Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Massa, Eugenio; Peron, Ana Paula; Porcu, Emilio

doi:10.3842/sigma.2017.088

Cited by 7 publications

(9 citation statements)

References 37 publications

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“…For instance, Lang and Schwab (2013) show that the rate of decay of the spectrum related to the function C determines the regularity properties of the associated Gaussian field in terms of interpolation spaces and Hölder continuity of sample paths. Spectral analysis on spheres became recently useful in contexts as diverse as spatial statistics (Guinness and Fuentes, 2016), equivalence of Gaussian measures and infill asymptotics (Arafat et al, 2018), approximation theory (Menegatto et al, 2006;Beatson et al, 2014;Ziegel, 2014;Massa et al, 2017) and spatial point processes (Møller et al, 2018).…”

Section: Random Fields On Spheres: Literature Reviewmentioning

confidence: 99%

Nonparametric Bayesian Modeling and Estimation of Spatial Correlation Functions for Global Data

et al. 2021

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We provide a nonparametric spectral approach to the modeling of correlation functions on spheres. The sequence of Schoenberg coefficients and their associated covariance functions are treated as random rather than assuming a parametric form. We propose a stick-breaking representation for the spectrum, and show that such a choice spans the support of the class of geodesically isotropic covariance functions under uniform convergence. Further, we examine the first order properties of such representation, from which geometric properties can be inferred, in terms of Hölder continuity, of the associated Gaussian random field. The properties of the posterior, in terms of existence, uniqueness, and Lipschitz continuity, are then inspected. Our findings are validated with MCMC simulations and illustrated using a global data set on surface temperatures.

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Section: Random Fields On Spheres: Literature Reviewmentioning

confidence: 99%

Nonparametric Bayesian Modeling and Estimation of Spatial Correlation Functions for Global Data

et al. 2021

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“…Lang and Schwab (2013) showed that the rate of decay of the d-Schoenberg sequence determines the regularity properties of the associated Gaussian field in terms of interpolation spaces and Hölder continuities of the sample paths. The d-Schoenberg sequences are useful in contexts as diverse as spatial statistics (Guinness and Fuentes, 2016), equivalence of Gaussian measures and infill asymptotics (Arafat et al, 2018), approximation theory (Menegatto et al, 2006;Beatson et al, 2014;Ziegel, 2014;Massa et al, 2017) and spatial point processes (Møller et al, 2018). In practice, the d-Schoenberg sequence of a parametric model is rarely known and it must be computed via numerical integration.…”

Section: Spectral Theorymentioning

confidence: 99%

The $\mathcal{F}$-family of covariance functions: A Matérn analogue for modeling random fields on spheres

Alegría¹,

Cuevas-Pacheco²,

Diggle³

et al. 2021

Preprint

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The Matérn family of isotropic covariance functions has been central to the theoretical development and application of statistical models for geospatial data. For global data defined over the whole sphere representing planet Earth, the natural distance between any two locations is the great circle distance. In this setting, the Matérn family of covariance functions has a restriction on the smoothness parameter, making it an unappealing choice to model smooth data. Finding a suitable analogue for modelling data on the sphere is still an open problem. This paper proposes a new family of isotropic covariance functions for random fields defined over the sphere. The proposed family has a parameter that indexes the mean square differentiability of the corresponding Gaussian field, and allows for any admissible range of fractal dimension. Our simulation study mimics the fixed domain asymptotic setting, which is the most natural regime for sampling on a closed and bounded set. As expected, our results support the analogous results (under the same asymptotic scheme) for planar processes that not all parameters can be estimated consistently. We apply the proposed model to a dataset of precipitable water content over a large portion of the Earth, and show that the model gives more precise predictions of the underlying process at unsampled locations than does the Matérn model using chordal distances.

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“…The application of such operators has consequences on the differentiability at the origin of the involved functions. Walks on spheres have been proposed by Beatson et al (), Ziegel () and Massa et al (). This last work extends the previous work to the case of complex spheres.…”

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 Complex Spheres and their Walks through Dimensions

Cited by 7 publications

References 37 publications

Nonparametric Bayesian Modeling and Estimation of Spatial Correlation Functions for Global Data

Nonparametric Bayesian Modeling and Estimation of Spatial Correlation Functions for Global Data

The $\mathcal{F}$-family of covariance functions: A Matérn analogue for modeling random fields on spheres

Modeling Temporally Evolving and Spatially Globally Dependent Data

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