A general space-time growth-interaction process for inferring and developing structure from partial observations
Seminar Room 1, Newton Institute
Not only have marked point processes received relatively little attention in the literature, but most analyses ignore the fact that in real life spatial structure often develops dynamically through time. We therefore develop a computationally fast and robust spatial-temporal process, based on stochastic immigration-death and deterministic growth-interaction. For this enables both single and multiple snap-shot marked point process data to be studied in considerable depth. Combining logistic and linear growth with (symmetric) disc-interaction and (asymmetric) area-interaction generates a wide variety of mark-point spatial structures. A maximum psuedo-likelihood approach is developed for parameter estimation at fixed times, and a least squares procedure for parameter estimation based on multiple time points.
A related problem in spatial statistics and stochastic geometry concerns the modelling and statistical analysis of hard particle systems involving discs or spheres. For successively filling remaining empty structure leads to a limiting maximum packing pattern whose structure depends on the given characteristics of the particles. Using our process to develop such patterns extends current methods, since a newly arrived particle is not immediately rejected if it does not fit into a specific gap, but can change size to adapt to the interaction pressure placed on it.
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