The Stick-Breaking and Ordering Representation of Compositional Data: Copulas and Regression models

Résumé

Compositional Data (CoDa) is usually viewed as data on the simplex and is studied via a log-ratio analysis, following the classical work of J. Aitchison (1986). We propose an alternative view of CoDa as stick breaking processes. The first stick-breaking approach gives rise to a view of CoDa as ordered statistics, from which we can derive “stick-ordered” distributions. The second approach is based on a rescaled stick-breaking transformation, and give rises to a geometric view of CoDa as a free unit cube. The latter allows to introduce copula and regression models, which are useful for studying the internal or external dependence of CoDa. We establish connections with other topics of statistics like i) spacings and order statistics, ii) Bayesian nonparametrics and Dirichlet distributions, iii) neutrality, iv) mixability.