Characterization theorems for the Gneiting class of space–time covariances

Zastavnyi, V. P.; Porcu, Emilio

doi:10.3150/10-bej278

Cited by 37 publications

(22 citation statements)

References 8 publications

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“…For n = 1 this proposition was proved in [23, Lemma 2.2 for ρ(x) = ‖x‖]. For any n ∈ N statement can be proved in a similar way as Lemma 2.2 in [23]. . .…”

Section: Characterization Theorems For Cross Covariances Based On Latmentioning

confidence: 71%

“…By Theorem 2.1 in Zastavnyi and Porcu [23] we get e −λh k (ϑ k ) ∈ Φ(E k ) for all λ > 0. This argument fixes the necessary part.…”

Section: Characterization Theorems For Cross Covariances Based On Latmentioning

confidence: 81%

“…From Bernstein's theorem it follows that statement 1 in Corollary 2 is equivalent to condition (11) for functions ϕ(t) = e −λt for all positive λ. Next, we must apply Proposition 5 with n = 1, d 1 = d, E 1 = R m × R k or Lemma 2.2 [23].…”

Section: Characterization Theorems For Cross Covariances Based On Latmentioning

confidence: 99%

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

Porcu

Zastavnyi

2011

Journal of Multivariate Analysis

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a b s t r a c tWe characterize some important classes of cross-covariance functions associated to vector valued random fields based on latent dimensions. We also give some results for mixture based models that allow for the construction of new cross-covariance models. In particular, we give a criterion for the permissibility of quasi-arithmetic operators in order to construct valid cross covariances.

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“…For n = 1 this proposition was proved in [23, Lemma 2.2 for ρ(x) = ‖x‖]. For any n ∈ N statement can be proved in a similar way as Lemma 2.2 in [23]. . .…”

Section: Characterization Theorems For Cross Covariances Based On Latmentioning

confidence: 71%

“…By Theorem 2.1 in Zastavnyi and Porcu [23] we get e −λh k (ϑ k ) ∈ Φ(E k ) for all λ > 0. This argument fixes the necessary part.…”

Section: Characterization Theorems For Cross Covariances Based On Latmentioning

confidence: 81%

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

Porcu

Zastavnyi

2011

Journal of Multivariate Analysis

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“…We remark that Gneiting (2002) presented a generalized version over the product space R d × R l and that, in the earlier literature, the spatial and temporal arguments have been inverted, but in the subsequent presentation, we prefer to work with such a parameterization. Zastavnyi and Porcu (2011) showed necessary and sufficient conditions for the positive definiteness of the Gneiting's class. The results below complete the picture of this class in the following way: we suppose that the spatial argument h is replaced with the great circle distance by restricting the function ψ in Equation (3) to the interval [0, π].…”

Section: Introductionmentioning

confidence: 99%

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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“…Another scale mixture approach allows for an adaptation of the Gneiting class (Gneiting, ; Zastavnyi & Porcu, ). Again, Porcu et al () show how to derive a space–time covariance of the type

ψ false(d_{GC}, u false) = \frac{σ^{2}}{g_{false[0, π false]} {false(d_{GC} false)}^{1 false/ 2}} f ((), \frac{u}{g_{false[0, π false]} false(d_{GC} false)}), false(d_{GC}, u false) \in false[0, π false] \times double-struckR .

…”

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 the Gneiting class of space–time covariances

Cited by 37 publications

References 8 publications

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

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

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

Modeling Temporally Evolving and Spatially Globally Dependent Data

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