Compositional regression using principal nested spheres

2026-03-21

AI Summary

Compositional data pose significant modeling challenges for conventional Euclidean regression methods due to the nonlinear geometry of the simplex. This work proposes a novel approach that embeds compositional data into the positive orthant of a sphere and then maps them, via Principal Nested Spheres (PNS), onto an interpretable intermediate space whose first dimension is circular and higher-order dimensions are Euclidean. Regression is performed in this transformed space, followed by an inverse mapping back to the original simplex. To our knowledge, this is the first application of PNS to compositional regression, effectively balancing geometric fidelity with statistical tractability. Simulation studies demonstrate superior performance, and real-world analysis of environmental chemical exposure data highlights the method’s interpretability and practical utility.

Technology Category

• Machine Learning: Learning with Manifolds
• Natural Language Processing: Interpretability, Analysis, and Evaluation of NLP Models
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• compositional regression
• simplex
• manifold-valued data
• nonlinear geometry
• Principal Nested Spheres

Innovation

• compositional regression
• Principal Nested Spheres
• manifold-valued data
• spherical embedding
• intermediate space