Bayes linear analysis for ordinary differential equations

Jones, Matthew and Goldstein, Michael and Randell, David and Jonathan, Philip (2021) Bayes linear analysis for ordinary differential equations. Computational Statistics and Data Analysis, 161: 107228. ISSN 0167-9473

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Differential equation models are used in a wide variety of scientific fields to describe the behaviour of physical systems. Commonly, solutions to given systems of differential equations are not available in closed-form; in such situations, the solution to the system is generally approximated numerically. The numerical solution obtained will be systematically different from the (unknown) true solution implicitly defined by the differential equations. Even if it were known, this true solution would be an imperfect representation of the behaviour of the real physical system that it was designed to represent. A Bayesian framework is proposed which handles all sources of numerical and structural uncertainty encountered when using ordinary differential equation (ODE) models to represent real-world processes. The model is represented graphically, and the graph proves to be useful tool, both for deriving a full prior belief specification and for inferring model components given observations of the real system. A general strategy for modelling the numerical discrepancy induced through choice of a particular solver is outlined, in which the variability of the numerical discrepancy is fixed to be proportional to the length of the solver time-step and a grid-refinement strategy is used to study its structure in detail. A Bayes linear adjustment procedure is presented, which uses a junction tree derived from the originally specified directed graphical model to propagate information efficiently between model components, lessening the computational demands associated with the inference. The proposed framework is illustrated through application to two examples: a model for the trajectory of an airborne projectile moving subject to gravity and air resistance, and a model for the coupled motion of a set of ringing bells and the tower which houses them.

Item Type:
Journal Article
Journal or Publication Title:
Computational Statistics and Data Analysis
Additional Information:
This is the author’s version of a work that was accepted for publication in Computational Statistics & Data Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computational Statistics & Data Analysis, 161, 2021 DOI: 10.1016/j.csda.2021.107228
Uncontrolled Keywords:
?? bayes linear analysisgraphical modelsnumerical discrepancystructural discrepancyuncertainty quantificationstatistics and probabilitycomputational mathematicscomputational theory and mathematicsapplied mathematics ??
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Deposited On:
03 Mar 2022 10:40
Last Modified:
08 May 2024 00:21