Spatial Point Processes and their Applications

A spatial point process is a random pattern of points in d-dimensional space (where usually d = 2 or d = 3 in applications). Spatial point processes are useful as statistical models in the analysis of observed patterns of points, where the points represent the locations of some object of study (e..g. trees in a forest, bird nests, disease cases, or petty crimes). Point processes play a special role in stochastic geometry, as the building blocks of more complicated random set models (such as the Boolean model), and as instructive simple examples of random sets. These lectures introduce basic concepts of spatial point processes, with a view toward applications, and with a minimum of technical detail. They cover methods for constructing, manipulating and analysing spatial point processes, and for analysing spatial point pattern data. Each lecture ends with a set of practical computer exercises, which the reader can carry out by downloading a free software package. Lecture 1 ('Point Processes') gives some motivation, defines point processes, explains how to construct point processes, and gives some important examples. Lecture 2 ('Moments') discusses means and higher moments for point processes, especially the intensity measure and the second moment measure, along with derived quantities such as the K-function and the pair correlation function. It covers the important Campbell formula for expectations. Lecture 3 ('Conditioning') explains how to condition on the event that the point process has a point at a specified location. This leads to the concept of the Palm distribution, and the related Campbell-Mecke formula. A dual concept is the conditional intensity, which provides many new results. Lecture 4 ('Mod-elling and Statistical Inference') covers the formulation of statistical models for point patterns, model-fitting methods, and statistical inference.