Bayesian Models and Methods for Binary Time Series

From the perspective of applied statistical modelling, binary time series analysis and forecasting are relatively undeveloped areas. This paper reports on preliminary investigations of the use of some Bayesian models, discussing a variety of mathematical and practical modelling issues. A flexible class of models is that based on logistic linear regressions which, with an emphasis on sequential forecasting, are provided as a subset of the class of dynamic generalised linear models (West, Harrison and Migon, 1985). Special cases are Markov chains, considered here in detail, and non-stationary Markov chains with time evolving transition probabilities. In Sections 2 and 3 we discuss the use of low order Markov chains to model the non-Markov structure of binary series derived as qualitative summaries of underlying quantitative processes. An example concerns binary data indicating when a real valued process exceeds a specified threshold level. Such clipped processes arise naturally in monitoring problems in, for example, river flow and dam water level management; pollution emission regulation; clinical measurements such as blood pressure, in patient care; financial and economic time series forecasting; and so forth. In the context of an underlying gaussian process generated by a simple, yet widely used, dynamic linear model we show how simple Markov models can approximate derived binary processes.