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June 1999 Approximate nonlinear filtering by projection on exponential manifolds of densities
Damiano Brigo, Bernard Hanzon, François Le Gland
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Bernoulli 5(3): 495-534 (June 1999).

Abstract

This paper introduces in detail a new systematic method to construct approximate finite-dimensional solutions for the nonlinear filtering problem. Once a finite-dimensional family is selected, the nonlinear filtering equation is projected in Fisher metric on the corresponding manifold of densities, yielding the projection filter for the chosen family. The general definition of the projection filter is given, and its structure is explored in detail for exponential families. Particular exponential families which optimize the correction step in the case of discrete-time observations are given, and an a posteriori estimate of the local error resulting from the projection is defined. Simulation results comparing the projection filter and the optimal filter for the cubic sensor problem are presented. The classical concept of assumed density filter (ADF) is compared with the projection filter. It is shown that the concept of ADF is inconsistent in the sense that the resulting filters depend on the choice of a stochastic calculus, i.e. the Itô or the Stratonovich calculus. It is shown that in the context of exponential families, the projection filter coincides with the Stratonovich-based ADF. An example is provided, which shows that this does not hold in general, for non-exponential families of densities.

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Damiano Brigo. Bernard Hanzon. François Le Gland. "Approximate nonlinear filtering by projection on exponential manifolds of densities." Bernoulli 5 (3) 495 - 534, June 1999.

Information

Published: June 1999
First available in Project Euclid: 27 February 2007

zbMATH: 0930.93074
MathSciNet: MR1693600

Keywords: assumed density filter , differential geometry and statistics , exponential family , finite-dimensional filter , Fisher metric , Hellinger metric , Nonlinear filtering , projection filter , Stratonovich stochastic differential equations

Rights: Copyright © 1999 Bernoulli Society for Mathematical Statistics and Probability

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Vol.5 • No. 3 • June 1999
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