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STREAMLINED SOLUTIONS TO MULTILEVEL SPARSE MATRIX PROBLEMS

Part of: Numerical linear algebra

Published online by Cambridge University Press:  01 June 2020

TUI H. NOLAN Open the ORCID record for TUI H. NOLAN [Opens in a new window]  and
MATT P. WAND Open the ORCID record for MATT P. WAND [Opens in a new window]
TUI H. NOLAN
Affiliation:
University of Technology Sydney, P.O. Box 123, Broadway, New South Wales2007, Australia email [email protected], [email protected]
MATT P. WAND*
Affiliation:
University of Technology Sydney, P.O. Box 123, Broadway, New South Wales2007, Australia email [email protected], [email protected]
*
✉[email protected]
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Abstract

We define and solve classes of sparse matrix problems that arise in multilevel modelling and data analysis. The classes are indexed by the number of nested units, with two-level problems corresponding to the common situation, in which data on level-1 units are grouped within a two-level structure. We provide full solutions for two-level and three-level problems, and their derivations provide blueprints for the challenging, albeit rarer in applications, higher-level versions of the problem. While our linear system solutions are a concise recasting of existing results, our matrix inverse sub-block results are novel and facilitate streamlined computation of standard errors in frequentist inference as well as allowing streamlined mean field variational Bayesian inference for models containing higher-level random effects.

Keywords

best linear unbiased predictionlinear mixed modelslongitudinal data analysispanel datasmall-area estimationvariational inference

MSC classification

Primary: 65F05: Direct methods for linear systems and matrix inversion
Type
Research Article
Information
The ANZIAM Journal , Volume 62 , Issue 1 , January 2020 , pp. 18 - 41
Copyright
© 2020 Australian Mathematical Society

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