A framework for analysing point patterns on nonconvex domains using visibility graphs and multidimensional scaling

Abstract

A point pattern is typically analysed to understand the first- and second-order properties of the underlying point process. These properties are usually inferred using estimation procedures that depend on interpoint distance and are thus sensitive to the choice of distance metric. Euclidean distance is conventionally used to quantify proximity between points, but it does not accurately reflect spatial relationships when points are constrained within irregular, nonconvex spatial domains. Herein, we propose a strategy to embed visibility graph distances into Euclidean metric space using multidimensional scaling. The aim is to simplify analyses, leverage well-developed methods based on Euclidean distance, and retain, as far as possible, the true proximity relationships on a nonconvex spatial domain. The kernel smoothed intensity estimate and the K-function are computed in this new spatial context and used to validate the effectiveness of the embedding strategy.