# Structured space-sphere point processes and $K$-functions

## TL;DR

This paper introduces models and a $K$-function extension for analyzing point processes on combined Euclidean and spherical spaces, enabling better model checking and fitting for complex spatial-spherical data.

## Contribution

It develops the space-sphere $K$-function and explores conditions for its separability, extending existing models and tools to space-sphere point processes.

## Key findings

- The space-sphere $K$-function effectively characterizes point processes on combined spaces.
- Separable structure assumptions allow the $K$-function to factor into spatial and spherical components.
- Illustrations demonstrate the $K$-function's utility on real and simulated datasets.

## Abstract

This paper concerns space-sphere point processes, that is, point processes on the product space of $\mathbb R^d$ (the $d$-dimensional Euclidean space) and $\mathbb S^k$ (the $k$-dimen\-sional sphere). We consider specific classes of models for space-sphere point processes, which are adaptations of existing models for either spherical or spatial point processes. For model checking or fitting, we present the space-sphere $K$-function which is a natural extension of the inhomogeneous $K$-function for point processes on $\mathbb R^d$ to the case of space-sphere point processes. Under the assumption that the intensity and pair correlation function both have a certain separable structure, the space-sphere $K$-function is shown to be proportional to the product of the inhomogeneous spatial and spherical $K$-functions. For the presented space-sphere point process models, we discuss cases where such a separable structure can be obtained. The usefulness of the space-sphere $K$-function is illustrated for real and simulated datasets with varying dimensions $d$ and $k$.