Bivariate Densities in Bayes Spaces: Orthogonal Decomposition and Spline Representation

TL;DR

This paper introduces an orthogonal decomposition method for bivariate densities in Bayes spaces, enabling separation of independent and dependent components, with a spline-based computational approach for practical analysis.

Contribution

It presents a novel orthogonal decomposition framework for bivariate densities in Bayes spaces and introduces a spline representation for computational implementation.

Findings

Decomposition separates independent and interactive parts of densities.

Framework enhances dependence modeling beyond copulas.

Spline representation facilitates practical computation.

Abstract

A new orthogonal decomposition for bivariate probability densities embedded in Bayes Hilbert spaces is derived. It allows one to represent a density into independent and interactive parts, the former being built as the product of revised definitions of marginal densities and the latter capturing the dependence between the two random variables being studied. The developed framework opens new perspectives for dependence modelling (which is commonly performed through copulas), and allows for the analysis of dataset of bivariate densities, in a Functional Data Analysis perspective. A spline representation for bivariate densities is also proposed, providing a computational cornerstone for the developed theory.