Preprint #96-11
Using Hidden Markov Chains and Empirical Bayes
Change-Point Estimation for Transect Data
by
Jay M. Ver Hoef and Noel Cressie
Abstract
Consider a lattice of locations in one dimension at which data are observed.
We model the data as a random hierarchical process. The hidden process is
assumed to have a (prior) distribution that is derived from a two-state
Markov chain. The states correspond to the mean values (high and low) of
the observed data. Conditional on the states, the observations are modeled,
for example, as independent Gaussian random variables with identical
variances. In this model, there are four free parameters: the Gaussian
variance, the high and low mean values, and the transition probability
in the Markov chain. A parametric empirical Bayes approach requires
estimation of these four parameters from the marginal (unconditional)
distribution of the data and we use the EM-algorithm to do this. From the
posterior of the hidden process, we use simulated annealing to find the
maximum a posteriori (MAP) estimate. Using a Gibbs sampler, we
also obtain the maximum marginal posterior probability (MMPP) estimate of
the hidden process. We use these methods to determine where change-points
occur in spatial transects through grassland vegetation, a problem of
considerable interest to plant ecologists.
Copies of preprints are available from the author upon request. Use
the preprint number (located at the top of the page) and
make the request directly to the author, Iowa State
University,
Department of Statistics, Snedecor Hall, Ames, IA 50011-1210.