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Ye J, Rey D, Kadakia N, et al. Systematic variational method for statistical nonlinear state and parameter estimation. Phys Rev E Stat Nonlin Soft Matter Phys. 2015;92(5):052901doi: 10.1103/PhysRevE.92.052901.
Ye, J., Rey, D., Kadakia, N., Eldridge, M., Morone, U. I., Rozdeba, P., Abarbanel, H. D., & Quinn, J. C. (2015). Systematic variational method for statistical nonlinear state and parameter estimation. Physical review. E, Statistical, nonlinear, and soft matter physics, 92(5), 052901. https://doi.org/10.1103/PhysRevE.92.052901
Ye, Jingxin, et al. "Systematic variational method for statistical nonlinear state and parameter estimation." Physical review. E, Statistical, nonlinear, and soft matter physics vol. 92,5 (2015): 052901. doi: https://doi.org/10.1103/PhysRevE.92.052901
Ye J, Rey D, Kadakia N, Eldridge M, Morone UI, Rozdeba P, Abarbanel HD, Quinn JC. Systematic variational method for statistical nonlinear state and parameter estimation. Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Nov;92(5):052901. doi: 10.1103/PhysRevE.92.052901. Epub 2015 Nov 02. PMID: 26651756.
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Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Nov;92(5):052901. doi: 10.1103/PhysRevE.92.052901. Epub 2015 Nov 02.

Systematic variational method for statistical nonlinear state and parameter estimation.

Physical review. E, Statistical, nonlinear, and soft matter physics

Jingxin Ye, Daniel Rey, Nirag Kadakia, Michael Eldridge, Uriel I Morone, Paul Rozdeba, Henry D I Abarbanel, John C Quinn

Affiliations

  1. Department of Physics, University of California, San Diego, La Jolla, California 92093-0374, USA.
  2. Department of Physics, University of California, San Diego, La Jolla, California 92093-0374, USA and Marine Physical Laboratory (Scripps Institution of Oceanography) University of California, San Diego La Jolla, California 92093-0374, USA.
  3. Intellisis Corporation, 10350 Science Center Drive, Suite 140 San Diego, California 92121, USA.

PMID: 26651756 DOI: 10.1103/PhysRevE.92.052901

Abstract

In statistical data assimilation one evaluates the conditional expected values, conditioned on measurements, of interesting quantities on the path of a model through observation and prediction windows. This often requires working with very high dimensional integrals in the discrete time descriptions of the observations and model dynamics, which become functional integrals in the continuous-time limit. Two familiar methods for performing these integrals include (1) Monte Carlo calculations and (2) variational approximations using the method of Laplace plus perturbative corrections to the dominant contributions. We attend here to aspects of the Laplace approximation and develop an annealing method for locating the variational path satisfying the Euler-Lagrange equations that comprises the major contribution to the integrals. This begins with the identification of the minimum action path starting with a situation where the model dynamics is totally unresolved in state space, and the consistent minimum of the variational problem is known. We then proceed to slowly increase the model resolution, seeking to remain in the basin of the minimum action path, until a path that gives the dominant contribution to the integral is identified. After a discussion of some general issues, we give examples of the assimilation process for some simple, instructive models from the geophysical literature. Then we explore a slightly richer model of the same type with two distinct time scales. This is followed by a model characterizing the biophysics of individual neurons.

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