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48
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 58 (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
Recursive Monte Carlo filters: Algorithms and theoretical analysis
, 2003
"... powerful tool to perform computations in general state space models. We discuss and compare the accept–reject version with the more common sampling importance resampling version of the algorithm. In particular, we show how auxiliary variable methods and stratification can be used in the accept–rejec ..."
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Cited by 40 (0 self)
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powerful tool to perform computations in general state space models. We discuss and compare the accept–reject version with the more common sampling importance resampling version of the algorithm. In particular, we show how auxiliary variable methods and stratification can be used in the accept–reject version, and we compare different resampling techniques. In a second part, we show laws of large numbers and a central limit theorem for these Monte Carlo filters by simple induction arguments that need only weak conditions. We also show that, under stronger conditions, the required sample size is independent of the length of the observed series. 1. State space and hidden Markov models. A general state space or hidden Markov model consists of an unobserved state sequence (Xt) and an observation sequence (Yt) with the following properties: State evolution: X0,X1,X2,... is a Markov chain with X0 ∼ a0(x)dµ(x) and XtXt−1 = xt−1 ∼ at(xt−1,x)dµ(x). Generation of observations: Conditionally on (Xt), the Yt’s are independent and Yt depends on Xt only with YtXt = xt ∼ bt(xt,y)dν(y). These models occur in a variety of applications. Linear state space models are equivalent to ARMA models (see, e.g., [16]) and have become popular Received January 2003; revised August 2004. AMS 2000 subject classifications. Primary 62M09; secondary 60G35, 60J22, 65C05. Key words and phrases. State space models, hidden Markov models, filtering and smoothing, particle filters, auxiliary variables, sampling importance resampling, central limit theorem. This is an electronic reprint of the original article published by the
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.
On the Optimality of Symbol by Symbol Filtering and Denoising
, 2003
"... We consider the problem of optimally recovering a finitealphabet discretetime stochastic process {X t } from its noisecorrupted observation process {Z t }. In general, the optimal estimate of X t will depend on all the components of {Z t } on which it can be based. We characterize nontrivial s ..."
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Cited by 14 (3 self)
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We consider the problem of optimally recovering a finitealphabet discretetime stochastic process {X t } from its noisecorrupted observation process {Z t }. In general, the optimal estimate of X t will depend on all the components of {Z t } on which it can be based. We characterize nontrivial situations (i.e., beyond the case where (X t , Z t ) are independent) for which optimum performance is attained using "symbol by symbol" operations (a.k.a.
Change Detection for Nonlinear Systems; A Particle
"... In this paper we present a change detection method for nonlinear stochastic systems based on Projection Particle Filtering. The statistic for this method is chosen in such a way that it can be calculated recursively while the computational complexity of the method remains constant with respect to ti ..."
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Cited by 13 (0 self)
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In this paper we present a change detection method for nonlinear stochastic systems based on Projection Particle Filtering. The statistic for this method is chosen in such a way that it can be calculated recursively while the computational complexity of the method remains constant with respect to time. We present some simulation results that show the advantages of this method compared to linearization techniques.