Estimation and Modelling Repeated Patterns in High Order Markov Chains with the Mixture Transition Distribution Model

Abstract

The mixture transition distribution (MTD) model was introduced by Raftery as a parsimonious model for high order Markov chains. It is flexible, can represent a wide range of dependence patterns, can be physically motivated, fits data well and is in several ways a discrete-valued analogue for the class of autoregressive time series models. However, estimation has presented difficulties because the parameter space is highly non-convex, being defined by a large number of non-linear constraints. Here we propose a computational algorithm for maximum likelihood estimation which is based on a way of reducing the large number of constraints. This also allows more structured versions of the model, e.g. those involving structural zeros, to be fitted quite easily. A way of fitting the model by using GLIM is also discussed. The algorithm is applied to a sequence of wind directions, and also to two sequences of deoxyribonucleic acid bases from introns from mouse genes. In each case, the MTD model fits better than the conventional Markov chain model, and for the wind data it provides superior out-of-sample predictions. A modification of the model to represent repeated patterns is proposed and a very parsimonious version of this modified model is successfully applied to data representing bird songs.