Different Aspects of Inference for Spatio-Temporal Point Processes
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.
Spatio-temporal marked point process
Maximum likelihood estimation
Least squares estimation
Pascal, Chalmers Tvärgata 3, Matematiska vetenskaper, Chalmers tekniska högskola
Opponent: Fil dr Lennart Norell, docent, universitetslektor, Enheten för tillämpad statistik och matematik, Svenska Lantbruksuniversitetet, Sverige