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

TL;DR

This paper develops and extends functional summary statistics for analyzing marked point processes on 3D convex shapes, enabling tests for independence between different types, with applications to simulated data and galaxy patterns.

Contribution

It introduces new methods for functional summary statistics on convex shapes and proposes testing procedures for independence in multi-type point processes.

Findings

Effective detection of dependencies in galaxy data

Extension of sphere-based methods to convex shapes

Proposed sampling methods for null distribution

Abstract

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.