Statistical Inference and Perfect Simulation for Point Processes Observed with Noise

Physics under supervision of professor Mats Rudemo. The most recent version of the papers in this thesis can always be obtained from my homepage It is usual to thank a lengthy list of persons in the preface of your thesis. I could certainly make such a list as I know and I have met a lot of wonderful, nice, and helpful persons during my Ph.D. studies. However, I am afraid that I would forget some of you, so I prefer to simply say THANK YOU without mentioning anybody. Your support, advice, and encouragement has been invaluable to me. I owe you all a great debt of gratitude. iii iv Summary The main themes of this thesis are spatial statistics and simulation algorithms. The thesis is split into ve papers that may be read independently. All ve papers deal with spatial models. Lund & Rudemo (1999), Lund et al. (1999), and Lund & Thhnnes (1999b) deal with the same new model for point processes observed with noise, and Lund et al. and analyse a new model for point processes observed with noise. Usually the analysis of spatial point patterns assume that the observed points (the true points) are a realization from a speciic model. In contrast our approach is to assume the observed pattern generated by thinning and displacement of the true points, and allow for contamination by points not belonging to the true pattern. Lund & Rudemo (1999) develop the model for point processes observed with noise. The likelihood function for an observation of a noise corrupted point pattern given the true positions is derived. As data for our analysis is indeed a realization of the underlying true process and its associated noise corrupted point pattern we need not consider a model for the underlying process. The parameters in the model describe how many of the true points are lost, how large the displacements are, and the number of contaminating surplus points. For estimation of the parameters in the noise model a deterministic, iterative, and approximative maximum likelihood estimation algorithm is developed. The likelihood function is a sum of an excessive large number of terms, and the algorithm works by nding large dominating terms. Alternative estimation methods are discussed. Lund et al. (1999) analyse the model developed in Lund & Rudemo (1999) with respect to the now unobserved true points. We assume a noisy observation of a true point pattern …