Fast and Exact Simulation of Large Gaussian Lattice Systems in ℝ<sup>2</sup>: Exploring the Limits

Gneiting, Tilmann; Ševčíková, Hana; Percival, Donald B.; Schlather, Martin; Jiang, Yindeng

doi:10.1198/106186006x128551

Cited by 62 publications

(72 citation statements)

References 23 publications

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“…The random component, e(x, y), is described by a Gaussian distribution with mean equal to 1, and its standard deviation, s e [R R (x, y)], is a function of radar rainfall values and spatial and temporal correlation functions, r s (Ds) and r t (Dt), as described in section 2. In general, the simulation of Gaussian random variables correlated in space or time can be performed according to wellestablished techniques (see Gneiting et al [2005] for a review). However, the simulation of spatially and temporally correlated Gaussian random fields presents a much higher degree of complexity [e.g., Gneiting et al, 2007].…”

Section: Ensemble Generatormentioning

confidence: 99%

“…In the literature, several methods have been proposed to generate correlated Gaussian random fields: matrix factorization, spectral turning bands, circulant embedding, and cutoff embedding [e.g., Mantoglou and Wilson, 1982;Davis, 1987;Dietrich and Newsam, 1993;Gneiting et al, 2005]. For our purpose, we selected the Cholesky decomposition method [e.g., Cressie, 1993], whose advantages include exactness (asymptotically, an ensemble of simulated field series has the correlation structure exactly as required) and an ability to account for the dependence of the variance on the given radar rainfall values to reproduce the estimated properties of the radar rainfall uncertainty model.…”

Section: Ensemble Generatormentioning

confidence: 99%

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Product‐error‐driven generator of probable rainfall conditioned on WSR‐88D precipitation estimates

Villarini

Krajewski

Ciach

et al. 2009

Water Resources Research

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[1] The existence of large errors in precipitation products delivered by the network of Weather Surveillance Radar, 1988 Doppler (WSR-88D) radars is broadly recognized. However, their quantitative characteristics remain poorly understood. Recently, the authors developed a functional-statistical model that quantifies the relation between radar rainfall and the corresponding true rainfall in a way that is applicable to the probabilistic quantitative precipitation estimation planned for future use by the U.S. National Weather Service. The model consists of a deterministic distortion function and a random uncertainty factor, both conditioned on given radar rainfall values. It also accounts for the spatiotemporal correlations in the random uncertainty factor. The model components were estimated on the basis of a 6-year-long data sample that considers the effects of seasons, range from radar, and time scales. In this study, the authors present two different applications of the aforementioned uncertainty model: (1) the estimation of rainfall probability maps and (2) the generation of radar rainfall ensembles. In the former, maps of the rainfall exceedance probability for any threshold are produced, given a radar rainfall map. We also present the analytical derivation of the exceedance probability maps at coarser spatial scales. In the latter, the users can generate ensembles of probable true rainfall fields that are consistent with the observed radar rainfall and its error structure. Simulation of the random component is based on the Cholesky decomposition method. Finally, the authors discuss possible uses of these applications in hydrology and hydroclimatology.

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Section: Ensemble Generatormentioning

confidence: 99%

Section: Ensemble Generatormentioning

confidence: 99%

Product‐error‐driven generator of probable rainfall conditioned on WSR‐88D precipitation estimates

Villarini

Krajewski

Ciach

et al. 2009

Water Resources Research

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“…We used the circulant embedding method, which is both fast and exact, for the simulation of a profile or surface associated to a Gaussian RF with a given correlation function. The computational features of this method, for lattice processes in R 2 , are illustrated in a recent paper by Gneiting et al [15].…”

Section: Simulation Studymentioning

confidence: 97%

A note on decoupling of local and global behaviours for the Dagum Random Field

Mateu

Porcu

Nicolis

2007

Probabilistic Engineering Mechanics

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“…In R, chol computes the upper-triangular matrix R. The computation has the same overall time complexity as eigenvector decomposition, namely O(M 3 N 3 ) for the decomposition (assuming a dense covariance matrix, see below), and then O(M 2 N 2 ) per sample (Golub and Van Loan 1989;Gneiting,Ševčíková, Percival, Schlather, and Jiang 2006). 5.…”

Section: Recover the Realization By Computing R Zmentioning

confidence: 99%

“…There is no way to guarantee this in a given application, though Møller et al (1998) and Møller and Waagepetersen (2004) remark that this has rarely been a problem for them in practice, provided a suitable level of discretization is employed, comments supported by our experiences. Wood and Chan (1994) and these later works recommend choosing larger lattice extensions as a possible remedy in the event negative eigenvalues are encountered, while Gneiting et al (2006) explore the use of appropriately modified covariance functions.…”

Section: Generationmentioning

confidence: 99%

On Circulant Embedding for Gaussian Random Fields inR

Davies¹,

Bryant²

2013

J. Stat. Soft.

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The high-dimensionality typically associated with discretized approximations to Gaussian random fields is a considerable hinderance to computationally efficient methods for their simulation. Many direct approaches require spectral decompositions of the associated covariance matrix and so are unable to complete the solving process in a timely fashion, if at all. However under certain conditions, we may construct block-circulant versions of the covariance matrix at hand thereby allowing access to fast-Fourier methods to perform the required operations with impressive speed. We demonstrate how circulant embedding and subsequent simulation can be performed directly in the R language. The approach is currently implemented in C for the R package RandomFields, and used in the recently released package lgcp. Motivated by applications dealing with spatial point processes we restrict attention to stationary Gaussian fields on R 2 , where sparsity of the covariance matrix cannot necessarily be assumed.

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Fast and Exact Simulation of Large Gaussian Lattice Systems in ℝ²: Exploring the Limits

Cited by 62 publications

References 23 publications

Product‐error‐driven generator of probable rainfall conditioned on WSR‐88D precipitation estimates

Product‐error‐driven generator of probable rainfall conditioned on WSR‐88D precipitation estimates

A note on decoupling of local and global behaviours for the Dagum Random Field

On Circulant Embedding for Gaussian Random Fields inR

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Fast and Exact Simulation of Large Gaussian Lattice Systems in ℝ2: Exploring the Limits

Cited by 62 publications

References 23 publications

Product‐error‐driven generator of probable rainfall conditioned on WSR‐88D precipitation estimates

Product‐error‐driven generator of probable rainfall conditioned on WSR‐88D precipitation estimates

A note on decoupling of local and global behaviours for the Dagum Random Field

On Circulant Embedding for Gaussian Random Fields inR

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Product

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Fast and Exact Simulation of Large Gaussian Lattice Systems in ℝ²: Exploring the Limits