Gneiting Class, Semi-Metric Spaces and Isometric Embeddings

Menegatto, V. A.; Oliveira, C. P.; Porcu, Emilio

doi:10.33205/cma.712049

Cited by 12 publications

(14 citation statements)

References 30 publications

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“…A.2 If f is a Stieltjies function and h a Bernstein function, then G α (kÁk 2 , |Á| 2 ) is positive definite for all α > 0 and for all d = 1, 2, …. This result was recently proved by Menegatto, Oliveira, and Porcu (2019). A.3 Assume f is a generalized Wendland function, W in Equation ( 6).…”

Section: Gneiting Class Across Different Metric Spacesmentioning

confidence: 84%

“…A bridge between Gneiting functions and semi‐metric spaces has been recently provided by Menegatto et al (2019).…”

Section: Classes Of Covariance Functionsmentioning

confidence: 99%

“…Generalizations to multivariate space-time processes have been considered in Alegría, Porcu, Furrer, and Mateu (2019). The richness in modeling stochastic processes over spheres or spheres cross time is reflected in the diversity of research in this area: Mathematical analysis (Barbosa & Menegatto, 2017;Beatson, Zu Castell, & Xu, 2014;Chen, Menegatto, & Sun, 2003;Gangolli, 1967;Guella, Menegatto, & Peron, 2016a, 2016bHannan, 1970;Menegatto, 1994Menegatto, , 1995Menegatto, Oliveira, & Peron, 2006;Schoenberg, 1942), probability theory (Baldi & Marinucci, 2006;Clarke, Alegria, & Porcu, 2018;Hansen, Thorarinsdottir, Ovcharov, & Gneiting, 2015;Lang & Schwab, 2013), spatial point processes (Møller, Nielsen, Porcu, & Rubak, 2018), spatial geostatistics (Christakos & Papanicolaou, 2000;Gerber, Mösinger, & Furrer, 2017;Gneiting, 2002b;Hitczenko & Stein, 2012;Huang, Zhang, & Robeson, 2012), space-time geostatistics (Berg & Porcu, 2017;Christakos, 1991aChristakos, , 2000Christakos, Hristopulos, & Bogaert, 2000;Porcu et al, 2016) and mathematical physics (Istas, 2005;Leonenko & Sakhno, 2012;Malyarenko, 2013).…”

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30 Years of space–time covariance functions

Porcu

Furrer

Nychka

2020

WIREs Computational Stats

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In this article, we provide a comprehensive review of space–time covariance functions. As for the spatial domain, we focus on either the d‐dimensional Euclidean space or on the unit d‐dimensional sphere. We start by providing background information about (spatial) covariance functions and their properties along with different types of covariance functions. While we focus primarily on Gaussian processes, many of the results are independent of the underlying distribution, as the covariance only depends on second‐moment relationships. We discuss properties of space–time covariance functions along with the relevant results associated with spectral representations. Special attention is given to the Gneiting class of covariance functions, which has been especially popular in space–time geostatistical modeling. We then discuss some techniques that are useful for constructing new classes of space–time covariance functions. Separate treatment is reserved for spectral models, as well as to what are termed models with special features. We also discuss the problem of estimation of parametric classes of space–time covariance functions. An outlook concludes the paper. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods Statistical and Graphical Methods of Data Analysis > Multivariate Analysis

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Section: Gneiting Class Across Different Metric Spacesmentioning

confidence: 84%

“…A bridge between Gneiting functions and semi‐metric spaces has been recently provided by Menegatto et al (2019).…”

Section: Classes Of Covariance Functionsmentioning

confidence: 99%

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confidence: 99%

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30 Years of space–time covariance functions

Porcu

Furrer

Nychka

2020

WIREs Computational Stats

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“…for some finite and positive measure µ w f on [0, ∞). Since Theorem 3.2(i) in [22] shows that the functions…”

Section: Functions In the Class Cnd(x × Y ρ σ)mentioning

confidence: 99%

A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces

Barbosa¹,

Menegatto²

2020

SIGMA

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This paper is concerned with the construction of positive definite functions on a cartesian product of quasi-metric spaces using generalized Stieltjes and complete Bernstein functions. The results we prove are aligned with a well-established method of T. Gneiting to construct space-time positive definite functions and its many extensions. Necessary and sufficient conditions for the strict positive definiteness of the models are provided when the spaces are metric.

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“…The boundedness of φ is required in order to make φ(0 + ) < ∞. The references [12,13,16,20] include some extensions and generalizations of this important result along with additional references on the topic. The paper proceeds as follows.…”

Section: Introductionmentioning

confidence: 99%

Matrix valued positive definite kernels related to the generalized Aitken's integral for Gaussians

Menegatto

Oliveira

2021

Constructive Mathematical Analysis

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We introduce a method to construct general multivariate positive definite kernels on a nonempty set XX that employs a prescribed bounded completely monotone function and special multivariate functions on XX. The method is consistent with a generalized version of Aitken's integral formula for Gaussians. In the case in which XX is a cartesian product, the method produces nonseparable positive definite kernels that may be useful in multivariate interpolation. In addition, it can be interpreted as an abstract multivariate version of the well-established Gneiting's model for constructing space-time covariances commonly highly cited in the literature. Many parametric models discussed in statistics can be interpreted as particular cases of the method.

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Gneiting Class, Semi-Metric Spaces and Isometric Embeddings

Cited by 12 publications

References 30 publications

30 Years of space–time covariance functions

30 Years of space–time covariance functions

A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces

Matrix valued positive definite kernels related to the generalized Aitken's integral for Gaussians

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