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.

Highlights

  • In the nonlinear filtering problem, one observes a system whose state is known to follow a given stochastic differential equation

  • When the observations are made in continuous time, the probability density follows a stochastic partial differential equation known as the Kushner–Stratonovich equation

  • We show that the projection filter for basic mixture manifolds in L2 metric is equivalent to a Galerkin method

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Summary

Introduction

In the nonlinear filtering problem, one observes a system whose state is known to follow a given stochastic differential equation. We will write down the stochastic ODE determined by the geometric approach when H = L2 and show how it leads to a numerical scheme for finding approximate solutions to the Kushner–Stratonovich equations in terms of a mixture of normal distributions. We will call this scheme the L2 normal mixture projection filter or the L2NM projection filter. 3, we introduce the geometric structure we need to project the filtering SPDE onto a finite-dimensional manifold of probability densities.

The nonlinear filtering problem with continuous time observations
Page 6 of 33
Families of distributions
Page 8 of 33
Two Hilbert spaces of probability distributions
The tangent space of a family of distributions
Page 10 of 33
The Fisher information metric
The direct L2 metric
The projection filter
Page 14 of 33
Equivalence with assumed density filters and Galerkin methods
Page 16 of 33
Numerical software design
Page 18 of 33
The case of normal mixture families
Page 20 of 33
Numerical results
The linear filter
The quadratic sensor
The cubic sensor
Page 26 of 33
Comparison with particle methods
Page 30 of 33
10 Comparison with robust Zakai implementation using Hermite functions
11 Conclusions
Page 32 of 33
Full Text
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