The turning arcs: a computationally efficient algorithm to simulate isotropic vector-valued Gaussian random fields on th
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The turning arcs: a computationally efficient algorithm to simulate isotropic vector-valued Gaussian random fields on the d-sphere Alfredo Alegría1 · Xavier Emery2,3 · Christian Lantuéjoul4 Received: 19 November 2019 / Accepted: 28 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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 by-product 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. Keywords Schoenberg sequence · Turning Bands · Gegenbauer polynomials · Central limit approximation · Berry-Esséen inequality
1 Introduction Spherically indexed Gaussian random fields have attracted a growing interest in recent decades. They are useful in the modeling of georeferenced variables arising in many branches of applied sciences, such as astronomy, climatology, oceanography, biology and geosciences, amongst many others. We refer the reader to Marinucci and Peccati (2011), Jeong et al. (2017) and Porcu et al. (2018) for recent reviews about this topic. In general, the space consists of a 2-dimensional sphere, but hyperspheres are sometimes met, Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11222-020-09952-8) contains supplementary material, which is available to authorized users.
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Alfredo Alegría [email protected]
1
Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile
2
Department of Mining Engineering, University of Chile, Santiago, Chile
3
Advanced Mining Technology Center, University of Chile, Santiago, Chile
4
Centre de Géosciences, MINES ParisTech, PSL University, Paris, France
e.g., in high-dimensional shape analysis (Dryden 2005; Mardia and Patrangenaru 2005). Simulation is crucial for the development of new applications in spatial statistics. It is well known that simulation algorithms based on the Cholesky decomposition of the covariance matrix (Ripley 1987) are computationally prohibitive when the sample size is large, since the order of computation of the Cholesky decomposition is equal to the cube of the sample size. As a result, the search for new efficient methods to simulate Gaussian random fields in spherical domains is of paramount importance. Within the class of isotropic random fields, i.e., random fields whose finite-di
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