The Mixture Transition Distribution Model for High-Order Markov Chains and Non-Gaussian Time Series

The mixture transition distribution model (MTD) was introduced in 1985 by Raftery for the modeling of high-order Markov chains with a finite state space. Since then it has been generalized and successfully applied to a range of situations, including the analysis of wind directions, DNA sequences and social behavior. Here we review the MTD model and the developments since 1985. We first introduce the basic principle and then we present several extensions, including general state spaces and spatial statistics. Following that, we review methods for estimating the model parameters. Finally, a review of different types of applications shows the practical interest of the MTD model.