Tree P\'olya Splitting distributions for multivariate count data

TL;DR

This paper introduces Tree Pólya Splitting distributions, a flexible new class for modeling complex dependence structures in multivariate count data, unifying several known distributions and demonstrated through ecological and microbiome datasets.

Contribution

The paper develops a novel class of multivariate distributions called Tree Pólya Splitting, unifying and extending existing models for count data with flexible dependence modeling.

Findings

Distributions can model positive, negative, or null dependence.

Theoretical properties include marginals, factorial moments, and dependence measures.

Demonstrated on ecological and microbiome datasets, outperforming some existing models.

Abstract

In this article, we develop a new class of multivariate distributions adapted for count data, called Tree P\'olya Splitting. This class results from the combination of a univariate distribution and singular multivariate distributions along a fixed partition tree. Known distributions, including the Dirichlet-multinomial, the generalized Dirichlet-multinomial and the Dirichlet-tree multinomial, are particular cases within this class. As we will demonstrate, these distributions are flexible, allowing for the modeling of complex dependence structures (positive, negative, or null) at the observation level. Specifically, we present the theoretical properties of Tree P\'olya Splitting distributions by focusing primarily on marginal distributions, factorial moments, and dependence structures (covariance and correlations). A dataset of abundance of Trichoptera is used, on one hand, as a benchmark to illustrate the theoretical properties developed in this article, and on the other hand, to demonstrate the interest of these types of models, notably by comparing them to other approaches for fitting multivariate data, such as the Poisson-lognormal model in ecology or singular multivariate distributions used in microbiome.