Projection filters are a set of algorithms based on stochastic analysis and information geometry, or the differential geometric approach to statistics, used to find approximate solutions for filtering problems for nonlinear state-space systems. The filtering problem consists of estimating the unobserved signal of a random dynamical system from partial noisy observations of the signal. The objective is computing the probability distribution of the signal conditional on the history of the noise-perturbed observations. This distribution allows for calculations of all statistics of the signal given the history of observations. If this distribution has a density, the density satisfies specific stochastic partial differential equations (SPDEs) called Kushner-Stratonovich equation, or Zakai equation. It is known that the nonlinear filter density evolves in an infinite dimensional function space. One can choose a finite dimensional family of probability densities, for example Gaussian densities, Gaussian mixtures, or exponential families, on which the infinite-dimensional filter density can be approximated. The basic idea of the projection filter is to use a geometric structure in the chosen spaces of densities to project the infinite dimensional SPDE of the optimal filter onto the chosen finite dimensional family, obtaining a finite dimensional stochastic differential equation (SDE) for the parameter of the density in the finite dimensional family that approximates the full filter evolution. To do this, the chosen finite dimensional family is equipped with a manifold structure as in information geometry. The projection filter was tested against the optimal filter for the cubic sensor problem. The projection filter could track effectively bimodal densities of the optimal filter that would have been difficult to approximate with standard algorithms like the extended Kalman filter. Projection filters are ideal for in-line estimation, as they are quick to implement and run efficiently in time, providing a finite dimensional SDE for the parameter that can be implemented efficiently. Projection filters are also flexible, as they allow fine tuning the precision of the approximation by choosing richer approximating families, and some exponential families make the correction step in the projection filtering algorithm exact. Some formulations coincide with heuristic based assumed density filters or with Galerkin methods. Projection filters can also approximate the full infinite-dimensional filter in an optimal way, beyond the optimal approximation of the SPDE coefficients alone, according to precise criteria such as mean square minimization. Projection filters have been studied by the Swedish Defense Research Agency and have also been successfully applied to a variety of fields including navigation, ocean dynamics, quantum optics and quantum systems, estimation of fiber diameters, estimation of chaotic time series, change point detection and other areas.
History and development The term "projection filter" was first coined in 1987 by Bernard Hanzon, and the related theory and numerical examples were fully developed, expanded and made rigorous during the Ph.D. work of Damiano Brigo, in collaboration with Bernard Hanzon and Francois LeGland. These works dealt with the projection filters in Hellinger distance and Fisher information metric, that were used to project the optimal filter infinite-dimensional SPDE on a chosen exponential family. The exponential family can be chosen so as to make the prediction step of the filtering algorithm exact. A different type of projection filters, based on an alternative projection metric, the direct L 2 {\displaystyle L^{2}} metric, was introduced in Armstrong and Brigo (2016). With this metric, the projection filters on families of mixture distributions coincide with filters based on Galerkin methods. Later on, Armstrong, Brigo and Rossi Ferrucci (2021) derive optimal projection filters that satisfy specific optimality criteria in approximating the infinite dimensional optimal filter. Indeed, the Stratonovich-based projection filters optimized the approximations of the SPDE separate coefficients on the chosen manifold but not the SPDE solution as a whole. This has been dealt with by introducing the optimal projection filters. The innovation here is to work directly with Ito calculus, instead of resorting to the Stratonovich calculus version of the filter equation. This is based on research on the geometry of Ito Stochastic differential equations on manifolds based on the jet bundle, the so-called 2-jet interpretation of Ito stochastic differential equations on manifolds.
Projection filters derivation Here the derivation of the different projection filters is sketched.
Stratonovich-based projection filters This is a derivation of both the initial filter in Hellinger/Fisher metric sketched by Hanzon and fully developed by Brigo, Hanzon and LeGland, and the later projection filter in direct L2 metric by Armstrong and Brigo (2016). It is assumed that the unobserved random signal X t ∈ R m {\displaystyle X_{t}\in \mathbb {R} ^{m}} is modelled by the Ito stochastic differential equation:
d X t = f ( X t , t ) d t + σ ( X t , t ) d W t {\displaystyle dX_{t}=f(X_{t},t)\,dt+\sigma (X_{t},t)\,dW_{t}}
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