Twenty-two families of multivariate covariance kernels on spheres, with their spectral representations and sufficient validity conditions

Emery, Xavier; Arroyo, Daisy; Mery, Nadia

doi:10.1007/s00477-021-02063-4

Cited by 9 publications

(5 citation statements)

References 60 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof Expressions for the Schoenberg coefficients are from Alegría et al (2020, Equation D.5). Following similar notation from Emery et al (2021), Alegría et al (2020) showed that

b_{m, d} = ρ true \sum_{ℓ = 0}^{\infty} α_{m + 2 ℓ} ξ_{m, d, ℓ}

, where

ξ_{m, d, ℓ}

is a constant independent of

ρ

and

ν, α_{k} = f false(ν, k false) \frac{2^{k}}{k!} {(||, Γ ()(, \frac{k + ı ν}{2}))}^{2}

, and

ı

is the imaginary unit. Abramowitz and Stegun (1970, Equation 6.1.25) have that

{(||, Γ ()(, \frac{k + ı ν}{2}))}^{2} = true \prod_{q = 0}^{\infty} {(][, 1 + \frac{ν^{2}}{{false(k + q false)}^{2}})}^{- 1} .

Under condition 1, the function in the brackets is negative definite and positive valued.…”

Section: Contributionsmentioning

confidence: 88%

“…Our spectral adaptive approach proposes a framework in which formulation of nonstationary covariance functions on spheres is achieved by an expansion of Gengenbauer polynomials with nonstationary coefficients. This introduces a many classes of nonstationary covariance functions which can be obtained through simple modifications of arguments presented in Emery et al (2021), two of which we elucidated upon here.…”

Section: Discussionmentioning

confidence: 99%

“…Emery et al (2021) recently addressed the problem on modelling multivariate processes on spheres and provided 22 families of stationary covariance functions with their associated sufficiency conditions. Here we show a generalized construction for nonstationary covariance functions and elucidate on two examples from Emery et al (2021) to illustrate how our method can adapt these kernels to nonstationarity.…”

Section: Contributionsmentioning

confidence: 99%

“…Following are two examples as corollaries. Arguments here are based on and modified from arguments in Emery et al (2021). Adaptations to other covariance functions presented in Emery et al (2021) can be made with minor modifications of the associated arguments presented therein.…”

Section: Spectral Adaptive Approachmentioning

confidence: 99%

“…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). Recently, Emery et al (2021) proposed 22 families of multivariate stationary covariance kernels on spheres. Stationary covariance functions on spheres are clearly useful.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Parametric nonstationary covariance functions on spheres

Blake

Porcu

Hammerling

2022

Stat

View full text Add to dashboard Cite

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.

show abstract

“…Proof Expressions for the Schoenberg coefficients are from Alegría et al (2020, Equation D.5). Following similar notation from Emery et al (2021), Alegría et al (2020) showed that

b_{m, d} = ρ true \sum_{ℓ = 0}^{\infty} α_{m + 2 ℓ} ξ_{m, d, ℓ}

, where

ξ_{m, d, ℓ}

is a constant independent of

ρ

and

ν, α_{k} = f false(ν, k false) \frac{2^{k}}{k!} {(||, Γ ()(, \frac{k + ı ν}{2}))}^{2}

, and

ı

is the imaginary unit. Abramowitz and Stegun (1970, Equation 6.1.25) have that

{(||, Γ ()(, \frac{k + ı ν}{2}))}^{2} = true \prod_{q = 0}^{\infty} {(][, 1 + \frac{ν^{2}}{{false(k + q false)}^{2}})}^{- 1} .

Under condition 1, the function in the brackets is negative definite and positive valued.…”

Section: Contributionsmentioning

confidence: 88%

Section: Discussionmentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

Section: Spectral Adaptive Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Parametric nonstationary covariance functions on spheres

Blake

Porcu

Hammerling

2022

Stat

View full text Add to dashboard Cite

show abstract

Spatio-Spectral Assessment of Some Isotropic Polynomial Covariance Functions on the Sphere

Piretzidis

Kotsakis

Mertikas

et al. 2023

International Association of Geodesy Symposia

View full text Add to dashboard Cite

In gravity field modeling, covariance functions are mainly associated with least squares collocation. Prior to the implementation of least squares collocation, the characteristics of the selected analytical covariance function need to be well understood. In this contribution, we study four polynomial covariance functions, i.e., the spherical, Askey, C2-Wendland and C4-Wendland models. All of them are defined on the sphere and correspond to isotropic, positive definite and compactly supported functions. We examine them in the spatial and spectral domains, and assess their characteristics, such as the correlation length, the curvature parameter, the spectral maximum and the spectral decay rate. We also provide analytical expressions and numerical estimates for these parameters.

show abstract

Compactly-Supported Isotropic Covariances on Spheres Obtained from Matrix-Valued Covariances in Euclidean Spaces

et al. 2022

Self Cite

View full text Add to dashboard Cite

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

show abstract

Twenty-two families of multivariate covariance kernels on spheres, with their spectral representations and sufficient validity conditions

Cited by 9 publications

References 60 publications

Parametric nonstationary covariance functions on spheres

Parametric nonstationary covariance functions on spheres

Spatio-Spectral Assessment of Some Isotropic Polynomial Covariance Functions on the Sphere

Compactly-Supported Isotropic Covariances on Spheres Obtained from Matrix-Valued Covariances in Euclidean Spaces

Contact Info

Product

Resources

About