303 research outputs found
Stochastic filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples
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.Comment: 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)
versio
Optimal projection filters with information geometry
We review the introduction of several types of projection filters. Projection structures coming from information geometry are used to obtain a finite dimensional filter in the form of a stochastic differential equation (SDE), starting from the exact infinite-dimensional stochastic partial differential equation (SPDE) for the optimal filter. We start with the Stratonovich projection filters based on the Hellinger distance as introduced and developed in Brigo, Hanzon and Le Gland (1998, 1999) [19, 20], where the SPDE is put in Stratonovich form before projection, hence the term “Stratonovich projection”. The correction step of the filtering algorithm can be made exact by choosing a suitable exponential family as manifold, there is equivalence with assumed density filters and numerical examples have been studied. Other authors further developed these projection filters and we present a brief literature review. A second type of Stratonovich projection filters was introduced in Armstrong and Brigo (2016) [6] where a direct L2 metric is used for projection. Projecting on mixtures of densities as a manifold coincides with Galerkin methods. All the above projection filters lack optimality, as the single vector fields of the Stratonovich SPDE are projected optimally but the SPDE solution as a whole is not approximated optimally by the projected SDE solution according to a clear criterion. This led to the optimal projection filters in Armstrong, Brigo and Rossi Ferrucci (2019, 2018) [10, 9], based on the Ito vector and Ito jet projections, where several types of mean square distances between the optimal filter SPDE solution and the sought finite dimensional SDE approximations are minimized, with numerical examples. After reviewing the above developments, we conclude with the remaining challenges
Optimal Projection Filters
We present the two new notions of projection of a stochastic differential
equation (SDE) onto a submanifold, as developed in Armstrong, Brigo e Rossi
Ferrucci (2019, 2018): the Ito-vector and Ito-jet projections. This allows one
to systematically and optimally develop low dimensional approximations to high
dimensional SDEs using differential geometric techniques. Our new projections
are based on optimality arguments and yield a well-defined ``optimal''
approximation to the original SDE in the mean-square sense. We also show that
the earlier Stratonovich projection satisfies an optimality criterion that is
more ad hoc and less natural than the criteria satisfied by the new
projections. As an application, we consider approximating the solution of the
non-linear filtering problem within a given manifold of densities, using either
the Hellinger or direct metrics and related Information Geometry
structures on the space of densities. The Stratonovich projection had yielded
the projection filters studied in Brigo, Hanzon and Le Gland (1998, 1999),
while the new projections lead to the optimal projection filters. The optimal
projection filters have been introduced in Armstrong, Brigo e Rossi Ferrucci
(2019), where numerical examples for the Gaussian case are given and where they
are compared to more traditional nonlinear filters.Comment: arXiv admin note: text overlap with arXiv:1610.0388
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