Analysis of spatial point patterns on surfaces
Creators: name: Ward, Scott
Publisher: name: Imperial College London
Year: 2021

description: With the advent of improved data acquisition technologies more complex spatial datasets can be collected at scale meaning theoretical and methodological developments in spatial statistics are imperative in order to analyse and generate meaningful conclusions. Spatial statistics has seen a plethora of applications in life sciences with particular emphasis on ecology, epidemiology and cell microscopy. Applications of these techniques provides researchers with insight on how the locations of objects of interest can be influenced by their neighbours and the environment. Examples include understanding the spatial distribution of trees observed within some window, and understanding how neighbouring trees and potentially soil contents can influence this. Whilst the literature for spatial statistics is rich the common assumption is that point processes are usually restricted to some d-dimensional Euclidean space, for example cell locations in a rectangular window of 2-dimensional Euclidean space. As such current theory is not capable of handling patterns which lie on more complex spaces, for example cubes and ellipsoids. Recent efforts have successfully extended methodology from Euclidean space to spheres by using the chordal distance (the shortest distance between any two points on a sphere) in place of the Euclidean distance. In this thesis we build on this work by considering point processes lying on more complex surfaces. Our first significant contribution discusses the construction of functional summary statistics for Poisson processes which lie on compact subsets of Rd which are off lower dimension. We map the process from its original space to the sphere where it is possible to take advantage of rotational symmetries which allow for well-defined summary statistics. These in turn can be used to determine whether an observed point patterns exhibits clustered or regular behaviour. Partnering this work we also provide a hypothesis testing procedure based on these functional summary statistics to determine whether an observed point pattern is complete spatially random. Two test statistics are proposed, one based on the commonly used L-function for planar processes and the other a standardisation of the K-function. These test statistics are compared in an extensive simulation study across ellipsoids of varying dimensions and processes which display differing levels of aggregation or regularity. Estimates of first order properties of a point process are extremely important. They can provide a graphical illustration of inhomogeneity and are useful in second order analysis. We demonstrate how kernel estimation can be extended from a Euclidean space to a Riemannian manifold where the Euclidean metric is now substituted for a Riemannian one. Many of the desirable properties for Euclidean kernel estimates carry over to the Riemannian setting. The issue of edge correction is also discussed and two criteria for bandwidth selection are proposed. These two selection criteria are explored through a simulation study. Finally, an important area of research in spatial statistics is exploring the interaction between different processes, for example how different species of plant spatially interact within some window. Under the framework of marked point processes we show that functional summary statistics for multivariate point patterns can be constructed on the sphere. This is extended to more general convex shapes through an appropriate mapping from the original shape to the sphere. A number of examples highlight that these summary statistics can capture independence, aggregation and repulsion between components of a multivariate process on both the sphere and more general surfaces.