Parameter estimation of partial differential equation models

Xun, X; Cao, J; Mallick, B; Maity, A; Carroll, RJ

Parameter estimation of partial differential equation models

Xun, X Cao, J Mallick, B Maity, A Carroll, RJ

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Publication Type:: Journal Article
Citation:: Journal of the American Statistical Association, 2013, 108 (503), pp. 1009 - 1020
Issue Date:: 2013-12-16

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Field	Value	Language
dc.contributor.author	Xun, X	en_US
dc.contributor.author	Cao, J	en_US
dc.contributor.author	Mallick, B	en_US
dc.contributor.author	Maity, A	en_US
dc.contributor.author	Carroll, RJ https://orcid.org/0000-0002-5465-9682	en_US
dc.date.issued	2013-12-16	en_US
dc.identifier.citation	Journal of the American Statistical Association, 2013, 108 (503), pp. 1009 - 1020	en_US
dc.identifier.issn	0162-1459	en_US
dc.identifier.uri	http://hdl.handle.net/10453/118044
dc.description.abstract	Partial differential equation (PDE) models are commonly used to model complex dynamic systems in applied sciences such as biology and finance. The forms of these PDE models are usually proposed by experts based on their prior knowledge and understanding of the dynamic system. Parameters in PDE models often have interesting scientific interpretations, but their values are often unknown and need to be estimated from the measurements of the dynamic system in the presence of measurement errors. Most PDEs used in practice have no analytic solutions, and can only be solved with numerical methods. Currently, methods for estimating PDE parameters require repeatedly solving PDEs numerically under thousands of candidate parameter values, and thus the computational load is high. In this article, we propose two methods to estimate parameters in PDE models: a parameter cascading method and a Bayesian approach. In both methods, the underlying dynamic process modeled with the PDE model is represented via basis function expansion. For the parameter cascading method, we develop two nested levels of optimization to estimate the PDE parameters. For the Bayesian method, we develop a joint model for data and the PDE and develop a novel hierarchical model allowing us to employ Markov chain Monte Carlo (MCMC) techniques to make posterior inference. Simulation studies show that the Bayesian method and parameter cascading method are comparable, and both outperform other available methods in terms of estimation accuracy. The two methods are demonstrated by estimating parameters in a PDE model from long-range infrared light detection and ranging data. Supplementary materials for this article are available online. © 2013 American Statistical Association.	en_US
dc.relation.ispartof	Journal of the American Statistical Association	en_US
dc.relation.isbasedon	10.1080/01621459.2013.794730	en_US
dc.subject.classification	Statistics & Probability	en_US
dc.title	Parameter estimation of partial differential equation models	en_US
dc.type	Journal Article
utslib.citation.volume	503	en_US
utslib.citation.volume	108	en_US
utslib.for	0104 Statistics	en_US
utslib.for	1403 Econometrics	en_US
utslib.for	1603 Demography	en_US
pubs.embargo.period	Not known	en_US
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Science
pubs.organisational-group	/University of Technology Sydney/Faculty of Science/School of Mathematical and Physical Sciences
utslib.copyright.status	closed_access
pubs.issue	503	en_US
pubs.publication-status	Published	en_US
pubs.volume	108	en_US

Abstract:

Partial differential equation (PDE) models are commonly used to model complex dynamic systems in applied sciences such as biology and finance. The forms of these PDE models are usually proposed by experts based on their prior knowledge and understanding of the dynamic system. Parameters in PDE models often have interesting scientific interpretations, but their values are often unknown and need to be estimated from the measurements of the dynamic system in the presence of measurement errors. Most PDEs used in practice have no analytic solutions, and can only be solved with numerical methods. Currently, methods for estimating PDE parameters require repeatedly solving PDEs numerically under thousands of candidate parameter values, and thus the computational load is high. In this article, we propose two methods to estimate parameters in PDE models: a parameter cascading method and a Bayesian approach. In both methods, the underlying dynamic process modeled with the PDE model is represented via basis function expansion. For the parameter cascading method, we develop two nested levels of optimization to estimate the PDE parameters. For the Bayesian method, we develop a joint model for data and the PDE and develop a novel hierarchical model allowing us to employ Markov chain Monte Carlo (MCMC) techniques to make posterior inference. Simulation studies show that the Bayesian method and parameter cascading method are comparable, and both outperform other available methods in terms of estimation accuracy. The two methods are demonstrated by estimating parameters in a PDE model from long-range infrared light detection and ranging data. Supplementary materials for this article are available online. © 2013 American Statistical Association.

Please use this identifier to cite or link to this item:

http://hdl.handle.net/10453/118044