An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach

By Finn Lindgren, Håvard Rue, Johan Lindström. 2011

2,822 citations recorded by the source corpus.

Subfields: Applied Mathematics, Economics and Econometrics, Environmental Engineering

View Paper: https://www.semanticscholar.org/paper/5cb1fb089cae0afd8016e92082ff075ec5f2f0a7

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Citations in the corpus

Nonparametric estimation of diffusions: a differential equations approach (2012, background): “The priors can be understood as the limit of the more familiar Gaussian Markov random fields (GMRF), which are treated in detail in Rue & Held (2005), and are close in spirit to the more recent work in Lindgren et al. (2011).”
Bayesian Spatial Modelling with R-INLA (2015, methodology): “However, as discussed in Lindgren, Rue, and Lindstr¨om (2011), one can express a large class of random ﬁeld models as solutions to continuous domain stochastic partial diﬀerential equations (SPDEs), and write down explicit links between the parameters of each SPDE and the elements of precision…”
Spatio-temporal modeling of particulate matter concentration through the SPDE approach (2012, methodology): “(10) of Lindgren et al (2011).”
On Sampling Spatially-Correlated Random Fields for Complex Geometries (2019, methodology): “A solution is oﬀered by stochastic partial diﬀerential equations (SPDEs) [13].”
The statistical finite element method (statFEM) for coherent synthesis of observation data and model predictions (2019, methodology): “Therefore, many of the numerical techniques developed for efficiently solving large-scale inverse problems, like the treatment of non-local covariance operators [43, 44], representation of random fields [13] or the sampling of posteriors [45], can all be adapted to statFEM.”
Constructing Priors that Penalize the Complexity of Gaussian Random Fields (2015, methodology): “The nuggets are iid (cid:14) i ∼ N ( 0 , σ 2N ) , and the spatial effect is constructed with the SPDE approach of Lindgren, Rue, and Lindström (2011) and the stationary version has two parameters: spatial range ρ and marginal variance of the spatial field σ 2 .”
Spatial models with explanatory variables in the dependence structure (2014, methodology): “Lindgren et al. (2011) show that an approximation to the solution of the SPDE in (1) can be obtained using a finite element method (FEM), a numerical technique for solving partial differential equations (see e.g. Brenner and Scott (2008)).”
A Case Study Competition Among Methods for Analyzing Large Spatial Data (2017, background): “…Lemos and Sansó 2009), fixed rank kriging (Cressie and Johannesson 2008; Kang andCressie 2011; Katzfuss and Cressie 2011), predictive processes (Banerjee et al. 2008; Finley et al. 2009), lattice kriging (Nychka et al. 2015) and stochastic partial differential equations (Lindgren et al. 2011).”
Multifidelity Information Fusion Algorithms for High-Dimensional Systems and Massive Data sets (2016, background): “Addressing this challenge has received great attention over the last decades and several methods have been proposed to alleviate the computational cost [16, 17, 18].”
Estimation and Inference for Generalized Geoadditive Models (2020, background): “For a fixed number of triangles, Lai and Schumaker (2007) and Lindgren, Rue, and Lindström (2011) recommend selecting the triangulation according to “max-min” criterion which maximizes the minimum angle of all the angles of the triangles in the triangulation.”
Space‐time modelling of trends in temperature series (2011, background): “The restriction to gridded data and predictions is unduly constraining in a world where computing resources are quite fast (see, e.g. Lindgren et al., 2011).”
Functional peaks‐over‐threshold analysis (2020, methodology): “The realism of simulated storms might be improved by using the methodology of Lindgren et al. (2011) to build physically-inspired non-stationary spatiotemporal dependence structures, using for instance the diffusion equation, and this would be computationally efficient and perhaps more realistic.”
INLA goes extreme: Bayesian tail regression for the estimation of high spatio-temporal quantiles (2018, background): “…s ) for simplicity, is assumed to be driven by a zero mean Gaussian process prior with precision τ s > 0 and Mat´ern correlation function (Stein, 1999; Lindgren et al. , 2011; Lindgren and Rue, 2015), i.e., cov { x ( s 1 ) , where K ν denotes the modiﬁed Bessel function of second kind of order ν…”