Testing for complete spatial randomness on three dimensional bounded convex shapes
File(s)
Published version
Supporting information
Author(s)
Ward, Scott
Cohen, Edward
Adams, niall
Type
Journal Article
Abstract
There is currently a gap in theory for point patterns that lie on the surface of objects, with researchers focusing on patterns that lie in a Euclidean space, typically planar and spatial data. Methodology for planar and spatial data thus relies on Euclidean geometry and is therefore inappropriate for analysis of point patterns observed in non-Euclidean spaces. Recently, there has been extensions to the analysis of point patterns on a sphere, however, many other shapes are left unexplored. This is in part due to the challenge of defining the notion of stationarity for a point process existing on such a space due to the lack of rotational and translational isometries. Here, we construct functional summary statistics for Poisson processes defined on convex shapes in three dimensions. Using the Mapping Theorem, a Poisson process can be transformed from any convex shape to a Poisson process on the unit sphere which has rotational symmetries that allow for functional summary statistics to be constructed. We present the first and second order properties of such summary statistics and demonstrate how they can be used to construct a test statistics to determine whether an observed pattern exhibits complete spatial randomness or spatial preference on the original convex space. We compare this test statistic with one constructed from an analogue L-function for inhomogeneous point processes on the sphere. A study of the Type I and II errors of our test statistics are explored through simulations on ellipsoids of varying dimensions.
Date Issued
2021-03-01
Date Acceptance
2020-12-18
Citation
Spatial Statistics, 2021, 41
ISSN
2211-6753
Publisher
Elsevier
Journal / Book Title
Spatial Statistics
Volume
41
Copyright Statement
© 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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Sponsor
Wellcome Trust
Grant Number
203799/Z/16/Z
Subjects
0104 Statistics
0801 Artificial Intelligence and Image Processing
Publication Status
Published
Article Number
ARTN 100489
Date Publish Online
2021-01-04