Characterization theorems for some classes of covariance functions associated to vector valued random fields

Porcu, Emilio; Zastavnyi, V. P.

doi:10.1016/j.jmva.2011.04.013

Cited by 45 publications

(36 citation statements)

References 20 publications

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“…The next result substantially rephrases a general result due to Porcu and Zastavnyi (2011) for the case of great circle distances. We omit the proof because it follows exactly the same arguments as in Porcu and Zastavnyi (2011).…”

Section: Some Examples and Parameterizationssupporting

confidence: 75%

Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere

Porcu¹,

Bevilacqua²,

Genton³

2016

Journal of the American Statistical Association

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148

183

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In this paper, we propose stationary covariance functions for processes that evolve temporally over a sphere, as well as cross-covariance functions for multivariate random fields defined over a sphere. For such processes, the great circle distance is the natural metric that should be used in order to describe spatial dependence. Given the mathematical difficulties for the construction of covariance functions for processes defined over spheres cross time, approximations of the state of nature have been proposed in the literature by using the Euclidean (based on map projections) and the chordal distances. We present several methods of construction based on the great circle distance and provide closed-form expressions for both spatio-temporal and multivariate cases. A simulation study assesses the discrepancy between the great circle distance, chordal distance and Euclidean distance based on a map projection both in terms of estimation and prediction in a space-time and a bivariate spatial setting, where the space is in this case the Earth. We revisit the analysis of Total Ozone Mapping Spectrometer (TOMS) data and investigate differences in terms of estimation and prediction between the aforementioned distance-based approaches. Both simulation and real data highlight sensible differences in terms of estimation of the spatial scale parameter. As far as prediction is concerned, the differences can be appreciated only when the interpoint distances are large, as demonstrated by an illustrative example.

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Section: Some Examples and Parameterizationssupporting

confidence: 75%

Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere

Porcu¹,

Bevilacqua²,

Genton³

2016

Journal of the American Statistical Association

Self Cite

148

183

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“…Recent literature (Gneiting et al 2010;Porcu and Zastavnyi 2011;Apanasovich and Genton 2010;Apanasovich et al 2012;Majumdar and Gelfand 2007;Kleiber and Porcu 2014;Hristopoulos and Porcu 2014;Alonso-Malaver et al 2013;Ruiz-Medina and Porcu 2014;Alonso-Malaver et al 2013) as well as earlier papers (Wackernagel 2003;Gaspari and Cohn 1999;Hoef and Barry 1998;Goulard and Voltz 1992) confirm that the construction of models for matrix-valued covariances is of real interest to the geostatistical community.On the one hand, such constructions can be challenging from the mathematical viewpoint; on the other hand, the requirement of compact support for the vector-valued case is even more challenging as shown in this paper. We believe that it is desirable to have structures that allow us to have (a) different compact supports that reflect different scales of dependence of the several components of the vector of random fields, and (b) a variable smoothing parameter that allows us to control the degree of differentiability at the origin.…”

Section: Introductionmentioning

confidence: 88%

“…The fact that the models described in this paper are well defined follows from a special case of an existence result in Porcu and Zastavnyi (2011) concerning the definition of mixtures of matrix-valued functions. This special case is given at Theorem A below.…”

Section: Background and Notationmentioning

confidence: 97%

Classes of compactly supported covariance functions for multivariate random fields

Daley

Porcu

Bevilacqua

2014

Stoch Environ Res Risk Assess

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The paper combines simple general methodologies to obtain new classes of matrix-valued covariance functions that have two important properties: (i) the domains of the compact support of the several components of the matrix-valued functions can vary between components; and (ii) the overall differentiability at the origin can also vary. These models exploit a class of functions called here the Wendland-Gneiting class; their use is illustrated via both a simulation study and an application to a North American bivariate dataset of precipitation and temperature. Because for this dataset, as for others, the empirical covariances exhibit a hole effect, the turning bands operator is extended to matrix-valued covariance functions so as to obtain matrix-valued covariance models with negative covariances.

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“…Also, let

C_{T} false(u; ξ false)

be a covariance for any ξ . Then, arguments in Porcu and Zastavnyi () and Gneiting et al () show that the function

ψ (d_{GC}, u) = \int_{{double-struckR}_{+}} ψ_{scriptS} (d_{GC}; ξ) C_{scriptT} (u; ξ) F (d ξ), (d_{GC}, u) \in [0, π] \times R

is a non‐separable geodesically isotropic and temporally symmetric space–time covariance function.…”

Section: Second‐order Approachesmentioning

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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Characterization theorems for some classes of covariance functions associated to vector valued random fields

Cited by 45 publications

References 20 publications

Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere

Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere

Classes of compactly supported covariance functions for multivariate random fields

Modeling Temporally Evolving and Spatially Globally Dependent Data

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