The<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e880" altimg="si246.svg"><mml:mi mathvariant="script">F</mml:mi></mml:math>-family of covariance functions: A Matérn analogue for modeling random fields on spheres

Alegría, Alfredo; Cuevas-Pacheco, Francisco; Diggle, Peter J.; Porcu, Emilio

doi:10.1016/j.spasta.2021.100512

Cited by 12 publications

(10 citation statements)

References 40 publications

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“…There is additional experimental evidence in the literature of obtaining good parameter estimates on the sphere using the chordal distance (e.g., Guinness and Fuentes, (2016); Jeong and Jun, (2015)). However, as noted recently in Alegría et al (2021), these findings are not consistent with claims presented elsewhere in the literature, such as in Banerjee (2005). Fundamentally, however, the chordal distance and the great‐circle distance are non‐linearly related, and so measuring distances in terms of the chordal distance for data collected on spheres can lead to distorting effects.…”

Section: Introductioncontrasting

confidence: 86%

“…Arguments from Gneiting (2013) and Miller and Samko (2001) imply the important observation that the Matérn covariance function is positive definite only if the smoothness parameter is restricted to

false(0,1 false/ 2 false]

. Alegría et al (2021) recently proposed a new class of stationary covariance functions on spheres, allowing parameterization of the assumed mean‐square differentiability, called the

scriptF

‐family. Their experiments show improved predictive performance against a classical Matérn covariance function using the chordal distance and the so‐called Circular Matérn introduced by Guinness and Fuentes (2016).…”

Section: Introductionmentioning

confidence: 99%

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Parametric nonstationary covariance functions on spheres

Blake

Porcu

Hammerling

2022

Stat

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Gaussian Processes are powerful tools for modelling spatial data. In this context, a significant amount of modelling focus is placed on specifying the covariance function, which is required to be symmetric and positive definite. Covariance functions have classically been defined and used in Euclidean space. However, as data collected from the globe becomes more prevalent, accounting for Earth's geometry becomes increasingly important. Using Euclidean distance can be suboptimal for these data. We survey the literature for recent developments related to construction of nonstationary covariance functions on spheres, which historically has been a challenging area. We present contributions in this effort by providing three general forms for families of parametric nonstationary covariance functions.

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Section: Introductioncontrasting

confidence: 86%

false(0,1 false/ 2 false]

. Alegría et al (2021) recently proposed a new class of stationary covariance functions on spheres, allowing parameterization of the assumed mean‐square differentiability, called the

scriptF

Section: Introductionmentioning

confidence: 99%

Parametric nonstationary covariance functions on spheres

Blake

Porcu

Hammerling

2022

Stat

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“…is the (i, j)-th entry of a positive definite function, N . We now apply the arguments in theorem 1 of Alegría et al (2021) to infer that…”

Section: New Classes Of Covariance Functionsmentioning

confidence: 99%

Multivariate Gaussian Random Fields over Generalized Product Spaces involving the Hypertorus

Bachoc,

Peron,

Porcu

2022

Preprint

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The paper deals with multivariate Gaussian random fields defined over generalized product spaces that involve the hypertorus. The assumption of Gaussianity implies the finite dimensional distributions to be completely specified by the covariance functions, being in this case matrix valued mappings. We start by considering the spectral representations that in turn allow for a characterization of such covariance functions. We then provide some methods for the construction of these matrix valued mappings. Finally, we consider strategies to evade radial symmetry (called isotropy in spatial statistics) and provide representation theorems for such a more general case.

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“…The following illustration (Figure 4.2) shows orthographic projections of one realization of a bivariate random field obtained by applying the turning arcs algorithm with the following parameters: κ with a zeta distribution with parameter 2; discretization of S 2 into 500 × 500 faces with regularly-spaced colatitudes and longitudes. The isotropic generalized F-covariance on S d is associated with the Schoenberg sequence {b F n,d : n ∈ N} defined as follows (Alegria et al, 2018):…”

Section: Choice Of the Distributions Of ε And κmentioning

confidence: 99%

The Turning Arcs: a Computationally Efficient Algorithm to Simulate Isotropic Vector-Valued Gaussian Random Fields on the $d$-Sphere

Alegría¹,

Emery²,

Lantuéjoul³

2020

Preprint

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Random fields on the sphere play a fundamental role in the natural sciences. This paper presents a simulation algorithm parenthetical to the spectral turning bands method used in Euclidean spaces, for simulating scalar-or vector-valued Gaussian random fields on the d-dimensional unit sphere. The simulated random field is obtained by a sum of Gegenbauer waves, each of which is variable along a randomly oriented arc and constant along the parallels orthogonal to the arc. Convergence criteria based on the Berry-Esséen inequality are proposed to choose suitable parameters for the implementation of the algorithm, which is illustrated through numerical experiments. A byproduct of this work is a closed-form expression of the Schoenberg coefficients associated with the Chentsov and exponential covariance models on spheres of dimensions greater than or equal to 2.

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TheF-family of covariance functions: A Matérn analogue for modeling random fields on spheres

Cited by 12 publications

References 40 publications

Parametric nonstationary covariance functions on spheres

Parametric nonstationary covariance functions on spheres

Multivariate Gaussian Random Fields over Generalized Product Spaces involving the Hypertorus

The Turning Arcs: a Computationally Efficient Algorithm to Simulate Isotropic Vector-Valued Gaussian Random Fields on the $d$-Sphere

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