arXiv subjects

Scott Ward

Publications and source records attributed to Scott Ward.

3 recordsLinked to original sources

arXiv2026

$K$-functions for point processes on complex surfaces

The $K$-function is a fundamental summary statistic for assessing clustering or regularity of point processes in two or three dimensional Euclidean space. In practice, however, many planar point patterns arise from projecting locations of objects on a surface in three dimensional space to two dimensional space. For example, when events or objects occur in a landscape their elevation is often ignored. This can lead to erroneous conclusions regarding properties of the point process generating the point pattern. There is not a unique way to extend the classical $K$-function to point patterns on a complex surface. In this paper we propose, explore, and discuss several approaches in terms of their theoretical and computational properties. The best performing approach, coined the surface area $K$-function, can be viewed is an analogue of the classical $K$-function replacing counts of points in Euclidean balls with counts of points in surface geodesic balls. However, an important distinction is that the argument of our surface area $K$-function is area instead of radius of geodesic balls. The performances of the various surface $K$-functions are compared in applications to simulated and real data.

stat.ME↗

arXiv2024

Functional summary statistics and testing for independence in marked point processes on the surface of three dimensional convex shapes

The fundamental functional summary statistics used for studying spatial point patterns are developed for marked homogeneous and inhomogeneous point processes on the surface of a sphere. These are extended to point processes on the surface of three dimensional convex shapes given the bijective mapping from the shape to the sphere is known. These functional summary statistics are used to test for independence between the marginals of multi-type spatial point processes with methods for sampling the null distribution proposed and discussed. This is illustrated on both simulated data and the RNGC galaxy point pattern, revealing attractive dependencies between different galaxy types.

stat.ME↗

arXiv2020

Testing for complete spatial randomness on three dimensional bounded convex shapes

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 test whether an observed pattern exhibits complete spatial randomness or spatial preference on the original convex space. A study of the Type I and II errors of our test statistics are explored through simulations on ellipsoids of varying dimensions.

math.ST↗