Dirichlet ARMA models for compositional time series

A compositional time series is a multivariate time series in which the observation vector at each time point is a set of proportions that sum to 1. Traditionally, such time series are modeled by taking a log-ratio transformation of the observations and then modeling them with a Gaussian vector autoregressive moving average (ARMA) model. In this paper, a new class of models is proposed by assuming that the proportions follow a time-varying Dirichlet distribution, and that the corresponding time-varying parameters, after a proper transformation, assume an ARMA-type of dynamic structure. The new model is referred to as the Dirichlet autoregressive moving average (DARMA) model. Under this model, after a proper transformation, the original data follow a vector ARMA model with a martingale difference sequence as its noise series. Two specific transformations are studied under the DARMA framework. Estimation procedures are developed and their numerical properties are investigated. Simulation studies and real examples are presented to demonstrate the properties of the proposed models, and comparisons are made with the existing modeling approaches.