Determinantal point process models on the sphere

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12 Citations (Scopus)

Abstract

We consider determinantal point processes on the d-dimensional unit sphere 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 S d ×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 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.

Original languageEnglish
JournalBernoulli
Volume24
Issue number2
Pages (from-to)1171-1201
Number of pages31
ISSN1350-7265
DOIs
Publication statusPublished - 2018

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Point Process
Process Model
Covariance Function
kernel
Moment
Spectral Representation
Eigenvalues and Eigenfunctions
Complex Functions
Unit Sphere
Figure
Determinant
Model
Simulation

Keywords

  • isotropic covariance function
  • joint intensities
  • quantifying repulsiveness
  • Schoenberg representation
  • spatial point process density
  • spectral representation

Cite this

@article{ba5096614f6b41fda8db291b20424765,
title = "Determinantal point process models on the sphere",
abstract = "We consider determinantal point processes on the d-dimensional unit sphere 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 S d ×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 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.",
keywords = "isotropic covariance function, joint intensities, quantifying repulsiveness, Schoenberg representation, spatial point process density, spectral representation",
author = "Jesper M{\o}ller and Morten Nielsen and Emilio Porcu and Rubak, {Ege Holger}",
year = "2018",
doi = "10.3150/16-BEJ896",
language = "English",
volume = "24",
pages = "1171--1201",
journal = "Bernoulli",
issn = "1350-7265",
publisher = "International Statistical Institute",
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}

Determinantal point process models on the sphere. / Møller, Jesper; Nielsen, Morten; Porcu, Emilio; Rubak, Ege Holger.

In: Bernoulli, Vol. 24, No. 2, 2018, p. 1171-1201.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Determinantal point process models on the sphere

AU - Møller, Jesper

AU - Nielsen, Morten

AU - Porcu, Emilio

AU - Rubak, Ege Holger

PY - 2018

Y1 - 2018

N2 - We consider determinantal point processes on the d-dimensional unit sphere 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 S d ×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 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.

AB - We consider determinantal point processes on the d-dimensional unit sphere 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 S d ×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 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.

KW - isotropic covariance function

KW - joint intensities

KW - quantifying repulsiveness

KW - Schoenberg representation

KW - spatial point process density

KW - spectral representation

U2 - 10.3150/16-BEJ896

DO - 10.3150/16-BEJ896

M3 - Journal article

VL - 24

SP - 1171

EP - 1201

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 2

ER -