A differential geometric approach to nonlinear filtering: the projection filter

TLDR: A convenient exponential family is proposed which allows one to simplify the projection filter equation and to define an a posteriori measure of the local error of the projections filter approximation.

Abstract: This paper presents a new and systematic method of approximating exact nonlinear filters with finite dimensional filters, using the differential geometric approach to statistics. The projection filter is defined rigorously in the case of exponential families. A convenient exponential family is proposed which allows one to simplify the projection filter equation and to define an a posteriori measure of the local error of the projection filter approximation. Finally, simulation results are discussed for the cubic sensor problem.

Figures

Figure 5: Mean from the projection lter and from the optimal lter between 0 and 5

Figure 6: Projection residual between 0 and 5

Figure 14: Projection lter density at 9.54

Figure 13: Optimal lter density at 9.54

Figure 4: True state and mean from the optimal lter between 5 and 10

Figure 3: True state and mean from the optimal lter between 0 and 5

Full text

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A dierential geometric approach to nonlinear ltering :
the projection lter
Damiano Brigo, Bernard Hanzon, François Le Gland
To cite this version:
Damiano Brigo, Bernard Hanzon, François Le Gland. A dierential geometric approach to nonlinear
ltering : the projection lter. [Research Report] 2598, INRIA Rennes - Bretagne Atlantique. 1995.
�hal-02101519�

ISSN 0249-6399
apport 

de recherche
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
A Differential Geometric Approach
to Nonlinear Filtering :
the Projection Filter
Damiano Brigo Bernard Hanzon, Francois Le Gland
N˚ 2598
Juin 1995
PROGRAMME 5

A Dierential Geometric Approach
to Nonlinear Filtering :
the Pro jection Filter
Damiano Brigo

Bernard Hanzon

, Francois Le Gland

Programme 5 | Traitement du signal, automatique et pro ductique
Pro jet AS
Rapport de recherche n2598 | Juin 1995 | 50 pages
Abstract:
This paper deals with a new and systematic method of approximating exact
nonlinear lters with nite dimensional lters. The metho d used here is based on the
dierential geometric approach to statistics. The pro jection lter is derived in the case
of exp onential families. A characterization of the lters is given in terms of an assumed
density principle. An a p osteriori measure of the performance of the pro jection lter is
dened. Applications to particular systems, and numerical schemes which can be used to
implement the pro jection lter are given in the nal part. The results of simulations for the
cubic sensor are discussed.
Key-words:
nite dimensional ltering, assumed density lter, pro jection lter, Fisher
information metric, dierential geometry and statistics.
(Resume : tsvp)
This work was partially supp orted by the Europ ean Economic Community, under the SCIENCE pro ject
System Identication
, pro ject numb er SC1*{CT92{0779, and by the Army Research Oce, under grant
DAAH04{95{1{0164. Damiano Brigo was also supp orted by an
Advanced Studying Fel lowship
of the Uni-
versity of Padua.

Department of Econometrics, Free University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam,
The Netherlands |
f
dbrigo,bhanzon
g
@econ.vu.nl

IRISA / INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France |
legland@irisa.fr
Unite´ de recherche INRIA Rennes
IRISA, Campus universitaire de Beaulieu, 35042 RENNES Cedex (France)
Te´le´phone : (33) 99 84 71 00 – Te´le´copie : (33) 99 84 71 71

Une Appro che du Filtrage Non{Lineaire
Fondee sur la Geometrie Dierentielle :
le Filtre par Pro jection
Resume :
Cet article prop ose une methode nouvelle et systematique p our l'approximation
d'un ltre non{lineaire exact par un ltre de dimension nie. La methode rep ose sur l'utili-
sation d'outils de geometrie dierentielle en statistique. L'equation du ltre par pro jection
est etablie dans le cas des familles exp onentielles, et on en donne une caracterisation en
tant que ltre de forme donnee. On denit egalement une mesure a p osteriori de la qualite
de l'approximation. Dans la derniere partie, on etudie quelques exemples, et on propose un
schema numerique p our la mise en uvre du ltre par pro jection. Finalement, on presente
des resultats de simulations p our le probleme du senseur cubique.
Mots-cle :
ltre de dimension nie, assumed density lter, pro jection lter, information
de Fisher, geometrie dierentielle et statistique

Frequently asked questions

Q1. What are the contributions in "A differential geometric approach to nonlinear filtering : the projection filter" ?

This paper deals with a new and systematic method of approximating exact nonlinear lters with nite dimensional lters. The projection lter is derived in the case of exponential families. An a posteriori measure of the performance of the projection lter is de ned. Applications to particular systems, and numerical schemes which can be used to implement the projection lter are given in the nal part. The results of simulations for the cubic sensor are discussed.