6 - Stochastic Models

Summary

The previous chapter outlined likelihood analysis of some standard models. Here we turn to data in which the dependence among the observations is more complex. We start by explaining how our earlier discussion extends to Markov processes in discrete and continuous time. We then extend this to more complex indexing sets and in particular to Markov random fields, in which basic concepts from graph theory play an important role. A special case is the multivariate normal distribution, an important model for data with several responses. We give some simple notions for time series, a very widespread form of dependent data, and then turn to point processes, describing models for rare events in passing.

Markov Chains

In certain applications interest is focused on transitions among a small number of states. A simple example is rainfall modelling, where a sequence … 010011 … indicates whether or not it has rained each day. Another is in panel studies of employment, where many individuals are interviewed periodically about their employment status, which might be full-time, part-time, home-worker, unemployed, retired, and so forth. Here interest will generally focus on how variables such as age, education, family events, health, and changes in the job market affect employment history for each interviewee, so that there are many short sequences of state data taken at unequal intervals, unlike the single long rainfall sequence. In each case, however, the key aspect is that transitions occur amongst discrete states, even though these typically are crude summaries of reality.