Isotropic Gaussian random fields on the sphere: regularity, fast simulation, and stochastic partial differential equations
TL;DR: In this article, rates of convergence of their finitely truncated Karhunen-Lo-ve expansions in terms of the covariance spectrum are established, and algorithmic aspects of fast sample generation via fast Fourier transforms on the sphere are indicated.
Abstract: Isotropic Gaussian random fields on the sphere are characterized by Karhunen-Lo\`{e}ve expansions with respect to the spherical harmonic functions and the angular power spectrum The smoothness of the covariance is connected to the decay of the angular power spectrum and the relation to sample H\"{o}lder continuity and sample differentiability of the random fields is discussed Rates of convergence of their finitely truncated Karhunen-Lo\`{e}ve expansions in terms of the covariance spectrum are established, and algorithmic aspects of fast sample generation via fast Fourier transforms on the sphere are indicated The relevance of the results on sample regularity for isotropic Gaussian random fields and the corresponding lognormal random fields on the sphere for several models from environmental sciences is indicated Finally, the stochastic heat equation on the sphere driven by additive, isotropic Wiener noise is considered, and strong convergence rates for spectral discretizations based on the spherical harmonic functions are proven
Citations
More filters
TL;DR: In this article, an analytic extension of the solution map on a tensor product of ellipses in the complex domain is proposed to estimate the Legendre coefficients of u. The analytic extension is based on the holomorphic version of the implicit function theorem in Banach spaces and can be applied to a large variety of parametric PDEs.
206 citations
TL;DR: This paper defines the notion of and proves a characterization theorem for m times mean square differentiable processes on d -dimensional spheres and proves that the resulting sphere-restricted process retains its differentiability properties, which has the important implication that the Matern family retains its full range of smoothness when applied to spheres so long as Euclidean distance is used.
87 citations
TL;DR: In this article, the authors provide an overview of statistical modeling techniques for space-time processes, where space is the sphere representing our planet and data sets are inherently global and temporally evolving.
Abstract: 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.
77 citations
TL;DR: While this article focuses primarily on Gaussian processes, many of the results are independent of the underlying distribution, as the covariance only depends on second‐moment relationships.
Abstract: In this article, we provide a comprehensive review of space–time covariance functions. As for the spatial domain, we focus on either the d‐dimensional Euclidean space or on the unit d‐dimensional sphere. We start by providing background information about (spatial) covariance functions and their properties along with different types of covariance functions. While we focus primarily on Gaussian processes, many of the results are independent of the underlying distribution, as the covariance only depends on second‐moment relationships. We discuss properties of space–time covariance functions along with the relevant results associated with spectral representations. Special attention is given to the Gneiting class of covariance functions, which has been especially popular in space–time geostatistical modeling. We then discuss some techniques that are useful for constructing new classes of space–time covariance functions. Separate treatment is reserved for spectral models, as well as to what are termed models with special features. We also discuss the problem of estimation of parametric classes of space–time covariance functions. An outlook concludes the paper.
64 citations
TL;DR: Chan and Lai as mentioned in this paper showed that the asymptotics of Piterbarg's approximation is similar to Pickands' approximation on the Euclidean space which involves pickands' constant.
Abstract: Let $X=\{X(x): x\in\mathbb{S}^N\}$ be a real-valued, centered Gaussian random field indexed on the $N$-dimensional unit sphere $\mathbb{S}^N$. Approximations to the excursion probability ${\mathbb{P}}\{\sup_{x\in\mathbb{S}^N}X(x)\ge u\}$, as $u\to\infty$, are obtained for two cases: (i) $X$ is locally isotropic and its sample functions are non-smooth and; (ii) $X$ is isotropic and its sample functions are twice differentiable. For case (i), the excursion probability can be studied by applying the results in Piterbarg (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), Mikhaleva and Piterbarg (Theory Probab. Appl. 41 (1997) 367--379) and Chan and Lai (Ann. Probab. 34 (2006) 80--121). It is shown that the asymptotics of ${\mathbb{P}}\{\sup_{x\in\mathbb {S}^N}X(x)\ge u\}$ is similar to Pickands' approximation on the Euclidean space which involves Pickands' constant. For case (ii), we apply the expected Euler characteristic method to obtain a more precise approximation such that the error is super-exponentially small.
43 citations
References
More filters
Posted Content•
18 Dec 2005
TL;DR: In this paper, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed and orthogonality on the unit circle is not discussed.
Abstract: In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed Orthogonal polynomials on the unit circle are not discussed
5,648 citations
"Isotropic Gaussian random fields on..." refers background in this paper
...! 2n‘! P(n;n) (‘ n) () (‘0+ n)! 2n‘0! P(n;n) (‘0 n) ()(1 ) n(1 + )nd = ‘‘0 2 2‘+ 1 (‘+ n)! (‘ n)! ; where the last equation follows from the orthogonality of the Jacobi polynomials (see, e.g., [21]) and Z 1 1 P(n;n) (‘ n) () 2 (1 )n(1 + )nd= 22n+1 2 ‘+ 1 ‘! ( n)!( + )! : 8 A. LANG AND CH. SCHWAB In conclusion we have shown that Z 1 1 I @n @n k() 2 (1 2)nd= X1 ‘=n A2 ‘ 2‘+ 1 2(4ˇ)2 (...
[...]
...n7), we rst have to dene the spherical harmonic functions on S2 which take a crucial role. We recall that the Legendre polynomials (P ‘;‘2N 0) are for example given by Rodrigues’ formula (see, e.g., [21]) P ‘() := 2 ‘ 1 ‘! @‘ @‘ (2 1)‘ for all ‘2N 0 and 2[ 1;1]. The Legendre polynomials dene the associated Legendre functions (P ‘m;‘2N 0;m= 0;:::;‘) by P ‘m() := ( 1)m(1 2)m=2 @m @m P ‘() for ...
[...]
Book•
26 Feb 1978
4,781 citations
Additional excerpts
...We define for n < η < n + 1 the interpolation space V η(−1, 1) with the real method of interpolation in the sense of [23] by V (−1, 1) = ( V (−1, 1), V (−1, 1) ) η−n,2 equipped with the norms ‖ · ‖V η(−1,1) given by ‖u‖(2)V η(−1,1) = ∫ ∞...
[...]
TL;DR: Higher Transcendental Functions Based on notes left by the late Prof. Harry Bateman, and compiled by the Staff of the Bateman Project as discussed by the authors, are presented in Table 1.
Abstract: Higher Transcendental Functions Based, in part, on notes left by the late Prof. Harry Bateman, and compiled by the Staff of the Bateman Project. Vol. 1. Pp. xxvi + 302. 52s. Vol. 2. Pp. xvii + 396. 60s. (London: McGraw-Hill Publishing Company, Ltd., 1953.)
4,428 citations
Book•
01 Jan 1983
TL;DR: In this article, the authors measure smoothness using Atoms and Pointwise Multipliers, Wavelets, Spaces on Lipschitz Domains, Wavelet and Sampling Numbers.
Abstract: How to Measure Smoothness.- Atoms and Pointwise Multipliers.- Wavelets.- Spaces on Lipschitz Domains, Wavelets and Sampling Numbers.- Anisotropic Function Spaces.- Weighted Function Spaces.- Fractal Analysis: Measures, Characteristics, Operators.- Function Spaces on Quasi-metric Spaces.- Function Spaces on Sets.
4,099 citations