Functional summary statistics for point processes on the sphere with an application to determinantal point processes

TL;DR

This paper develops and analyzes functional summary statistics for point processes on the sphere, focusing on their properties, especially for determinantal point processes, and explores their applications in modeling spatial patterns.

Contribution

It introduces new functional summary statistics for spherical point processes and examines their application to determinantal point processes, highlighting their repulsive and aggregative properties.

Findings

DPPs exhibit repulsiveness on the sphere

Functional summary statistics effectively characterize spatial patterns

Application of DPPs with dependent thinnings models complex spatial structures

Abstract

We study point processes on $\mathbb S^d$, the $d$-dimensional unit sphere $\mathbb S^d$, considering both the isotropic and the anisotropic case, and focusing mostly on the spherical case $d=2$. The first part studies reduced Palm distributions and functional summary statistics, including nearest neighbour functions, empty space functions, and Ripley's and inhomogeneous $K$-functions. The second part partly discusses the appealing properties of determinantal point process (DPP) models on the sphere and partly considers the application of functional summary statistics to DPPs. In fact DPPs exhibit repulsiveness, but we also use them together with certain dependent thinnings when constructing point process models on the sphere with aggregation on the large scale and regularity on the small scale. We conclude with a discussion on future work on statistics for spatial point processes on the sphere.