Isotropy, symmetry, separability and strict positive definiteness for covariance functions: A critical review

Iaco, Sandra De; Posa, D.; Cappello, Claudia; Maggio, Sabrina

doi:10.1016/j.spasta.2018.09.003

Cited by 24 publications

(9 citation statements)

References 88 publications

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“…A recent review in De Iaco, Posa, Cappello, and Maggio (2019) digs into other descriptive properties: Explicit distinction is made between partial, additive and total separability, as well as the concepts of axial, full and quadrant symmetries on the plane. Some tests on separability of space–time covariance functions can be found in Scaccia and Martin (2005, 2002, 2011), Fuentes (2006); Bevilacqua, Mateu, Porcu, Zhang, and Zini (2010), Mitchell, Genton, and Gumpertz (2006), Constantinou, Kokoszka, and Reimherr (2017), Lu and Zimmerman (2005), Aston, Pigoli, and Tavakoli (2017), Li et al (2007), De Iaco, Posa, & Myers, 2013, De Iaco, Palma, & Posa, 2016), and Cappello, De Iaco, and Posa (2018).…”

Section: Properties Of Covariance Functionsmentioning

confidence: 99%

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: Properties Of Covariance Functionsmentioning

confidence: 99%

30 Years of space–time covariance functions

Porcu

Furrer

Nychka

2020

WIREs Computational Stats

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“…Before going into details of the typical characteristics of the Gaussian covariance model, it is worth highlighting some general results concerning the product of covariance functions and some consequences in terms of isotropy and strict positive definiteness (De Iaco & Posa, 2018; De Iaco et al , 2019). • Property 1.…”

Section: Specific Features Of Gaussian Covariance Functionsmentioning

confidence: 99%

On Some Characteristics of Gaussian Covariance Functions

Iaco

Posa

Cappello

et al. 2020

Int Statistical Rev

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Summary The concepts of isotropy/anisotropy and separability/non‐separability of a covariance function are strictly related. If a covariance function is separable, it cannot be isotropic or geometrically anisotropic, except for the Gaussian covariance function, which is the only model both separable and isotropic. In this paper, some interesting results concerning the Gaussian covariance model and its properties related to isotropy and separability are given, and moreover, some examples are provided. Finally, a discussion on asymmetric models, with Gaussian marginals, is furnished and the strictly positive definiteness condition is discussed.

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“…Moreover, it can describe the spatio-temporal behavior of a space-time random field which is formed by a product of two independent spatial and temporal non-zero mean random fields (Cappello et al. 2019 ). Note that the temporal part might be chosen as any covariance model originating from time series analysis, e.g.…”

Section: Introductionmentioning

confidence: 99%

Blind recovery of sources for multivariate space-time random fields

Muehlmann

Iaco

Nordhausen

2022

Stoch Environ Res Risk Assess

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With advances in modern worlds technology, huge datasets that show dependencies in space as well as in time occur frequently in practice. As an example, several monitoring stations at different geographical locations track hourly concentration measurements of a number of air pollutants for several years. Such a dataset contains thousands of multivariate observations, thus, proper statistical analysis needs to account for dependencies in space and time between and among the different monitored variables. To simplify the consequent multivariate spatio-temporal statistical analysis it might be of interest to detect linear transformations of the original observations that result in straightforward interpretative, spatio-temporally uncorrelated processes that are also highly likely to have a real physical meaning. Blind source separation (BSS) represents a statistical methodology which has the aim to recover so-called latent processes, that exactly meet the former requirements. BSS was already successfully used in sole temporal and sole spatial applications with great success, but, it was not yet introduced for the spatio-temporal case. In this contribution, a reasonable and innovative generalization of BSS for multivariate space-time random fields (stBSS), under second-order stationarity, is proposed, together with two space-time extensions of the well-known algorithms for multiple unknown signals extraction (stAMUSE) and the second-order blind identification (stSOBI) which solve the formulated problem. Furthermore, symmetry and separability properties of the model are elaborated and connections to the space-time linear model of coregionalization and to the classical principal component analysis are drawn. Finally, the usefulness of the new methods is shown in a thorough simulation study and on a real environmental application.

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Isotropy, symmetry, separability and strict positive definiteness for covariance functions: A critical review

Cited by 24 publications

References 88 publications

30 Years of space–time covariance functions

30 Years of space–time covariance functions

On Some Characteristics of Gaussian Covariance Functions

Blind recovery of sources for multivariate space-time random fields

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