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A sparse linear algebra algorithm for fast computation of prediction variances with Gaussian Markov random fields

Journal Article


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Abstract


  • Gaussian Markov random fields are used in a large number of disciplines in machine vision and spatial statistics. The models take advantage of sparsity in matrices introduced through the Markov assumptions, and all operations in inference and prediction use sparse linear algebra operations that scale well with dimensionality. Yet, for very high-dimensional models, exact computation of predictive variances of linear combinations of variables is generally computationally prohibitive, and approximate methods (generally interpolation or conditional simulation) are typically used instead. A set of conditions isestablished under which the variances of linear combinations of random variables can be computed exactly using the Takahashi recursions. The ensuing computational simplification has wide applicability and may be used to enhance several software packages where model fitting is seated in a maximum-likelihood framework. The resulting algorithm is ideal for use in a variety of spatial statistical applications, including LatticeKrig modelling, statistical downscaling, and fixed rank kriging. It can compute hundreds of thousands exact predictive variances of linear combinations on a standard desktop with ease, even when large spatial GMRF models are used.

Publication Date


  • 2018

Citation


  • Zammit-Mangion, A. & Rougier, J. (2018). A sparse linear algebra algorithm for fast computation of prediction variances with Gaussian Markov random fields. Computational Statistics and Data Analysis, 123 116-130.

Scopus Eid


  • 2-s2.0-85042867014

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2146&context=eispapers1

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/1144

Number Of Pages


  • 14

Start Page


  • 116

End Page


  • 130

Volume


  • 123

Place Of Publication


  • Netherlands

Abstract


  • Gaussian Markov random fields are used in a large number of disciplines in machine vision and spatial statistics. The models take advantage of sparsity in matrices introduced through the Markov assumptions, and all operations in inference and prediction use sparse linear algebra operations that scale well with dimensionality. Yet, for very high-dimensional models, exact computation of predictive variances of linear combinations of variables is generally computationally prohibitive, and approximate methods (generally interpolation or conditional simulation) are typically used instead. A set of conditions isestablished under which the variances of linear combinations of random variables can be computed exactly using the Takahashi recursions. The ensuing computational simplification has wide applicability and may be used to enhance several software packages where model fitting is seated in a maximum-likelihood framework. The resulting algorithm is ideal for use in a variety of spatial statistical applications, including LatticeKrig modelling, statistical downscaling, and fixed rank kriging. It can compute hundreds of thousands exact predictive variances of linear combinations on a standard desktop with ease, even when large spatial GMRF models are used.

Publication Date


  • 2018

Citation


  • Zammit-Mangion, A. & Rougier, J. (2018). A sparse linear algebra algorithm for fast computation of prediction variances with Gaussian Markov random fields. Computational Statistics and Data Analysis, 123 116-130.

Scopus Eid


  • 2-s2.0-85042867014

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2146&context=eispapers1

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/1144

Number Of Pages


  • 14

Start Page


  • 116

End Page


  • 130

Volume


  • 123

Place Of Publication


  • Netherlands