Modeling the spread of plant disease using a sequence of binary random fields with absorbing states

Abstract

The presence or absence of some event at locations on a spatial lattice can often be modeled using a binary Markov random field model. If observations are repeated over time at the same locations this model structure can be extended to include neighbors in both space and time, as long as conditions sufficient to allow the construction of a joint distribution from the set of specified conditionals are satisfied. One of those conditions involves the relation between the support of the joint distribution and the individual supports of marginal distributions at each location. When a sequence of binary random fields involves an absorbing state and more than two points in time, the support conditions needed to construct a model for the entire sequence as a Markov random field are not satisfied. An application involving the spread of a soybean disease across a field motivates development of a model that avoids the difficulties caused by absorbing states over time, by formulating the overall model as a sequence of conditional Markov random fields which themselves follow a Markov property in time.