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 objective is computing the probability distribution of the signal conditional on the history of the noise-perturbed observations.
It is known that the nonlinear filter density evolves in an infinite dimensional function space.
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.
[3] To do this, the chosen finite dimensional family is equipped with a manifold structure as in information geometry.
[2][6] 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.
[2] 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.
[3] Some formulations coincide with heuristic based assumed density filters[3] or with Galerkin methods.
[7] Projection filters have been studied by the Swedish Defense Research Agency[1] 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.
[8] The term "projection filter" was first coined in 1987 by Bernard Hanzon,[9] 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.
[10][2][3] 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.
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.
This is a derivation of both the initial filter in Hellinger/Fisher metric sketched by Hanzon[9] and fully developed by Brigo, Hanzon and LeGland,[10][2] and the later projection filter in direct L2 metric by Armstrong and Brigo (2016).
Validity of all regularity conditions necessary for the results to hold will be assumed, with details given in the references.
These are Stratonovich SPDEs whose solutions evolve in infinite dimensional function spaces.
) will not evolve inside any finite dimensional family of densitities, The projection filter idea is approximating
For this version of the projection filter one is satisfied with dealing with the two vector fields separately.
The projected equation thus reads which can be written as where it has been crucial that Stratonovich calculus obeys the chain rule.
By substituting the definition of the operators F and G we obtain the fully explicit projection filter equation in direct metric:
, one can see that the correction step at each new observation is exact, as the related Bayes formula entails no approximation.
separately, although this does not imply it provides the best approximation for the filter SPDE solution as a whole.
A further benefit of the Ito vector projection is that it minimizes the order 1 Taylor expansion in
The detailed calculations are lengthy and laborious,[7] but the resulting approximation achieves
[7] Jones and Soatto (2011) mention projection filters as possible algorithms for on-line estimation in visual-inertial navigation,[12] mapping and localization, while again on navigation Azimi-Sadjadi and Krishnaprasad (2005)[13] use projection filters algorithms.
The projection filter has been also considered for applications in ocean dynamics by Lermusiaux 2006.
Further applications to quantum systems are considered in Gao, Zhang and Petersen (2019).
[17] Ma, Zhao, Chen and Chang (2015) refer to projection filters in the context of hazard position estimation, while Vellekoop and Clark (2006)[18] generalize the projection filter theory to deal with changepoint detection.
Broecker and Parlitz (2000)[20] study projection filter methods for noise reduction in chaotic time series.
Zhang, Wang, Wu and Xu (2014) [21] apply the Gaussian projection filter as part of their estimation technique to deal with measurements of fiber diameters in melt-blown nonwovens.