Open Access
May 2018 Determinantal point process models on the sphere
Jesper Møller, Morten Nielsen, Emilio Porcu, Ege Rubak
Bernoulli 24(2): 1171-1201 (May 2018). DOI: 10.3150/16-BEJ896


We consider determinantal point processes on the $d$-dimensional unit sphere $\mathbb{S}^{d}$. These are finite point processes exhibiting repulsiveness and with moment properties determined by a certain determinant whose entries are specified by a so-called kernel which we assume is a complex covariance function defined on $\mathbb{S}^{d}\times\mathbb{S}^{d}$. We review the appealing properties of such processes, including their specific moment properties, density expressions and simulation procedures. Particularly, we characterize and construct isotropic DPPs models on $\mathbb{S}^{d}$, where it becomes essential to specify the eigenvalues and eigenfunctions in a spectral representation for the kernel, and we figure out how repulsive isotropic DPPs can be. Moreover, we discuss the shortcomings of adapting existing models for isotropic covariance functions and consider strategies for developing new models, including a useful spectral approach.


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Jesper Møller. Morten Nielsen. Emilio Porcu. Ege Rubak. "Determinantal point process models on the sphere." Bernoulli 24 (2) 1171 - 1201, May 2018.


Received: 1 October 2015; Revised: 1 July 2016; Published: May 2018
First available in Project Euclid: 21 September 2017

zbMATH: 06778362
MathSciNet: MR3706791
Digital Object Identifier: 10.3150/16-BEJ896

Keywords: isotropic covariance function , joint intensities , quantifying repulsiveness , Schoenberg representation , spatial point process density , ‎spectral representation

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 2 • May 2018
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