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Different Aspects of Inference for Spatio-Temporal Point Processes

Författare och institution:
Ottmar Cronie (Institutionen för matematiska vetenskaper, Chalmers/GU)
Utgiven i serie vid Göteborgs universitet:
Preprint - Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University, ISSN 1652-9715
Antal sidor:
Chalmers University of Technology
Datum för examination:
Tidpunkt för examination:
Pascal, Chalmers Tvärgata 3, Matematiska vetenskaper, Chalmers tekniska högskola
Fil dr Lennart Norell, docent, universitetslektor, Enheten för tillämpad statistik och matematik, Svenska Lantbruksuniversitetet, Sverige
Fulltextlänk (lokalt arkiv):
Sammanfattning (abstract):
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.
Ämne (baseras på Högskoleverkets indelning av forskningsämnen):
Matematik ->
Sannolikhetsteori och statistik ->
Matematisk statistik
Lantbruksvetenskap, skogsbruk och fiske ->
Skogsvetenskap ->
Asymptotic normality, Consistency, Edge correction, Immigration-death process, Least squares estimation, Maximum likelihood estimation, Spatio-temporal marked point process, Transition probability.
Ytterligare information:
ISSN: 1652-9715 Rapport-nummer (NO): 2010:5
Postens nummer:
Posten skapad:
2010-03-02 17:53
Posten ändrad:
2010-03-04 14:42

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