The analysis of marked point patterns evolving through space and time

A maximum pseudo-likelihood approach has previously been developed for fitting pairwise interaction models to patterns generated by growth–interaction processes that are sampled at fixed time points. This approach is now extended, not only by estimating the parameters of the process through time, but also by employing least squares estimation since likelihood based approaches are much more computationally demanding. First, simple stochastic models are used to demonstrate that least squares methods are as powerful as likelihood-based approaches, as well as being mathematically and computationally simpler. The algorithm generates simulations of the deterministic growth–interaction and stochastic immigration–death process, and through these the parameter estimates are determined. Logistic and linear growth are then combined with (symmetric) disc-interaction and (asymmetric) area-interaction processes, and between them these generate a variety of mark–point spatial structures. A robustness study shows that the procedure works well in that the presence, structure and strength of a growth–interaction process can be determined even when an incorrectly presumed model is employed. Thus, the technique is likely to prove to be very useful in general practical applications where the underlying process generating mechanism is almost certain to be unknown. Finally, the procedure is applied to the analysis of a new Swedish pine forest data set for which tree location and diameter at breast height were recorded in 1985, 1990 and 1996.