Parametric Lie group structures on the probabilistic simplex and generalized Compositional Data

TL;DR

This paper introduces parametric quotient Lie group structures on the probabilistic simplex, enabling a unified geometric framework for compositional data and extending classical statistical methods to more general equivalence relations.

Contribution

It develops a novel set of parametric quotient Lie group structures on the probabilistic simplex, connecting geometric and statistical approaches for compositional data.

Findings

Reveals Lie group structures underlying compositional data

Enables extension of classical statistical methods to generalized data

Provides a geometric framework for scale-invariant data analysis

Abstract

In this paper we build a set of parametric quotient Lie group structures on the probabilistic simplex that can be extended to real vector space structures. In particular, we rediscover the main mathematical objects generally used when treating compositional data as elements associated to the quotient Lie group with respect to the equivalence relation induced by the scale invariance principle. This perspective facilitates the adaptation of the statistical methods used for classical compositional data to data that follows a more general equivalence relation.