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39
Stability and Uniform Approximation of Nonlinear Filters Using the Hilbert Metric, and Application to Particle Filters
, 2002
"... this article, we use the approach based on the Hilbert metric to study the asymptotic behavior of the optimal filter, and to prove as in [9] the uniform convergence of several particle filters, such as the interacting particle filter (IPF) and some other original particle filters. A common assumptio ..."
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Cited by 57 (5 self)
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this article, we use the approach based on the Hilbert metric to study the asymptotic behavior of the optimal filter, and to prove as in [9] the uniform convergence of several particle filters, such as the interacting particle filter (IPF) and some other original particle filters. A common assumption to prove stability results, see e.g. in [9, Theorem 2.4], is that the Markov transition kernels are mixing, which implies that the hidden state sequence is ergodic. Our results are obtained under the assumption that the nonnegative kernels describing the evolution of the unnormalized optimal filter, and incorporating simultaneously the Markov transition kernels and the likelihood functions, are mixing. This is a weaker assumption, see Proposition 3.9, which allows to consider some cases, similar to the case studied in [6], where the hidden state sequence is not ergodic, see Example 3.10. This point of view is further developped by Le Gland and Oudjane in [22] and by Oudjane and Rubenthaler in [28]. Our main contribution is to study also the stability of the optimal filter w.r.t. the model, when the local error is propagated by mixing kernels, and can be estimated in the Hilbert metric, in the total variation norm, or in a weaker distance suitable for random probability distributions. AMS 1991 subject classifications. Primary 93E11, 93E15, 62E25; secondary 60B10, 60J27, 62G07, 62G09, 62L10
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 ...
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.
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
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...
Change Detection in Partially Observed Nonlinear Dynamic Systems with Unknown Change Parameters
 in American Control Conference (ACC
, 2004
"... We study the change detection problem in partially observed nonlinear dynamic systems. We assume that the change parameters are unknown and the change could be gradual (slow) or sudden (drastic). For most nonlinear systems, no finite dimensional filters exist and approximation filtering methods like ..."
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Cited by 16 (14 self)
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We study the change detection problem in partially observed nonlinear dynamic systems. We assume that the change parameters are unknown and the change could be gradual (slow) or sudden (drastic). For most nonlinear systems, no finite dimensional filters exist and approximation filtering methods like the Particle Filter are used. Even when change parameters are unknown, drastic changes can be detected easily using the increase in tracking (output) error or the negative log of observation likelihood (OL). But slow changes usually get missed. We propose in this paper, a statistic for slow change detection which turns out to be the same as the Kerridge Inaccuracy between the posterior state distribution and the normal system prior. We show asymptotic convergence (under certain assumptions) of the bounding, modeling and particle filtering errors in its approximation using a particle filter optimal for the normal system. We also demonstrate using the bounds on the errors that our statistic works in situations where observation likelihood (OL) fails and vice versa.
R.Liptser, Stability of nonlinear filters in nonmixing
"... The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by, so called, mixing condition. The ..."
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Cited by 14 (6 self)
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The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by, so called, mixing condition. The latter is formulated in terms of the transition probability density of the signal. The most restrictive requirement of the mixing condition is a uniform positiveness of this density. We show that this requirement might be weakened regardless of an observation process structure.
Exponential Stability in Discrete Time Filtering for NonErgodic Signals
 System and Control Letters
, 1999
"... In this paper we prove exponential asymptotic stability for discrete time filters for signals arising as solutions of ddimensional stochastic difference equations. The observation process is the signal corrupted by an additive white noise of su#ciently small variance. The model for the signal admit ..."
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Cited by 13 (5 self)
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In this paper we prove exponential asymptotic stability for discrete time filters for signals arising as solutions of ddimensional stochastic difference equations. The observation process is the signal corrupted by an additive white noise of su#ciently small variance. The model for the signal admits nonergodic processes. We show that almost surely, the total variation distance between the optimal filter and an incorrectly initialized filter converges to 0 exponentially fast as time approaches #. Key Words: nonlinear filtering, asymptotic stability, measure valued processes. # Research suppored by the NSF grant DMI 9812857. 1 1 Introduction The central problem of nonlinear filtering is to study the conditional distribution of a signal process at any time instant given noisy observations on the signal available up until that time. If the signalobservation pair is Markov, the conditional distribution process, referred to hereafter as the optimal filter , is determined completely ...
Markov Property and Ergodicity of the Nonlinear Filter
, 1999
"... In this paper we first prove, under quite general conditions, that the nonlinear filter and the pair: (signal,filter) are FellerMarkov processes. The state space of the signal is allowed to be non locally compact and the observation function: h can be unbounded. Our proofs in contrast to those of K ..."
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Cited by 11 (4 self)
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In this paper we first prove, under quite general conditions, that the nonlinear filter and the pair: (signal,filter) are FellerMarkov processes. The state space of the signal is allowed to be non locally compact and the observation function: h can be unbounded. Our proofs in contrast to those of Kunita(1971,1991), Stettner(1989) do not depend upon the uniqueness of the solutions to the filtering equations. We then obtain conditions for existence and uniqueness of invariant measures for the nonlinear filter and the pair process. These results extend those of Kunita and Stettner, which hold for locally compact state space and bounded h, to our general framework. Finally we show that the recent results of OconePardoux [11] on asymptotic stability of the nonlinear filter, which use the KunitaStettner setup, hold for the general situation considered in this paper. Key Words: nonlinear filtering, invariant measures, asymptotic stability, measure valued processes. AMS Classification:60 ...