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

Lindgren, Finn LU ; Rue, Havard and Lindström, Johan LU (2011) In Journal of the Royal Statistical Society. Series B: Statistical Methodology 73(4). p.423-498
Abstract
Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse... (More)
Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in R-2 only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matern class, provide an explicit link, for any triangulation of R-d, between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Approximate Bayesian inference, Covariance functions, Gaussian fields, Gaussian Markov random fields, Latent Gaussian models, Sparse matrices, Stochastic partial differential equations
in
Journal of the Royal Statistical Society. Series B: Statistical Methodology
volume
73
issue
4
pages
423 - 498
publisher
Wiley-Blackwell
external identifiers
  • wos:000293566900001
  • scopus:79961050814
ISSN
1369-7412
DOI
10.1111/j.1467-9868.2011.00777.x
project
MERGE
BECC
language
English
LU publication?
yes
id
aa9806a6-1f3e-49cf-8464-9e4e97d2139d (old id 2065439)
date added to LUP
2011-08-29 16:48:41
date last changed
2017-11-19 03:52:11
@article{aa9806a6-1f3e-49cf-8464-9e4e97d2139d,
  abstract     = {Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in R-2 only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matern class, provide an explicit link, for any triangulation of R-d, between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere.},
  author       = {Lindgren, Finn and Rue, Havard and Lindström, Johan},
  issn         = {1369-7412},
  keyword      = {Approximate Bayesian inference,Covariance functions,Gaussian fields,Gaussian Markov random fields,Latent Gaussian models,Sparse matrices,Stochastic partial differential equations},
  language     = {eng},
  number       = {4},
  pages        = {423--498},
  publisher    = {Wiley-Blackwell},
  series       = {Journal of the Royal Statistical Society. Series B: Statistical Methodology},
  title        = {An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach},
  url          = {http://dx.doi.org/10.1111/j.1467-9868.2011.00777.x},
  volume       = {73},
  year         = {2011},
}