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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 26 (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.
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.
Monte Carlo algorithms and asymptotic problems in nonlinear filtering
 To Appear in Stochastics in Finite/Infinite Dimensions (Volume in honor of Gopinath Kallianpur
, 1999
"... This paper is an extension of [4], which dealt with a wide variety of approximations to optimal nonlinear filters over long time intervals, where ..."
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Cited by 6 (2 self)
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This paper is an extension of [4], which dealt with a wide variety of approximations to optimal nonlinear filters over long time intervals, where
Stability of nonlinear filters in nonmixing case
, 2004
"... 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 an ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by the, so called, mixing conditi ..."
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Cited by 6 (1 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 an ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by the, 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 the uniform positiveness of this density. We show that it might be relaxed regardless of an observation process structure. 1. Introduction and the main result. This paper addresses the stability problem of the nonlinear filtering equation with respect to its initial condition. We consider a homogeneous ergodic Markov chain (Xn)n≥0 with values in S ⊆ Rd regarded as a signal to be filtered from observation of (Yn)n≥1,
Asymptotic Stability, Ergodicity and Other Asymptotic Properties of the Nonlinear Filter
, 2002
"... In this work we study connections between various asymptotic properties of the nonlinear filter. It is assumed that the signal has a unique invariant probability measure. The key property of interest is expressed in terms of a relationship between the observation # field and the tail # field of the ..."
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Cited by 5 (0 self)
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In this work we study connections between various asymptotic properties of the nonlinear filter. It is assumed that the signal has a unique invariant probability measure. The key property of interest is expressed in terms of a relationship between the observation # field and the tail # field of the signal, in the stationary filtering problem. This property can be viewed as the permissibility of the interchange of the order of the operations of maximum and countable intersection for certain # fields. Under suitable conditions, it is shown that the above property is equivalent to various desirable properties of the filter such as (a) uniqueness of invariant measure for the signal, (b) uniqueness of invariant measure for the pair (signal, filter), (c) a finite memory property of the filter , (d) a property of finite time dependence between the signal and observation # fields and (e) asymptotic stability of the filter. Previous works on the asymptotic stability of the filter for a variety of filtering models then identify a rich class of filtering problems for which the above equivalent properties hold.
Ergodic properties of the Nonlinear Filter
 Stochastic Processes and their Applications, 95:1–24
, 2000
"... In a recent work [5] various Markov and ergodicity properties of the nonlinear filter, for the classical model of nonlinear filtering, were studied. It was shown that under quite general conditions, when the signal is a FellerMarkov process with values in a complete separable metric space E then th ..."
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Cited by 2 (1 self)
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In a recent work [5] various Markov and ergodicity properties of the nonlinear filter, for the classical model of nonlinear filtering, were studied. It was shown that under quite general conditions, when the signal is a FellerMarkov process with values in a complete separable metric space E then the pair process (signal, filter) is also a FellerMarkov process with state space E P(E), where P(E) is the space of probability measures on E. Furthermore, it was shown that if the signal has a unique invariant measure then, under appropriate conditions, uniqueness of the invariant measure for the above pair process holds within a certain restricted class of invariant measures. In many asymptotic problems concerning approximate filters [6, 7] it is desirable to have the uniqueness of the invariant measure to hold in the class of all invariant measures. In this paper we first show that for a rich class of filtering problems, when the signal has a unique invariant measure, the property of...