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Structured Space-Sphere Point Processes and K-Functions

  • Jesper MøllerEmail author
  • Heidi S. Christensen
  • Francisco Cuevas-Pacheco
  • Andreas D. Christoffersen
Article
  • 7 Downloads

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 

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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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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark

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