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40
Exponential Stability for Nonlinear Filtering
, 1996
"... We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficient. C ..."
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Cited by 53 (2 self)
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We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficient. Criteria for exponential stability and explicit bounds on the rate are given in the specific cases of a diffusion process on a compact manifold, and discrete time Markov chains on both continuous and discretecountable state spaces. R'esum'e Nous 'etudions la stabilit'e du filtre optimal par raport `a ses conditions initiales. Le taux de d'ecroissance exponentielle est calcul'e dans un cadre g'en'eral, pour temps discret et temps continu, en terme du coefficient de contraction de Birkhoff. Des crit`eres de stabilit'e exponentielle et des bornes explicites sur le taux sont calcul'ees pour les cas particuliers d'une diffusion sur une vari'ete compacte, ainsi que pour des chaines de Markov sur ...
Discrete Filtering Using Branching and Interacting Particle Systems
, 1998
"... The stochastic filtering problem deals with the estimation of the current state of a signal process given the information supplied by an associate process, usually called the observation process. We describe a particle algorithm designed for solving numerically discrete filtering problems. The algor ..."
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Cited by 34 (3 self)
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The stochastic filtering problem deals with the estimation of the current state of a signal process given the information supplied by an associate process, usually called the observation process. We describe a particle algorithm designed for solving numerically discrete filtering problems. The algorithm involves the use of a system of n particles which evolve (mutate) in correlation with each other (interact) according to law of the signal process and, at fixed times, give birth to a number of offsprings depending on the observation process. We present several possible branching mechanisms and prove, in a general context the convergence of the particle systems (as n tends to 1) to the conditional distribution of the signal given the observation. We then apply the result to the discrete filtering and give several example when the results can be applied. AMS Subject Classification (1991): 93E11, 60G57, 65U05 Keywords: Filtering, Particle Systems, Branching Algorithms, Interacting Algor...
A Robustification Approach to Stability and to Uniform Particle Approximation of Nonlinear Filters: The Example of PseudoMixing Signals
, 2002
"... We propose a new approach to study the stability of the optimal filter w.r.t. its initial condition, by introducing a "robust" filter, which is exponentially stable and which approximates the optimal filter uniformly in time. The "robust" filter is obtained here by truncation of the likelihood funct ..."
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Cited by 30 (3 self)
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We propose a new approach to study the stability of the optimal filter w.r.t. its initial condition, by introducing a "robust" filter, which is exponentially stable and which approximates the optimal filter uniformly in time. The "robust" filter is obtained here by truncation of the likelihood function, and the robustification result is proved under the assumption that the Markov transition kernel satisfies a pseudomixing condition (weaker than the usual mixing condition), and that the observations are "sufficiently good". This robustification approach allows us to prove also the uniform convergence of several particle approximations to the optimal filter, in some cases of nonergodic signals.
Central Limit Theorem for Non Linear Filtering and Interacting Particle Systems
 Ann. Appl. Probab
, 1999
"... Several random particle systems approaches were recently suggested to solve numerically non linear filtering problems. The present analysis is concerned with genetictype interacting particle systems. Our aim is to study the fluctuations on path space of such particle approximating models. Keywords ..."
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Cited by 26 (6 self)
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Several random particle systems approaches were recently suggested to solve numerically non linear filtering problems. The present analysis is concerned with genetictype interacting particle systems. Our aim is to study the fluctuations on path space of such particle approximating models. Keywords : Central Limit, Interacting random processes, Filtering, Stochastic approximation. code A.M.S : 60F05, 60G35, 93E11, 62L20. 1 Introduction 1.1 Background and motivations The Non Linear Filtering problem consists in recursively computing the conditional distributions of a non linear signal given its noisy observations. This problem has been extensively studied in the literature and, with the notable exception of the linearGaussian situation or wider classes of models (B`enes filters [2]) optimal filters have no finitely recursive solution (ChaleyatMaurel /Michel [7]). Although Kalman filtering ([26],[29]) is a popular tool in handling estimation problems its optimality heavily depends on...
Asymptotic stability of the Wonham filter: ergodic and nonergodic signals
 SIAM J. Control Optim
"... Abstract. Stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the ..."
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Cited by 25 (13 self)
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Abstract. Stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the nonergodic case, the stability is implied by identifiability conditions, formulated explicitly in terms of the transition intensities matrix and the observation structure. Key words. Nonlinear filtering, stability, Wonham filter
Nonlinear Filtering: Interacting Particle Resolution
, 1996
"... This paper cover stochastic particle methods for the numerical solving of the nonlinear filtering equations based upon the simulation of interacting particle systems. The main contribution of this paper is to prove the convergences of such approximations to the optimal filter, yielding what seemed t ..."
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Cited by 25 (5 self)
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This paper cover stochastic particle methods for the numerical solving of the nonlinear filtering equations based upon the simulation of interacting particle systems. The main contribution of this paper is to prove the convergences of such approximations to the optimal filter, yielding what seemed to be the first convergence results for such approximations of the nonlinear filtering equations. This new treatment was influenced primarily by the development of genetic algorithms (J.H. Holland [11], R. Cerf [2]) and secondarily by the papers of H.Kunita and L.Stettner ([12], [13]). Such interacting particle resolutions encompass genetic algorithms, incidentally our models provide essential insight for the analysis of genetic algorithms with a non homogeneous fitness function with respect to time.
Large Deviations for Interacting Particle Systems. Applications to Non Linear Filtering
 Stochastic Processes and their Applications
, 1997
"... The non linear filtering problem consists in computing the conditional distributions of a Markov signal process given its noisy observations. The dynamical structure of such distributions can be modelled by a measure valued dynamical Markov process. Several random particle approximations were recent ..."
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Cited by 20 (5 self)
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The non linear filtering problem consists in computing the conditional distributions of a Markov signal process given its noisy observations. The dynamical structure of such distributions can be modelled by a measure valued dynamical Markov process. Several random particle approximations were recently suggested to approximate recursively in time the socalled non linear filtering equations. We present an interacting particle system approach and we develop large deviations principles for the empirical measures of the particle systems. We end this paper extending the results to an interacting particle system approach which includes branchings. Keywords : Large deviations, Interacting random processes, Filtering, Stochastic approximation. code A.M.S : 60F10, 60H10, 60G35, 93E11, 62L20. 1 Introduction 1.1 Background and motivations The non linear filtering problem consists in computing the conditional distribution of a signal given its noisy observation. Roughly speaking, a basic model...
Approximation and Limit Results for Nonlinear Filters over an Infinite Time Interval: Part II, Random Sampling Algorithms
"... The paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval. Since the optimal filter can rarely be constructed, one needs to compute with numerically feasible approximations. The signal model can be a jumpdiffusion, reflected or no ..."
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Cited by 19 (8 self)
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The paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval. Since the optimal filter can rarely be constructed, one needs to compute with numerically feasible approximations. The signal model can be a jumpdiffusion, reflected or not. The observations can be taken either in discrete or continuous time. The cost of interest is the pathwise error per unit time over a long time interval. In a previous paper of the authors [2], it was shown, under quite reasonable conditions on the approximating filter and on the signal and noise processes that (as time, bandwidth, process and filter approximation, etc.) go to their limit in any way at all, the limit of the pathwise average costs per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter were used. When suitable approximating filters cannot be readily constructed due to excessive computational requirem...
Robustness of Nonlinear Filters over the Infinite Time Interval
 SIAM J. on Control and Optimization
, 1997
"... Nonlinear filtering is one of the classical areas of stochastic control. From the point of view of practical usefulness, it is important that the filter not be too sensitive to the assumptions made on the initial distribution, the transition function of the underlying signal process and the model fo ..."
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Cited by 18 (7 self)
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Nonlinear filtering is one of the classical areas of stochastic control. From the point of view of practical usefulness, it is important that the filter not be too sensitive to the assumptions made on the initial distribution, the transition function of the underlying signal process and the model for the observation. This is particularly acute if the filter is of interest over a very long or potentially infinite time interval. Then the effects of small errors in the model which is used to construct the filter might accumulate to make the output useless for large time. The problem of asymptotic sensitivity to the initial condition has been treated in several papers. We are concerned with this as well as with the sensitivity to the signal model, uniformly over the infinite time interval. It is conceivable that the effects of even small errors in the model will accumulate so that the filter will eventually be useless. The robustness is shown for three classes of problems. For the first tw...
Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems
 Ann. Appl. Prob
, 1996
"... In the paper we study interacting particle approximations of discrete time and measure valued dynamical systems. Such systems have arisen in such diverse scientific disciplines as physics and signal processing. We give conditions for the socalled particle density profiles to converge to the desired ..."
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Cited by 17 (6 self)
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In the paper we study interacting particle approximations of discrete time and measure valued dynamical systems. Such systems have arisen in such diverse scientific disciplines as physics and signal processing. We give conditions for the socalled particle density profiles to converge to the desired distribution when the number of particles is growing. The strength of our approach is that is applicable to a large class of measure valued dynamical system arising in engineering and particularly in nonlinear filtering problems. Our second objective is to use these results to solve numerically the nonlinear filtering equation. Examples arising in fluid mechanics are also given. 1 Introduction 1.1 Measure valued processes Let (E; fi(E)) be a locally compact and separable metric space, endowed with a Borel oefield, state space. Denote by P(E) be the space of all probability measures on E with the weak topology. The aim of this work is the design of a stochastic particle system approach fo...