This thesis is concerned with some practical and theoretical problems in the field of spatial stochastic point processes. The work described falls into two main areas: Chapters 1-3 describe some techniques for analysing spatial patterns, while, in Chapters 4-7, properties of certain stochastic models for spatial patterns are investigated, theoretically and empirically. We consider simple point processes in the plane, strictly stationary under rigid motions of the plane; many of the techniques and results can be generalised to more than two dimensions. Chapter 1 discusses the use of empty space in analysing spatial patterns; the method described estimates the probability that a randomly placed set of a given shape and size is empty of points of the pattern. Chapter 2 describes the use of histograms to estimate g(t)/2 for a spatial pattern; this technique is of use during the earlier stages of the analysis of a pattern. In Chapter 3, empty space techniques are generalised for the analysis of 'multitype' point patterns, and are compared with second-order methods for analysing such patterns. All the techniques of Chapters 1-3 are illustrated by examples. Chapters 4-7 deal mainly with models for 'hard core' point patterns. Chapter 4 contains a review of various models for hard core processes. Chapters 5 and 6 are concerned with spatial birth and death processes which are used to simulate realisations of certain models for point patterns, notably Kelly-Ripley models. In Chapter 5, coupling techniques from Markov Chain theory are used to obtain theoretical convergence results for such processes. Chapter 6 describes a computer algorithm for simulating hard core birth and death processes at high packing densities. In Chapter 7, this algorithm is modified to simulate the 'SSI' process and to investigate the complete packing of non-overlapping discs in a rectangular container.
|Date of Award||1981|