# Structured Space-Sphere Point Processes and *K*-Functions

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## 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*-dimensional 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*.

## Keywords

First and second order separability Functional summary statistic Log Gaussian Cox process Pair correlation function Shot noise Cox process## Mathematics Subject Classification (2010)

60G55 62M30## Preview

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## Notes

### Acknowledgements

The authors are grateful to Jiří Dvořák for helpful comments and to Ali H. Rafati for collecting the pyramidal cell data.

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