The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running

Lindgren, Finn; Bolin, David; Rue, Haåvard

doi:10.1016/j.spasta.2022.100599

Cited by 65 publications

(41 citation statements)

References 113 publications

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“…The latter may be particularly suitable to mitigate the effects of boundary conditions (see [15]). Second, the techniques in this paper may be extended to generalized Matérn fields on compact Riemannian fields, see [24,23,28,22]. Finally, in Bayesian inverse problems, it is straightforward to extend the techniques to nonlinear forward problems within a Newton-based solver for the MAP estimate [31].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…Notice that, [1] uses the Fourier approach which is limited to periodic settings and [2] uses an eigendecomposition of the Laplacian that is not scalable to large problems. There are some techniques in computational and spatial statistics that are relevant to this discussion; however for a detailed review, see [23]. In [28], a method was proposed for sampling from generalized Matérn fields on compact Riemannian manifolds using the so-called matrix transfer technique.…”

Section: Related Workmentioning

confidence: 99%

“…Based on this connection, they provided extensions beyond the Matérn model and showed how the SPDE approach can be applied on manifolds, for nonstationary random fields. In a review paper [23], 10 years since the publication of [24], the authors trace the recent developments in the SPDE approach to Gaussian and non-Gaussian random fields. Recent work (e.g., [7,8]) allows for noninteger values of the exponent; see discussion below.…”

Section: Motivation and Introductionmentioning

confidence: 99%

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Efficient algorithms for Bayesian Inverse Problems with Whittle--Matérn Priors

Antil¹,

Saibaba²

2022

Preprint

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This paper tackles efficient methods for Bayesian inverse problems with priors based on Whittle-Matérn Gaussian random fields. The Whittle-Matérn prior is characterized by a mean function and a covariance operator that is taken as a negative power of an elliptic differential operator. This approach is flexible in that it can incorporate a wide range of prior information including non-stationary effects, but it is currently computationally advantageous only for integer values of the exponent. In this paper, we derive an efficient method for handling all admissible noninteger values of the exponent. The method first discretizes the covariance operator using finite elements and quadrature, and uses preconditioned Krylov subspace solvers for shifted linear systems to efficiently apply the resulting covariance matrix to a vector. This approach can be used for generating samples from the distribution in two different ways: by solving a stochastic partial differential equation, and by using a truncated Karhunen-Loève expansion. We show how to incorporate this prior representation into the infinitedimensional Bayesian formulation, and show how to efficiently compute the maximum a posteriori estimate, and approximate the posterior variance. Although the focus of this paper is on Bayesian inverse problems, the techniques developed here are applicable to solving systems with fractional Laplacians and Gaussian random fields. Numerical experiments demonstrate the performance and scalability of the solvers and their applicability to model and real-data inverse problems in tomography and a time-dependent heat equation.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Motivation and Introductionmentioning

confidence: 99%

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Efficient algorithms for Bayesian Inverse Problems with Whittle--Matérn Priors

Antil¹,

Saibaba²

2022

Preprint

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“…where C is a diagonal matrix with its (j, j) Bakka et al (2018) and Lindgren et al (2021). Finally, the vector…”

Section: Construction Of Z(•)mentioning

confidence: 99%

Realistic and Fast Modeling of Spatial Extremes over Large Geographical Domains

Hazra¹,

Huser²,

Bolin³

2021

Preprint

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Various natural phenomena, such as precipitation, generally exhibit spatial extremal dependence at short distances only, while the dependence usually fades away as the distance between sites increases arbitrarily. However, the available models proposed in the literature for spatial extremes, which are based on max-stable or Pareto processes or comparatively less computationally demanding "sub-asymptotic" models based on Gaussian location and/or scale mixtures, generally assume that spatial extremal dependence persists across the entire spatial domain. This is a clear limitation when modeling extremes over large geographical domains, but surprisingly, it has been mostly overlooked in the literature. In this paper, we develop a more realistic Bayesian framework based on a novel Gaussian scale mixture model, where the Gaussian process component is defined by a stochastic partial differential equation that yields a sparse precision matrix, and the random scale component is modeled as a lowrank Pareto-tailed or Weibull-tailed spatial process determined by compactly supported basis functions. We show that our proposed model is approximately tail-stationary despite its nonstationary construction in terms of basis functions, and we demonstrate that it can capture a wide range of extremal dependence structures as a function of distance. Furthermore, the inherently sparse structure of our spatial model allows fast Bayesian computations, even in high spatial dimensions, based on a customized Markov chain Monte Carlo algorithm, which prioritize calibration in the tail. In our application, we fit our model to analyze heavy monsoon rainfall data in Bangladesh. Our study indicates that the proposed model outperforms some natural alternatives, and that the model fits precipitation extremes satisfactorily well. Finally, we use the fitted model to draw inferences on long-term return levels for marginal precipitation at each site, and for spatial aggregates.

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“…The reason for considering this particular SPDE is that when (2) is considered on R d , it has Gaussian random fields with the covariance function (1) as stationary solutions (Whittle, 1963). The method of generalizing the Matérn fields to Riemaniann manifolds by defining Whittle-Matérn field as solutions to (2) specified on the manifold was proposed by Lindgren et al (2011), and has since then been extended to a number of scenarios (see Lindgren et al, 2022, for a recent review), including non-stationary (Bakka et al, 2019;Hildeman et al, 2021) and non-Gaussian (Bolin, 2014;Bolin and Wallin, 2020) models.…”

mentioning

confidence: 99%

Gaussian Whittle-Matérn fields on metric graphs

Bolin¹,

Simas²,

Wallin³

2022

Preprint

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We define a new class of Gaussian processes on compact metric graphs such as street or river networks. The proposed models, the Whittle-Matérn fields, are defined via a fractional stochastic partial differential equation on the compact metric graph and are a natural extension of Gaussian fields with Matérn covariance functions on Euclidean domains to the non-Euclidean metric graph setting. Existence of the processes, as well as their sample path regularity properties are derived. The model class in particular contains differentiable Gaussian processes. To the best of our knowledge, this is the first construction of a valid differentiable Gaussian field on general compact metric graphs. We then focus on a model subclass which we show contains processes with Markov properties. For this case, we show how to evaluate finite dimensional distributions of the process exactly and computationally efficiently. This facilitates using the proposed models for statistical inference without the need for any approximations. Finally, we derive some of the main statistical properties of the model class, such as consistency of maximum likelihood estimators of model parameters and asymptotic optimality properties of linear prediction based on the model with misspecified parameters.

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The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running

Cited by 65 publications

References 113 publications

Efficient algorithms for Bayesian Inverse Problems with Whittle--Matérn Priors

Efficient algorithms for Bayesian Inverse Problems with Whittle--Matérn Priors

Realistic and Fast Modeling of Spatial Extremes over Large Geographical Domains

Gaussian Whittle-Matérn fields on metric graphs

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