Statistics Seminars 2005-2006
3 February 2006, L401, 2:15pm
Professor Eric Renshaw,
The Generation of Space-Time
Growth-Interaction Processes:
Inferring Structure from Partial Observations
Abstract
Not only have marked point processes received relatively little
attention in the literature in comparison to the study of purely point
processes, but virtually all analyses ignore the fact that in real life spatial
structure often develops dynamically through time. We shall therefore develop a
computationally fast and robust spatial-temporal process, based on stochastic
immigration-death and deterministic growth-interaction. For this enables marked
point process data, taken at both single and multiple snap-shot instants, 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, followed by the
construction of a least squares procedure which enables parameter estimation
based on multiple time points. Examples are shown for both simulated and real
data; robustness studies indicate that the procedure works well even when a
wrong model is employed, as is virtually bound to happen in practice.
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. The underlying
mathematics is impossibly hard and progress can only be made through
simulation. Using our process to develop high intensity packing patterns
extends current methods, since a newly arrived particle is not immediately
rejected if it does not fit into a specific gap, but changes size to suit the
interaction pressure placed on it; deletion only occurs when size becomes
non-positive. The range of packing structures that can be generated appears to
be large, and progress is currently being made to understand the roles that
individual parameters play in pattern formation
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