Gaussian Markov Random Fields: Theory and Applications

By Håvard Rue, Leonhard Held. 2005

2,517 citations recorded by the source corpus.

Subfields: Economics and Econometrics, Environmental Engineering, Statistics and Probability

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Citations in the corpus

Effective Bayesian Modeling of Groups of Related Count Time Series (2014, background): “The second line of this equation expresses a prior over jointly normally-distributed variables with a highly structured (and very sparse) precision matrix, a Gaussian Markov random field (GMRF, see Rue & Held, 2005), to which we now turn.”
Fast, Exact and Multi-scale Inference for Semantic Image Segmentation with Deep Gaussian CRFs (2016, methodology): “ for x. In contrast, the Gauss-Seidel [23] method always uses the most current estimate of x i as given by: x(k+1) i 1 a ii 8 < : b i X j<i a ijx (k+1) j X j>i a ijx (k) j 9 = ; . (13) As in [24], the Gaussian Markov Random Field (GMRF) in its canonical form is expressed as ˇ(x) /exp 1 2 x ”
Excursion and contour uncertainty regions for latent Gaussian models (2012, background): “See Rue and Held (2005) for an introduction to GMRFs, and note that GMRFs are used also for modelling in continuous space, e.”
Spatio-temporal modeling of particulate matter concentration through the SPDE approach (2012, methodology): “Following the notation of Rue and Held (2005), we have that this conditional independence relation can be written as The key point is that this conditional independence property is strictly related to the precision matrix Q .”
Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors (2014, background): “This also includes the case where R is not of full rank, like in popular intrinsic models, spline and thin-plate spline models (Rue and Held, 2005).”
Bayesian structured additive distributional regression with an application to regional income inequality in Germany (2015, methodology): “For the structured spatial effect, we assume spatial correlations defined implicitly by assuming a Gaussian Markov random field prior [Rue and Held (2005)] for a suitable neighbourhood structure derived from the spatial orientation of the data.”
Matérn Cross-Covariance Functions for Multivariate Random Fields (2010, background): “Gaussian Markov random fields provide an alternative that is particularly suited to lattice data (Besag 1974; Mardia 1988; Rue and Held 2005; Sain and Cressie 2007).”
Bayesian calibration of a large‐scale geothermal reservoir model by a new adaptive delayed acceptance Metropolis Hastings algorithm (2011, methodology): “The permeability distribution is parametrized by a pixel-based random field representation [Cressie, 1993; Rue and Held, 2005] that has the same resolution as the coarse grid.”
Randomized Iterative Methods for Linear Systems (2015, background): “In machine learning, applications arise for the problem of ﬁnding optimal conﬁgura-tions in Gaussian Markov Random Fields [32], in graph-based semi-supervised learning and other graph-Laplacian problems [2], least-squares SVMs, Gaussian processes and more.”
Discriminative Decorrelation for Clustering and Classification (2012, methodology): “We then incorporate assumptions of translation invariance by modeling Σ with a spatial autocorrelation function [14]:”
Gaussian Conditional Random Field Network for Semantic Segmentation (2016, background): “In contrast to discrete graphical models, Gaussian graphical models [41, 50] are simpler models, and have inference procedures that are guaranteed to converge to the global optimal solution.”
Deep learning based brain tumor segmentation: a survey (2020, background): “Recently, conditional random fields (CRF) and Gaussian Markov Random Fields (MRF) have been used for post-processing by inferring the pixel pairs given prior information such as pixel intensity distributions and spatial distance [108], [109].”
Distributed Estimation of Variance in Gaussian Graphical Model via Belief Propagation: Accuracy Analysis and Improvement (2015, neutral): “Gaussian graphical model, variance accuracy analysis.”
Geostatistical inference under preferential sampling (2010, background): “Y 1 ,... , Y n / (Rue and Held, 2005).”
Graphical models for extremes (2018, background): “The normal distribution Wk\k factorizes in the classical sense on the subgraph G\k, and efficient simulation algorithms exist if the graph is sparse (e.g. Rue and Held (2005)).”
Estimation and prediction of the spatial occurrence of fish species using Bayesian latent Gaussian models (2013, methodology): “…have proposed an alternative approach by using an approximate stochastic weak solution to a Stochastic Partial Differential Equation (SPDE) as a Gaussian Markov Random Field (GMRF, Rue and Held 2005; Rue et al 2009) approximation to continuous Gaussian Fields with Matérn covariance structure.”