Modelling and Inference for Spatio-Temporal Marked Point Processes
Abstract: This thesis deals with inference problems related to the growth-interaction process (GI-process). The GI-process is a continuous time spatio-temporal point process with dynamic interacting marks (closed disks), in which the immigration-death process (ID-process) 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 and the main area of application of the GI-process is the dynamical modelling of the trees in forest stands. The parameters related to the development of the marks are estimated using the least-squares (LS) approach. The death rate, which is assumed to be a function of the mark sizes, and the arrival intensity and are estimated by (approximate) maximum likelihood (ML) methods. We also propose three edge correction methods for discretely sampled (marked) spatio-temporal point processes. The edge correction methods together with the LS approach are applied to fit the GI-process to a forest stand of Scots pines. We derive the transition probabilities of the (Markovian) ID-process, which form the likelihood function of its two parameters. We further reduce the ML-problem from two dimensions to one dimension. Given an equidistant sampling scheme and some conditions for the parameter space, we manage to prove the consistency and the asymptotic normality of the ML-estimators. The results are also evaluated numerically. Measurements of locations and radii at breast height (rbh) made at 3 different time points of the individual trees in 10 Swedish Scots pine stands, are modelled spatio-temporally by the GI-process. A new location assignment strategy and a more flexible function for the open-growth (growth in absence of competition) are suggested in order to improve the fit. A linear relationship is found between the site productivity index (fertility) and the sizes of the trees. This relationship is exploited in the estimation of the carrying capacity parameter (theoretical upper bound for the radii). We also test the goodness-of-fit of the fitted model in terms of prediction. By adding scaled continuous white noise to the mark growth equations, we obtain a system of stochastic differential equation (SDEs) for the mark growth. We consider the case where there is no interaction present and the mark SDEs are independent Cox-Ingersoll-Ross SDEs. Closed form expressions are available both for the transition densities and the stationary distributions. Under the assumption that the mark processes are stationary, consistency and asymptotic normality of the ML-estimators of the parameters are proved.
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