A review of multivariate distributions for count data derived from the Poisson distribution

• Published in: Wiley interdisciplinary reviews. Computational statistics Vol. 9; no. 3; pp. np - n/a
• Main Authors: Inouye, David I., Yang, Eunho, Allen, Genevera I., Ravikumar, Pradeep
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 01-05-2017; Wiley Subscription Services, Inc
• Subjects: Copulas; Counting; Crime; Criminal statistics; Data; Data analysis; Data processing; Distribution; Empirical analysis; Graphical methods; Graphical models; High dimensional; Mathematical models; Modelling; Multivariate; Multivariate analysis; Poisson; Poisson distribution; Poisson distributions; Statistical analysis; Statistical methods; Statistics; Texts; Traffic accidents; Traffic models; Wire; Copulas; Poisson; High Dimensional; Graphical Models; Multivariate
• ISSN: 1939-5108; 1939-0068
• DOI: 10.1002/wics.1398

The Poisson distribution has been widely studied and used for modeling univariate count‐valued data. However, multivariate generalizations of the Poisson distribution that permit dependencies have been far less popular. Yet, real‐world, high‐dimensional, count‐valued data found in word counts, genomics, and crime statistics, for example, exhibit rich dependencies and motivate the need for multivariate distributions that can appropriately model this data. We review multivariate distributions derived from the univariate Poisson, categorizing these models into three main classes: (1) where the marginal distributions are Poisson, (2) where the joint distribution is a mixture of independent multivariate Poisson distributions, and (3) where the node‐conditional distributions are derived from the Poisson. We discuss the development of multiple instances of these classes and compare the models in terms of interpretability and theory. Then, we empirically compare multiple models from each class on three real‐world datasets that have varying data characteristics from different domains, namely traffic accident data, biological next generation sequencing data, and text data. These empirical experiments develop intuition about the comparative advantages and disadvantages of each class of multivariate distribution that was derived from the Poisson. Finally, we suggest new research directions as explored in the subsequent Discussion section. WIREs Comput Stat 2017, 9:e1398. doi: 10.1002/wics.1398
This article is categorized under:
Statistical and Graphical Methods of Data Analysis > Multivariate Analysis
This paper reviews and empirically compares multivariate models derived from the Poisson distribution which can be categorized into three model classes based on primary modeling assumptions.