dorsal/arxiv
View SchemaA Variational Formulation of Optimal Nonlinear Estimation
| Authors | Gregory L. Eyink |
|---|---|
| Categories | |
| ArXiv ID | physics/0011049 |
| URL | https://arxiv.org/abs/physics/0011049 |
Abstract
We propose a variational method to solve all three estimation problems for nonlinear stochastic dynamical systems: prediction, filtering, and smoothing. Our new approach is based upon a proper choice of cost function, termed the {\it effective action}. We show that this functional of time-histories is the unique statistically well-founded cost function to determine most probable histories within empirical ensembles. The ensemble dispersion about the sample mean history can also be obtained from the Hessian of the cost function. We show that the effective action can be calculated by a variational prescription, which generalizes the ``sweep method'' used in optimal linear estimation. An iterative numerical scheme results which converges globally to the variational estimator. This scheme involves integrating forward in time a ``perturbed'' Fokker-Planck equation, very closely related to the Kushner-Stratonovich equation for optimal filtering, and an adjoint equation backward in time, similarly related to the Pardoux-Kushner equation for optimal smoothing. The variational estimator enjoys a somewhat weaker property, which we call ``mean optimality''. However, the variational scheme has the principal advantage---crucial for practical applications---that it admits a wide variety of finite-dimensional moment-closure approximations. The moment approximations are derived reductively from the Euler-Lagrange variational formulation and preserve the good structural properties of the optimal estimator.
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"abstract": "We propose a variational method to solve all three estimation problems for\nnonlinear stochastic dynamical systems: prediction, filtering, and smoothing.\nOur new approach is based upon a proper choice of cost function, termed the\n{\\it effective action}. We show that this functional of time-histories is the\nunique statistically well-founded cost function to determine most probable\nhistories within empirical ensembles. The ensemble dispersion about the sample\nmean history can also be obtained from the Hessian of the cost function. We\nshow that the effective action can be calculated by a variational prescription,\nwhich generalizes the ``sweep method\u0027\u0027 used in optimal linear estimation. An\niterative numerical scheme results which converges globally to the variational\nestimator. This scheme involves integrating forward in time a ``perturbed\u0027\u0027\nFokker-Planck equation, very closely related to the Kushner-Stratonovich\nequation for optimal filtering, and an adjoint equation backward in time,\nsimilarly related to the Pardoux-Kushner equation for optimal smoothing. The\nvariational estimator enjoys a somewhat weaker property, which we call ``mean\noptimality\u0027\u0027. However, the variational scheme has the principal\nadvantage---crucial for practical applications---that it admits a wide variety\nof finite-dimensional moment-closure approximations. The moment approximations\nare derived reductively from the Euler-Lagrange variational formulation and\npreserve the good structural properties of the optimal estimator.",
"arxiv_id": "physics/0011049",
"authors": [
"Gregory L. Eyink"
],
"categories": [
"physics.data-an",
"math-ph",
"math.MP",
"nlin.CD"
],
"title": "A Variational Formulation of Optimal Nonlinear Estimation",
"url": "https://arxiv.org/abs/physics/0011049"
},
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