An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach
(2011) In Journal of the Royal Statistical Society. Series B: Statistical Methodology 73(4). p.423498 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 R2 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 Rd, 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 nonstationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a nonstationary model defined on a sphere. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/2065439
 author
 Lindgren, Finn ^{LU} ; Rue, Havard and Lindström, Johan ^{LU}
 organization
 publishing date
 2011
 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
 WileyBlackwell
 external identifiers

 wos:000293566900001
 scopus:79961050814
 ISSN
 13697412
 DOI
 10.1111/j.14679868.2011.00777.x
 project
 MERGE
 BECC
 language
 English
 LU publication?
 yes
 id
 aa9806a61f3e49cf84649e4e97d2139d (old id 2065439)
 date added to LUP
 20110829 16:48:41
 date last changed
 20171119 03:52:11
@article{aa9806a61f3e49cf84649e4e97d2139d, 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 R2 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 Rd, 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 nonstationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a nonstationary model defined on a sphere.}, author = {Lindgren, Finn and Rue, Havard and Lindström, Johan}, issn = {13697412}, 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 = {423498}, publisher = {WileyBlackwell}, 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.14679868.2011.00777.x}, volume = {73}, year = {2011}, }