Compositional regression using principal nested spheres

TL;DR

This paper introduces a novel regression framework for compositional data using Principal Nested Spheres, enabling effective analysis of manifold-valued responses with improved interpretability.

Contribution

It develops a new regression method for compositional responses by embedding data in spherical space and applying PNS, addressing nonlinear geometry challenges.

Findings

PNS-based regression performs well in simulations.

The method provides interpretable results in environmental data.

Fitted values can be accurately mapped back to the simplex.

Abstract

Regression with compositional responses is challenging due to the nonlinear geometry of the simplex and the limitations of Euclidean methods. We propose a regression framework for manifold-valued data based on mappings to statistically tractable intermediate spaces. For compositional data, responses are embedded in the positive orthant of the sphere and analysed using Principal Nested Spheres (PNS), yielding a cylindrical intermediate space with a circular leading score and Euclidean higher-order scores. Regression is performed in this intermediate space and fitted values are mapped back to the simplex. A simulation study demonstrates good performance of PNS-based regression. An application to environmental chemical exposure data illustrates the interpretability and practical utility of the method.