In this article, we develop a new class of multivariate distributions adapted for count data, called Tree P{\'o}lya Splitting. This class results from the combination of a univariate distribution and singular multivariate distributions along a fixed partition tree. As we will demonstrate, these distributions are flexible, allowing for the modeling of complex dependencies (positive, negative, or null) at the observation level. Specifically, we present the theoretical properties of Tree P{\'o}lya Splitting distributions by focusing primarily on marginal distributions, factorial moments, and dependency structures (covariance and correlations). The abundance of 17 species of Trichoptera recorded at 49 sites is used, on one hand, to illustrate the theoretical properties developed in this article on a concrete case, and on the other hand, to demonstrate the interest of this type of models, notably by comparing them to classical approaches in ecology or microbiome.

翻译：本文提出了一类适用于计数数据的新型多元分布——树Pólya分裂分布（Tree Pólya Splitting）。该类分布通过将单变量分布与沿固定分割树的奇异多元分布相结合而得到。我们将证明这些分布具有灵活性，能够在观测层面建模复杂相关性（正相关、负相关或零相关）。具体而言，我们重点从边际分布、阶乘矩及依赖结构（协方差与相关性）三个方面阐述树Pólya分裂分布的理论性质。利用49个采样点记录的17种毛翅目昆虫丰度数据，一方面在具体案例中验证本文发展的理论性质，另一方面通过对比生态学或微生物组学中的经典方法，展示该类模型的应用价值。