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Exploiting dynamical coherence: A geometric approach to parameter estimation in nonlinear models

Smith, Leonard A.; Cuéllar, Milena C.; Du, Hailiang; Judd, Kevin


Leonard A. Smith

Milena C. Cuéllar

Kevin Judd


Parameter estimation in nonlinear models is a common task, and one for which there is no general solution at present. In the case of linear models, the distribution of forecast errors provides a reliable guide to parameter estimation, but in nonlinear models the facts that predictability may vary with location in state space, and that the distribution of forecast errors is expected not to be Normal, means that parameter estimation based on least squares methods will result in systematic errors. A new approach to parameter estimation is presented which focuses on the geometry of trajectories of the model rather than the distribution of distances between model forecast and the observation at a given lead time. Specifically, we test a number of candidate trajectories to determine the duration for which they can shadow the observations, rather than evaluating a forecast error statistic at any specific lead time(s). This yields insights into both the parameters of the dynamical model and those of the observational noise model. The advances reported here are made possible by extracting more information from the dynamical equations, and thus improving the balance between information gleaned from the structural form of the equations and that from the observations. The technique is illustrated for both flows and maps, applied in 2-, 3-, and 8-dimensional dynamical systems, and shown to be effective in a case of incomplete observation where some components of the state are not observed at all. While the demonstration of effectiveness is strong, there remain fundamental challenges in the problem of estimating model parameters when the system that generated the observations is not a member of the model class. Parameter estimation appears ill defined in this case.


Smith, L. A., Cuéllar, M. C., Du, H., & Judd, K. (2010). Exploiting dynamical coherence: A geometric approach to parameter estimation in nonlinear models. Physics Letters A, 374(26), 2618-2623.

Journal Article Type Article
Acceptance Date Apr 9, 2010
Online Publication Date Apr 21, 2010
Publication Date 2010-06
Deposit Date Jul 31, 2018
Journal Physics Letters A
Print ISSN 0375-9601
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 374
Issue 26
Pages 2618-2623