Determinantal point process models on the sphere

Møller, Jesper; Nielsen, Morten; Porcu, Emilio; Rubak, Ege

doi:10.3150/16-bej896

Cited by 36 publications

(46 citation statements)

References 28 publications

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“…Point process techniques routinely deal with data observed in a restricted subset of the spatial domain. However, statistical techniques for point processes on the sphere (Ripley, ; Bahcall & Soneira, ; Scott & Tout, ; Raskin, ; Robeson et al, ; Møller et al, ) are still surprisingly underdeveloped. In principle, this is just a matter of adapting the existing methodology for point patterns in two‐dimensional space (Diggle, ; Møller & Waagepetersen, ; Baddeley et al, ) to the sphere.…”

Section: Introductionmentioning

confidence: 99%

Point pattern analysis on a region of a sphere

Lawrence

Baddeley

Milne

et al. 2016

Stat

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Section: Introductionmentioning

confidence: 99%

Point pattern analysis on a region of a sphere

Lawrence

Baddeley

Milne

et al. 2016

Stat

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“…Instead, properties of Legendre polynomials (see Szegö, ) imply that correlations based on geodesic distance can attain any value between −1 and +1. Another argument in favour of the great circle distance is that the differentiability of a given covariance function depending on the great circle distance can be modeled by imposing a given rate of decay of the associated 2‐Schoenberg coefficients

{false{b_{k} false}}_{k = 0}^{\infty}

, as shown in Møller et al () or even by modeling the rate of decay of the associated 2‐Schoenberg functions in , as shown by Clarke et al ().…”

Section: Second‐order Approachesmentioning

confidence: 99%

“…Notable attempts have been made by Guinness and Fuentes (), with partial success. Møller et al () suggest modeling the n ‐Schoenberg coefficients (see Appendix A for details) while imposing a given rate of decay, but this does not allow for explicit closed forms. Compact Supports with Differentiable Temporal Margins. A potential drawback of the space–time construction as in Theorem is that it only allows for temporal margins that are non‐differentiable at the origin.…”

Section: Research Problemsmentioning

confidence: 99%

Modeling Temporally Evolving and Spatially Globally Dependent Data

Porcu

Alegría

Furrer

2018

Int Statistical Rev

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Summary The last decades have seen an unprecedented increase in the availability of data sets that are inherently global and temporally evolving, from remotely sensed networks to climate model ensembles. This paper provides an overview of statistical modeling techniques for space–time processes, where space is the sphere representing our planet. In particular, we make a distintion between (a) second order‐based approaches and (b) practical approaches to modeling temporally evolving global processes. The former approaches are based on the specification of a class of space–time covariance functions, with space being the two‐dimensional sphere. The latter are based on explicit description of the dynamics of the space–time process, that is, by specifying its evolution as a function of its past history with added spatially dependent noise. We focus primarily on approach (a), for which the literature has been sparse. We provide new models of space–time covariance functions for random fields defined on spheres cross time. Practical approaches (b) are also discussed, with special emphasis on models built directly on the sphere, without projecting spherical coordinates onto the plane. We present a case study focused on the analysis of air pollution from the 2015 wildfires in Equatorial Asia, an event that was classified as the year's worst environmental disaster. The paper finishes with a list of the main theoretical and applied research problems in the area, where we expect the statistical community to engage over the next decade.

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“…Some exceptions can be found in Møller et al (2015), where it is shown that the 1-Schoenberg coefficients associated to Wendland functions have oscillating behaviors. This makes extremely difficult to apply our results for our cases and opens future researches oriented to finding other criteria for equivalence of Gaussian measures over the sphere.…”

Section: Discussionmentioning

confidence: 99%

“…Let p i ∈ (0, 1) and τ i > 0, i = 1, 2 be the parameters of the negative Binomial distribution on S d (Møller et al, 2015), given by…”

Section: Multiquadric Covariance Functionmentioning

confidence: 99%

Mathematical Developments on Isotropic Positive Definite Functions on Spheres

Moahmmed¹

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Abstract:In this thesis, we investigate some problems related to positive definite functions on the sphere. As known, positive definite functions play an important role in representation theory, probability theory, stochastic processes, harmonic analysis and machine learning. This role is emphasized a long the thesis.First, equivalence of Gaussian measures represents a fundamental tool to establish the properties of maximum likelihood estimators as well as kriging predictions under fixed domain asymptotics. Classical results for the equivalence and orthogonality of measures associated to Gaussian fields on bounded sets of R d are available in the literature. The present work considers Gaussian fields defined over spheres of R d+1 , with covariance functions depending on the great circle distance. We provide necessary and sufficient conditions for the equivalence of two Gaussian measures with two different covariance models with associated d-Schoenberg sequences. As an example we study equivalence of Gaussian measures for some parametric families of covariance functions valid on spheres. A simulation study explores the consistency of the maximum likelihood estimator associated to the covariance parameters of some covariance models on the sphere. We face Problems 1 and 3 proposed in the essay by Gneiting (2013b) and related to the d-Schoenberg coefficients in the series expansion of members of Ψ d for a given d. Such problems have precise implications for the simulation of Gaussian fields on spheres as well as for applications to geostatistical data, with special emphasis to atmospheric sciences. We also show how to deduce the 2-Schoenberg coefficients of given parametric families of members of the class Ψ d , called respectively exponential and Askey families, for which the 1-coefficients were available but the 2-dimensional case was still elusive.Third, we propose and define a family of marked point processes in a noncompact semisimple Lie groups. We first generate Lévy processes via marked point processes by using jump-diffusion processes. We then build a family of Markov processes in a maximal compact subgroup of a given semisimple Lie group. With all my heart, thank to you all.

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Determinantal point process models on the sphere

Cited by 36 publications

References 28 publications

Point pattern analysis on a region of a sphere

Point pattern analysis on a region of a sphere

Modeling Temporally Evolving and Spatially Globally Dependent Data

Mathematical Developments on Isotropic Positive Definite Functions on Spheres

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