Fast and exact simulation of univariate and bivariate Gaussian random fields

Abstract

Circulant embedding is a powerful algorithm for fast simulation of stationary Gaussian random fields on a rectangular grid in R n , which works perfectly for compactly supported covariance functions. Cut-off circulant embedding techniques have been developed for univariate random fields for dimensions up to R 3 and rely on the modification of a covariance function outside the simulation window, such that the modified covariance function is compactly supported. In this paper, we propose extensions of the cut-off approach for univariate and bivariate Gaussian random fields. In particular, we provide a method for simulating bivariate fields with a powered exponential model and the Matérn model for certain sets of parameters. Movie 2. Simulations from the bivariate Matérn model [To ensure that the visuanimation will play properly, use Adobe Acrobat Reader to view the pdf file.].holds. The covariance function C is stationary and isotropic if additionally C.h 1 / D C.h 2 / whenever kh 1 k D kh 2 k; that is, marginal covariances and cross-covariances depend only on the distance between the locations. Hereinafter, we write C.r/ instead of C.h/ with r D khk, whenever C.h/ is stationary and isotropic.where W is the one-dimensional discrete Fourier transform matrix of size 2N 2N with entries w pq D e 2 ipq=2N , p, q D 0, : : : , 2N 1, and W is the conjugate transpose of W. The eigenvalues in the diagonal matrix ƒ form the Figure 1. Toeplitz matrix and its circulant embedding matrix.

Joint asymptotics for estimating the fractal indices of bivariate Gaussian processes

“…In particular, we prove that √ nα 11 and √ nα 22 are asymptotically uncorrelated if (α 11 + α 22 )/2 < α 12 , while they are asymptotically correlated if (α 11 + α 22 )/2 = α 12 . Our results are applicable to a wide class of bivariate Gaussian processes, including the bivariate Matérn model introduced by Gneiting, Kleiber and Schlather [19], the bivariate powered exponential model and bivariate Cauchy model of Moreva and Schlather [30], and a class of bivariate models introduced by Du and Ma [14].…”

“…In recent years, multivariate (or vector-valued) Gaussian processes and random fields have become popular in modeling multivariate spatial datasets (see, e.g., [17,36]). Several classes of multivariate spatial models were introduced in [4,13,14,19,26,30,33]. Two of the challenges in multivariate modeling are to specify the cross-dependence structures and to quantify the effect of the crossdependence on the estimation and prediction performance.…”

A parametric model bridging between bounded and unbounded variograms

“…To gain speed, we use a parallel version of the Cholesky decomposition in the upcoming version of the package RandomFieldsUtils (Schlather et al ) of R (R Core Team, ). Simulations are performed with the package RandomFields (Schlather et al ) and the same random seed for all parameter combinations. A visuanimation (Genton et al 2015, ) of the graphs of γ α , β and the corresponding random fields is given in Movie .…”

“…The turning bands method, based on the locally stationary representation of γ α , β and the Cholesky decomposition, can be used to obtain approximate realizations of Z on a large number of arbitrary locations in dimensions larger than 1. Exact and fast simulations of Z on a grid are provided for some parameter values by the circulant embedding method and its modifications, see, for example, Wood and Chan (), Gneiting et al (), Stein (), Dietrich and Newsam () and Moreva and Schlather (), and Schlather et al () for its implementation.…”

The Shkarofsky‐Gneiting class of covariance models for bivariate Gaussian random fields

“…The literature on multivariate covariance models for spatial processes has become ubiquitous. Recent works (Alonso‐Malaver, Porcu, & Giraldo, ; Apanasovich, Genton, & Sun, ; Daley, Porcu, & Bevilacqua, ; Gneiting, Kleiber, & Schlather, ; Hristopoulos & Porcu, ; Kleiber & Nychka, ; Majumdar & Gelfand, ; Moreva & Schlather, ; Porcu, Daley, Buhmann, & Bevilacqua, ) as well as earlier papers (Goulard & Voltz, ; Ver Hoef & Barry, ; Wackernagel, ) confirm that the construction of models for matrix‐valued covariances is of real interest to the geostatistical community. Multivariate covariances are also of interest to many other branches of applied mathematics, probability theory, and statistics, and the reader is referred to Genton and Kleiber (20), and the references therein, for a thorough review, as well as to the discussion by Bevilacqua, Hering, and Porcu ().…”