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

Andrew Zammit-Mangion*, Jonathan Rougier

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

3 Citations (Scopus)
194 Downloads (Pure)

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.

Original languageEnglish
Pages (from-to)116-130
Number of pages15
JournalComputational Statistics and Data Analysis
Volume123
Early online date15 Feb 2018
DOIs
Publication statusPublished - 1 Jul 2018

Keywords

  • Conditional dependence
  • GMRF
  • Lattice spatial model
  • Sparse inverse subset
  • Takahashi equations

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