multivariate circular data (also known as toroidal data), rendering spurious the use of PCA on it due to the periodicity of its support. This paper introduces Toroidal Ridge PCA (TR-PCA), a novel construction of PCA for bivariate circular data that leverages the concept of density ridges as a flexible first principal component analog. Two reference bivariate circular distributions, the bivariate sine von Mises and the bivariate wrapped Cauchy, are employed as the parametric distributional basis of TR-PCA. Efficient algorithms are presented to compute density ridges for these two distribution models. A complete PCA methodology adapted to toroidal data (including scores, variance decomposition, and resolution of edge cases) is introduced and implemented in the companion R package ridgetorus. The usefulness of TR-PCA is showcased with a novel case study involving the analysis of ocean currents on the coast of Santa Barbara.