Stochastic filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples

Abstract

We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite dimensional Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric and on a family of normal mixtures. We compare this method to earlier projection methods based on the Hellinger distance/Fisher metric and exponential families, and we compare the L2 mixture projection filter with a particle method with the same number of parameters, using the Levy metric. We prove that for a simple choice of the mixture manifold the L2 mixture projection filter coincides with a Galerkin method, whereas for more general mixture manifolds the equivalence does not hold and the L2 mixture filter is more general. We study particular systems that may illustrate the advantages of this new filter over other algorithms when comparing outputs with the optimal filter. We finally consider a specific software design that is suited for a numerically efficient implementation of this filter and provide numerical examples.

Updated and expanded version published in the Journal reference below. Preprint updates: January 2016 (v3) added projection of Zakai Equation and difference with projection of Kushner-Stratonovich (section 4.1). August 2014 (v2) added Galerkin equivalence proof (Section 5) to the March 2013 (v1) version

References in corpus (3)

• Complete Real Time Solution of the General Nonlinear Filtering Problem without Memory
• Hermite spectral method to 1D forward Kolmogorov equation and its application to nonlinear filtering problems
• Stochastic filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples

Cited by in corpus (4)

• Stochastic filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples
• Coordinate-free Stochastic Differential Equations as Jets
• Projection based dimensionality reduction for measure valued evolution equations in statistical manifolds
• Optimal approximations of the Fokker-Planck-Kolmogorov equation: projection, maximum likelihood eigenfunctions and Galerkin methods