An extension of a theorem of Schoenberg to products of spheres

Guella, Jean Carlo; Menegatto, V. A.; Peron, Ana Paula

doi:10.1215/17358787-3649260

Cited by 29 publications

(38 citation statements)

References 18 publications

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“…Positive definite functions as well as strictly positive definite functions in several contexts have been deeply studied by the mathematical analysis literature, and the reader is referred to the works by Menegatto et al (see Chen et al [11], Menegatto and Peron [26], Guella et al [19], and references therein). The use of positive definite functions on real spheres for geostatisticians has arrived recently, thanks to the survey by Gneiting [17] and the recent developments by Berg and Porcu [7] and Porcu et al [28].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The class Ψ(Ω 2q ) is parenthetical to the class Ψ d introduced by Schoenberg [40], and we refer the reader to the recent review in Gneiting [17] for a thorough description of the properties of this class. Further, the class Ψ d represents the building block for extension to product spaces, and the reader is referred to Berg and Porcu [7] as well as to Guella et al [19] for recent efforts in this direction. The classes Ψ(Ω 2q ) are nested, with the following inclusion relation being strict:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Massa¹,

Peron²,

Porcu³

2017

SIGMA

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Abstract. We provide walks through dimensions for isotropic positive definite functions defined over complex spheres. We show that the analogues of Montée and , 1965], allow for similar walks through dimensions. We show that the Montée operators also preserve, up to a constant, strict positive definiteness. For the Descente operators, we show that strict positive definiteness is preserved under some additional conditions, but we provide counterexamples showing that this is not true in general. We also provide a list of parametric families of (strictly) positive definite functions over complex spheres, which are important for several applications.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Massa¹,

Peron²,

Porcu³

2017

SIGMA

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“…Because real phenomena on the globe are notably non‐stationary, it is necessary to use those models as building blocks for more sophisticated constructions. In particular, the development of non‐stationary models is necessary and a promising direction for research is to extend the work of Guella et al () who define kernels over products of n ‐dimensional spheres. Another direction for research might be to take into account differences in local geometry of the sphere representing planet Earth.…”

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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“…Positive definite functions on real and complex spheres have a long history, that starts with the seminal paper by Schoenberg [41]. Positive definiteness is crucial to many branches of mathematical analysis [4,5,9,17,21,22,23,24,33,34,36,41] and statistics [3,7,10,11,12,13,14,18,25,26,27,28,29,30,31,38,40]. Recent reviews on positive definite functions on either spheres or product spaces involving spheres can be found in [19] and in [39] as well.…”

Section: Introductionmentioning

confidence: 99%

Relations between Schoenberg Coefficients on Real and Complex Spheres of Different Dimensions

Bissiri

Menegatto

Porcu

2019

SIGMA

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Positive definite functions on spheres have received an increasing interest in many branches of mathematics and statistics. In particular, the Schoenberg sequences in the spectral representation of positive definite functions have been studied by several mathematicians in the last years. This paper provides a set of relations between Schoenberg sequences defined over real as well as complex spheres of different dimensions. We illustrate our findings describing an application to strict positive definiteness.

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An extension of a theorem of Schoenberg to products of spheres

Cited by 29 publications

References 18 publications

Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Modeling Temporally Evolving and Spatially Globally Dependent Data

Relations between Schoenberg Coefficients on Real and Complex Spheres of Different Dimensions

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