Bayesian Dynamic Dirichlet Models

Abstract

The main purpose of this article is the study of statistical models for compositional data, which are characterized by random vectors defined on the open standard (k − 1)-simplex. Each coordinate of represents the share, in percentage, of each one of the k categories that represent a given phenomenon. We propose a new dynamic model, the Dynamic Dirichlet Model (DDM), for describing time series of compositional data. DDM includes, as submodels, the Beta Dynamic model, static Dirichlet regression, and a competitor of the static Beta regression. We design both on-line and off-line approaches for the estimation of the parameters in the model. The on-line version is adequate for recursive estimation while the off-line one, which is based on stochastic simulation via Markov Chain Monte Carlo (MCMC), can be used when there are some specific unknown parameters in the model. We discuss the practical use of the proposed model in describing the past behavior of the series, as well as in the prediction process. We also discuss the application of DDM in a static context. This particular case is important because when the latent states in the model do not vary over time, DDM takes the form of a Dirichlet regression model. Some DDM submodels, such as the dynamic Beta model and the static Beta regression, are also discussed.

Keywords:

• Bayesian approach
• Beta distribution
• Compositional data
• Dirichlet distribution
• Dynamic models
• Logistic-Normal distribution
• Time series

Mathematics Subject Classification:

• 91B82
• 62F15

Acknowledgments

The authors would like to thank the anonymous referees for their careful reading of the manuscript and for their constructive suggestions that considerably improved the article.

Additional information

Funding

The authors also thank the Office to Improve University Research (CAPES-Brazil), for financial support via Project PROCAD-NF 2008. Cibele Q. da-Silva received support from CNPq-Brazil, and BPPesq. Helio S. Migon received support from the National Research Council (CNPq-Brazil, BPPesq) and the Rio de Janeiro State Research Foundation (Faperj) together with CAPES via the Pronex project.