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32
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 ...
An introduction to quantum filtering
, 2006
"... Abstract. This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation ..."
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Cited by 24 (13 self)
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Abstract. This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation
Weighted Stochastic Sobolev Spaces and Bilinear SPDE's Driven by Spacetime White Noise
"... : In this paper we develop basic elements of Malliavin calculus on a weighted L 2(\Omega\Gamma2 This class of generalized Wiener functionals is a Hilbert space. It turns out to be substantially smaller than the space of Hida distributions while large enough to accommodate solutions of bilinear sto ..."
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Cited by 19 (6 self)
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: In this paper we develop basic elements of Malliavin calculus on a weighted L 2(\Omega\Gamma2 This class of generalized Wiener functionals is a Hilbert space. It turns out to be substantially smaller than the space of Hida distributions while large enough to accommodate solutions of bilinear stochastic PDE's. As an example, we consider a stochastic advectiondiffusion equation driven by spacetime white noise in IR d . It is known that for d ? 1, this equation has no solutions in L 2(\Omega\Gamma4 In contrast, it is shown in the paper that in an appropriately weighted L 2(\Omega\Gamma there is a unique solution to the stochastic advectiondiffusion equation for any d 1. In addition we present explicit formulas for the HermiteFourier coefficients in the Wiener chaos expansion of the solution. 2 1 Introduction Consider a stochastic partial differential equation of the form: ( @u @t = Lu + u \Delta W u(0; x) = u 0 (x); (1:1) where W is a white noise on [0; T ] \Theta IR...
On Unscented Kalman Filtering for State Estimation of ContinuousTime Nonlinear Systems
, 2007
"... This article considers the application of the unscented Kalman filter (UKF) to continuoustime filtering problems, where both the state and measurement processes are modeled as stochastic differential equations. The mean and covariance differential equations which result in the continuoustime lim ..."
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Cited by 17 (7 self)
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This article considers the application of the unscented Kalman filter (UKF) to continuoustime filtering problems, where both the state and measurement processes are modeled as stochastic differential equations. The mean and covariance differential equations which result in the continuoustime limit of the UKF are derived. The continuousdiscrete unscented Kalman filter is derived as a special case of the continuoustime filter, when the continuoustime prediction equations are combined with the update step of the discretetime unscented Kalman filter. The filter equations are also transformed into sigmapoint differential equations, which can be interpreted as matrix square root versions of the filter equations.
3D Object Tracking Using ShapeEncoded Particle Propagation
 In: International Conference on Computer Vision
, 2001
"... We present a comprehensive treatment of 3D object tracking by posing it as a nonlinear state estimation problem. The measurements are derived using the outputs of shapeencoded filters. The nonlinear state estimation is performed by solving the Zakai equation, and we use the branching particle propa ..."
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Cited by 13 (2 self)
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We present a comprehensive treatment of 3D object tracking by posing it as a nonlinear state estimation problem. The measurements are derived using the outputs of shapeencoded filters. The nonlinear state estimation is performed by solving the Zakai equation, and we use the branching particle propagation method for computing the solution. The unnormalized conditional density for the solution to the Zakai equation is realized by the weight of the particle. We first sample a set of particles approximating the initial distribution of the state vector conditioned on the observations, where each particle encodes the set of geometric parameters of the object. The weight of the particle represents geometric and temporal fit, which is computed bottomup from the raw image using a shapeencoded filter. The particles branch so that the mean number of offspring is proportional to the weight. Time update is handled by employing a secondorder motion model, combined with local stochastic search to minimize the prediction error. The prediction adjustment suggested by system identification theory is empirically verified to contribute to global stability. The amount of diffusion is effectively adjusted using a Kalman updating of the covariance matrix. We have successfully applied this method to human head tracking, where we estimate head motion and compute structure using simple head and facial feature models. 1
A Discrete Invitation to Quantum Filtering and Feedback Control
, 2009
"... The engineering and control of devices at the quantum mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a nov ..."
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Cited by 10 (2 self)
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The engineering and control of devices at the quantum mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory while remaining completely within the setting of finitedimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.
Finitedimensional filters with nonlinear drift XIV: Classification of finitedimensional estimation algebras of maximal rank with artitrary state space dimension and mitter conjecture, submitted for publication
"... Abstract. The idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed by Brockett and Mitter independently. For this approach, one needs to know explicitly the structure of these estimation algebras in order to construct finite dimensional nonlinear fil ..."
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Cited by 7 (1 self)
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Abstract. The idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed by Brockett and Mitter independently. For this approach, one needs to know explicitly the structure of these estimation algebras in order to construct finite dimensional nonlinear filters. Therefore Brockett proposed to classify all finite dimensional estimation algebras. Chiou and Yau [ChYa1] classify all finite dimensional estimation algebras of maximal rank with dimension of the state space less than or equal to two. The purpose of this paper is to give a new result on classification of all finite dimensional estimation algebras of maximal rank with state space dimension less than or equal to five. 1. Introduction. The
Implicit particle filters for data assimilation
, 2010
"... Implicit particle filters for data assimilation update the particles by first choosing probabilities and then looking for particle locations that assume them, guiding the particles one by one to the high probability domain. We provide a detailed description of these filters, with illustrative exampl ..."
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Cited by 7 (7 self)
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Implicit particle filters for data assimilation update the particles by first choosing probabilities and then looking for particle locations that assume them, guiding the particles one by one to the high probability domain. We provide a detailed description of these filters, with illustrative examples, together with new, more general, methods for solving the algebraic equations and with a new algorithm for parameter identification. 1
Estimation and control for linear, partially observable systems with nonGaussian initial distribution, Stochastic processes and their
 Applications
, 1983
"... The nonlinear filtering problem of estimating the state of a linear stochastic system from noisy observations is solved for a broad class of probability distributions of the initial state. It is shown that the conditional density of the present state, given the past observations, is a mixture of Gau ..."
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Cited by 4 (0 self)
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The nonlinear filtering problem of estimating the state of a linear stochastic system from noisy observations is solved for a broad class of probability distributions of the initial state. It is shown that the conditional density of the present state, given the past observations, is a mixture of Gaussian distributions, and is parametrically determined by two sets of sufficient statistics which satisfy stochastic DES; this result leads to a generalization of the KalmanBucy filter to a structure with a conditional mean vector, and additional sufficient statistics that obey nonlinear equations, and determine a generalized (random) Kalman gain. The theory is used to solve explicitly a control problem with quadratic running and terminal costs, and bounded controls. 1.
Robustness and Convergence of Approximations to Nonlinear Filters for JumpDiffusions
 Computational and Applied Math
, 1996
"... The paper treats numerical approximations to the nonlinear filtering problem for jumpdiffusion processes. This is a key problem in stochastic systems analysis. The processes are defined, and the optimal filters described. In the general nonlinear case, the optimal filters cannot be computed, and s ..."
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Cited by 4 (2 self)
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The paper treats numerical approximations to the nonlinear filtering problem for jumpdiffusion processes. This is a key problem in stochastic systems analysis. The processes are defined, and the optimal filters described. In the general nonlinear case, the optimal filters cannot be computed, and some numerical approximation is needed. Then the weak conditions that are required for the convergence of the approximations are given and the convergence is proved. Examples of useful approximations which satisfy the conditions are given. Quite weak conditions are given under which the approximating filter is continuous in the observation function, and it is shown that our canonical methods satisfy the conditions. Such continuity is essential if the approximations are to be used with confidence on actual physical data. Finally, we prove the convergence of monte carlo methods for approximating the optimal filters, and also show that the optimal filter is continuous in the parameters of the si...