Dimension reduction of multivariate densities in Bayes spaces

2026-06-17

AI Summary

This work proposes a dimensionality reduction and structural decomposition approach for multivariate probability density functions characterized by relative and constrained properties, formulated within the Bayesian paradigm. By embedding density functions into a Hilbert space via the centered log-ratio (clr) transformation and applying functional principal component analysis (FPCA), the method achieves effective dimensionality reduction. An orthogonal decomposition is further introduced to disentangle independent and interactive components. The core contribution lies in establishing an optimal variance decomposition framework for multivariate densities in the Bayes space, which endows FPCA outcomes with clear geometric and statistical interpretability. Experimental results on housing and geological datasets demonstrate the method’s efficacy in producing interpretable low-dimensional representations and accurately identifying constituent components.

Abstract

The Bayes space provides a Hilbert space structure for analysing probability density functions (PDFs), equipping them with a geometry that reflects their relative and constrained nature. A key tool in this framework is the centred logratio (clr) transformation, which establishes an isometric isomorphism between the Bayes space and (a subspace of) the classical $L^2$ space. This makes it possible to apply functional data analysis (FDA) techniques, particularly functional principal component analysis (FPCA), to both univariate and multivariate density data in the context of dimension reduction. For multivariate PDFs, embedding them in the Bayes space enables an orthogonal decomposition into independent and interactive components. Furthermore, the independent part can be decomposed into mutually orthogonal geometric marginals. This structure provides more profound insights into the sources of variation in multivariate densities. We show that this decomposition of the total variance is optimal in a PCA sense, impacting the interpretation of the eigenfunctions and scores resulting from FPCA. We demonstrate that applying FPCA directly to multivariate densities is equivalent in a certain sense to applying multivariate FPCA to their decomposed form, with the resulting eigenfunctions and scores decomposing accordingly. The unique decomposition based on these theoretical results is applied to housing and geological empirical data respectively, demonstrating the interpretability and practical value of this approach.

Problem

• Bayes space
• dimension reduction
• multivariate densities
• functional principal component analysis
• clr transformation

Innovation

• Bayes space
• centered logratio transformation
• functional PCA
• orthogonal decomposition
• multivariate density