TY - JOUR

T1 - A sparse linear algebra algorithm for fast computation of prediction variances with Gaussian Markov random fields

AU - Zammit-Mangion, Andrew

AU - Rougier, Jonathan

PY - 2018/7/1

Y1 - 2018/7/1

N2 - 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.

AB - 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.

KW - Conditional dependence

KW - GMRF

KW - Lattice spatial model

KW - Sparse inverse subset

KW - Takahashi equations

UR - http://www.scopus.com/inward/record.url?scp=85042867014&partnerID=8YFLogxK

U2 - 10.1016/j.csda.2018.02.001

DO - 10.1016/j.csda.2018.02.001

M3 - Article (Academic Journal)

AN - SCOPUS:85042867014

VL - 123

SP - 116

EP - 130

JO - Computational Statistics & Data Analysis

JF - Computational Statistics & Data Analysis

SN - 0167-9473

ER -