Different Aspects of Inference for Spatio-Temporal Point Processes

Licentiatavhandling, 2010

This thesis deals with inference problems related to the Renshaw-Särkkä growth interaction model (RS-model). It is a continuous time spatio-temporal point process with time dependent interacting marks, in which the immigrationdeath process (a continuous time Markov chain) controls the arrivals of new marked points as well as their potential life-times. The data considered are marked point patterns sampled at fixed time points. First we propose three edge correction methods for discretely sampled (marked) spatio-temporal point processes. These are all based on the idea of placing an approximated expected behaviour of our process at hand (based on simulated realisations) outside the study region, which in turn interacts with the data during the estimation. We study the methods and evaluate them numerically in the context of the RS-model. The parameters related to the development of the marks are estimated using the least-squares approach. Secondly, we propose (approximate) maximum likelihood (ML) estimators for the two parameters of the immigration-death process; the arrival intensity and the death rate. The arrival intensity is assumed to be constant and the death rate is assumed to be proportional to a function of the current mark size of a point. The arrival intensity estimator is constructed to compensate for the (unobserved) individuals arriving and dying between two sampled time points. When assumed that the death rate is constant we can derive the transition probabilities of the immigration-death process. These in turn give us the exact likelihood of its parameter pair. We are able to reduce the likelihood maximisation problem from two dimensions to one dimension. Furthermore, under the condition that the parameter pair lies in some compact subset of the positive part of the real plane, we manage to show the consistency and the asymptotic normality of its ML-estimator under an equidistant sampling scheme. These results are also evaluated numerically.