The Rational SPDE Approach for Gaussian Random Fields With General Smoothness

Bolin, David; Kirchner, Kristin

doi:10.1080/10618600.2019.1665537

Cited by 72 publications

(63 citation statements)

References 35 publications

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“…Nevertheless, extending the approach to fields with general smoothness would increase the flexibility. As previously mentioned, this could probably be done by using the rational SPDE approach (Bolin and Kirchner, 2019), and extending that method to multivariate type G fields is thus an interesting topic for future research.…”

Section: Discussionmentioning

confidence: 99%

“…As is standard in the SPDE approach, we assume that the smoothness parameters are fixed and known. It should be noted that models with general smoothness parameters probably could be estimated from data by using the rational SPDE approach (Bolin and Kirchner, 2019). However, we leave the adaptation of this approach to the multivariate type G setting for future research.…”

Section: Parameter Estimationmentioning

confidence: 99%

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Multivariate Type G Matérn Stochastic Partial Differential Equation Random Fields

Bolin

Wallin

2019

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary For many applications with multivariate data, random‐field models capturing departures from Gaussianity within realizations are appropriate. For this reason, we formulate a new class of multivariate non‐Gaussian models based on systems of stochastic partial differential equations with additive type G noise whose marginal covariance functions are of Matérn type. We consider four increasingly flexible constructions of the noise, where the first two are similar to existing copula‐based models. In contrast with these, the last two constructions can model non‐Gaussian spatial data without replicates. Computationally efficient methods for likelihood‐based parameter estimation and probabilistic prediction are proposed, and the flexibility of the models suggested is illustrated by numerical examples and two statistical applications.

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Section: Discussionmentioning

confidence: 99%

Section: Parameter Estimationmentioning

confidence: 99%

Multivariate Type G Matérn Stochastic Partial Differential Equation Random Fields

Bolin

Wallin

2019

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…The parsimonious fractional approximation, which is implemented in R‐INLA for the stationary Matérn model, is not applicable for the more general nonstationary models. However, Bolin and Kirchner () propose a rational SPDE method that is computable for any α > 0, and which has a higher accuracy than the parsimonious approximation for Matérn model. It combines the FEM approximation in space with a rational approximation of the function x − α /2 in order to compute an approximation of u ( s ) on the form u = Px , where

bold-italicx \sim scriptN ((),, 0, {boldQ}^{‐ 1})

, and P and Q are sparse matrices.…”

Section: Adding Complexity To the Spatial Effectmentioning

confidence: 99%

Spatial modeling with R‐INLA: A review

Bakka

Rue

Fuglstad

et al. 2018

WIREs Computational Stats

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304

246

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Coming up with Bayesian models for spatial data is easy, but performing inference with them can be challenging. Writing fast inference code for a complex spatial model with realistically‐sized datasets from scratch is time‐consuming, and if changes are made to the model, there is little guarantee that the code performs well. The key advantages of R‐INLA are the ease with which complex models can be created and modified, without the need to write complex code, and the speed at which inference can be done even for spatial problems with hundreds of thousands of observations. R‐INLA handles latent Gaussian models, where fixed effects, structured and unstructured Gaussian random effects are combined linearly in a linear predictor, and the elements of the linear predictor are observed through one or more likelihoods. The structured random effects can be both standard areal model such as the Besag and the BYM models, and geostatistical models from a subset of the Matérn Gaussian random fields. In this review, we discuss the large success of spatial modeling with R‐INLA and the types of spatial models that can be fitted, we give an overview of recent developments for areal models, and we give an overview of the stochastic partial differential equation (SPDE) approach and some of the ways it can be extended beyond the assumptions of isotropy and separability. In particular, we describe how slight changes to the SPDE approach leads to straight‐forward approaches for nonstationary spatial models and nonseparable space–time models. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory Statistical Models > Bayesian Models Data: Types and Structure > Massive Data

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“…Finally, we observe that PDE-based sampling requires a discretization of the fractional elliptic SPDE; this is often performed together with the white noise discretization. Recent studies on the finite element discretization and error analysis of semilinear and fractional elliptic SPDEs with white noise forcing can be found in [60] and [6,7,8], respectively. For the spatial discretization, classical piecewise linear [7,8,16,40] or mixed finite elements [43,44] have been employed.…”

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confidence: 99%

Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields

Khristenko¹,

Scarabosio²,

Swierczynski³

et al. 2019

SIAM/ASA J. Uncertainty Quantification

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We consider the generation of samples of a mean-zero Gaussian random field with Matérn covariance function. Every sample requires the solution of a differential equation with Gaussian white noise forcing, formulated on a bounded computational domain. This introduces unwanted boundary effects since the stochastic partial differential equation is originally posed on the whole R d , without boundary conditions. We use a window technique, whereby one embeds the computational domain into a larger domain, and postulates convenient boundary conditions on the extended domain. To mitigate the pollution from the artificial boundary it has been suggested in numerical studies to choose a window size that is at least as large as the correlation length of the Matérn field. We provide a rigorous analysis for the error in the covariance introduced by the window technique, for homogeneous Dirichlet, homogeneous Neumann, and periodic boundary conditions. We show that the error decays exponentially in the window size, independently of the type of boundary condition. We conduct numerical experiments in 1D and 2D space, confirming our theoretical results.

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The Rational SPDE Approach for Gaussian Random Fields With General Smoothness

Cited by 72 publications

References 35 publications

Multivariate Type G Matérn Stochastic Partial Differential Equation Random Fields

Multivariate Type G Matérn Stochastic Partial Differential Equation Random Fields

Spatial modeling with R‐INLA: A review

Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields

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