Enhanced point pattern analysis on nonconvex spatial domains

Abstract

Point pattern analysis is the study of the spatial arrangement of points in space, usually two-dimensional space. The points arise from a stochastic mechanism, termed a point process, whose characteristics are of scientific interest. The properties of point patterns are characterised using statistical measures that are a function of the study area and distance. Consequently, the domain in which points are observed and the distance metric used to quantify proximity between points plays an important role. Convex domains with the Euclidean distance are often used. This choice of domain and distance measure, however, makes an implicit assumption that all points are connected in a space without obstacles. In real-world applications, points may be constrained by their environments, thus a convex window and the Euclidean distance may not correctly capture spatial proximity relationships and restrictions imposed by the domain’s geometry. In this thesis, a presentation of methodology that accounts for the nonconvex structure of the spatial domain in point pattern analysis is provided. Firstly, consideration is given to the selection of nonconvex windows (when unknown) for point patterns realised from a process that is governed by a covariate. The proposed algorithm uses a weighted distance-based outlier scoring scheme that considers the distribution of covariates at observed data point locations. The robustness of the algorithm is demonstrated through a simulation study. Subsequently, a framework is developed to quantify proximity relationships using a graph theoretic approach based on visibility graphs. This characterisation of distance is used to extend first- and second-order point pattern measures for appropriate use on nonconvex domains. Finally, we provide an implementation strategy to efficiently compute summary measures based on a query to the visibility graph.