Changing reference measure in Bayes spaces with applications to functional data analysis

TL;DR

This paper develops a mathematical framework for changing the reference measure in Bayes spaces, impacting the geometry and enabling advanced functional data analysis techniques with applications to income data.

Contribution

It introduces a new weighting scheme and a centered log-ratio transformation for Bayes spaces, facilitating the use of standard functional data analysis tools with weighted distributions.

Findings

Effective weighting scheme for distributional data.

Centered log-ratio transformation enables use of existing functional data tools.

Successful application to Italian income data.

Abstract

Probability density functions (PDFs) can be understood as continuous compositions by the theory of Bayes spaces. The origin of a Bayes space is determined by a given reference measure. This can be easily changed through the well-known chain rule which has an impact on the geometry of the Bayes space. This work provides a mathematical framework for setting a reference measure. It is used to develop a weighting scheme on the bounded domain of distributional data. The impact on statistical analysis is shown from the perspective of simplicial functional principal component analysis. Moreover, a novel centered log-ratio transformation is proposed to map a weighted Bayes spaces into an unweighted $L^2$ space, enabling to use most tools developed in functional data analysis (e.g. clustering, regression analysis, etc.) while accounting for the weighting strategy. The potential of our proposal is shown through simulation and on a real case study using Italian income data.