We point out a general problem in fitting continuous time spatially explicit models to a temporal sequence of spatial data observed at discrete times. To illustrate the problem, we examined the continuous time Markov model for forest gap dynamics. A forest is assumed to be apportioned into discrete cells (or sites) arranged in a regular square lattice. Each site is characterized as either a gap or a non-gap site according to the vegetation height of trees. The model incorporates the influence of neighboring sites on transition rate: transition rate from a non-gap to a gap site increases linearly with the number of neighbors that are currently in the gap state, and vice versa. We fitted the model to the spatiotemporal data of canopy height observed at the permanent plot in Barro Colorado Island (BCI). When we used the approximate maximum likelihood method to estimate the parameters of the model, the estimated transition rates included a large bias---in particular, the strength of interaction between nearby sites was underestimated. This bias originated from the assumption that each transition between two observation times is independent. The interaction between sites at local scale creates a long chain of transitions within a single census interval, which violates the independence of each transition. We show that a computer-intensive method, called Monte Carlo bias correction (MCBC), is very effective in removing the bias included in the estimate. The global and local gap densities measuring spatial aggregation of gap sites were computed from simulated and real gap dynamics to assess the model. When the approximate likelihood estimates were applied to the model, the predicted local gap density was clearly lower than the observed one. The use of MCBC estimates, suggesting a strong interaction between sites, improved this discrepancy.
Key Words: Gap dynamics; Markov model; Maximum likelihood method; Bias correction; Parameter estimation; Spatial data