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

Compositional Data (CoDa) is usually viewed as data on the simplex and is studied via a log-ratio analysis, following the classical work of Aitchison (1986). We propose to bring to the fore an alternative view of CoDa as a stick breaking process, an approach which originates from Bayesian nonparametrics. 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. These stick-breaking representations allow to effectively and simply deal with CoDa with zeroes. We establish connections with other topics of probability and statistics like i) spacings and order statistics, ii) Bayesian nonparametrics and Dirichlet distributions, iii) neutrality, iv) hazard rates and the product integral, v) mixability.