Approximate Nonlinear Filtering by Projection on Exponential Manifolds of Densities

Abstract

This paper introduces in detail a new systematic method to construct approximate ®nite-dimensional solutions for the nonlinear ®ltering problem.Once a ®nite-dimensional family is selected, the nonlinear ®ltering equation is projected in Fisher metric on the corresponding manifold of densities, yielding the projection ®lter for the chosen family.The general de®nition of the projection ®lter 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 de®ned.Simulation results comparing the projection ®lter and the optimal ®lter for the cubic sensor problem are presented.The classical concept of assumed density ®lter (ADF) is compared with the projection ®lter.It is shown that the concept of ADF is inconsistent in the sense that the resulting ®lters depend on the choice of a stochastic calculus, i.e. the Ito à or the Stratonovich calculus.It is shown that in the context of exponential families, the projection ®lter 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.