500 research outputs found
On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering
The purpose of this review is to present a comprehensive overview of the
theory of ensemble Kalman-Bucy filtering for linear-Gaussian signal models. We
present a system of equations that describe the flow of individual particles
and the flow of the sample covariance and the sample mean in continuous-time
ensemble filtering. We consider these equations and their characteristics in a
number of popular ensemble Kalman filtering variants. Given these equations, we
study their asymptotic convergence to the optimal Bayesian filter. We also
study in detail some non-asymptotic time-uniform fluctuation, stability, and
contraction results on the sample covariance and sample mean (or sample error
track). We focus on testable signal/observation model conditions, and we
accommodate fully unstable (latent) signal models. We discuss the relevance and
importance of these results in characterising the filter's behaviour, e.g. it's
signal tracking performance, and we contrast these results with those in
classical studies of stability in Kalman-Bucy filtering. We provide intuition
for how these results extend to nonlinear signal models and comment on their
consequence on some typical filter behaviours seen in practice, e.g.
catastrophic divergence
Approximate Kalman-Bucy filter for continuous-time semi-Markov jump linear systems
The aim of this paper is to propose a new numerical approximation of the
Kalman-Bucy filter for semi-Markov jump linear systems. This approximation is
based on the selection of typical trajectories of the driving semi-Markov chain
of the process by using an optimal quantization technique. The main advantage
of this approach is that it makes pre-computations possible. We derive a
Lipschitz property for the solution of the Riccati equation and a general
result on the convergence of perturbed solutions of semi-Markov switching
Riccati equations when the perturbation comes from the driving semi-Markov
chain. Based on these results, we prove the convergence of our approximation
scheme in a general infinite countable state space framework and derive an
error bound in terms of the quantization error and time discretization step. We
employ the proposed filter in a magnetic levitation example with markovian
failures and compare its performance with both the Kalman-Bucy filter and the
Markovian linear minimum mean squares estimator
The geometry of low-rank Kalman filters
An important property of the Kalman filter is that the underlying Riccati
flow is a contraction for the natural metric of the cone of symmetric positive
definite matrices. The present paper studies the geometry of a low-rank version
of the Kalman filter. The underlying Riccati flow evolves on the manifold of
fixed rank symmetric positive semidefinite matrices. Contraction properties of
the low-rank flow are studied by means of a suitable metric recently introduced
by the authors.Comment: Final version published in Matrix Information Geometry, pp53-68,
Springer Verlag, 201
Filtering of SPDEs: The Ensemble Kalman Filter and related methods
This paper is concerned with the derivation and mathematical analysis of
continuous time Ensemble Kalman Filters (EnKBFs) and related data assimilation
methods for Stochastic Partial Differential Equations (SPDEs) with finite
dimensional observations. The signal SPDE is allowed to be nonlinear and is
posed in the standard abstract variational setting. Its coefficients are
assumed to satisfy global one-sided Lipschitz conditions. We first review
classical filtering algorithms in this setting, namely the
Kushner--Stratonovich and the Kalman--Bucy filter, proving a law of total
variance. Then we consider mean-field filtering equations, deriving both a
Feedback Particle Filter and a mean-field EnKBF for nonlinear signal SPDEs. The
second part of the paper is devoted to the elementary mathematical analysis of
the EnKBF in this infinite dimensional setting, showing the well posedness of
both the mean-field EnKBF and its interacting particle approximation. Finally
we prove the convergence of the particle approximation. Under the additional
assumption that the observation function is bounded, we even recover explicit
and (nearly) optimal rates
Convergence of discrete time Kalman filter estimate to continuous time estimate
This article is concerned with the convergence of the state estimate obtained
from the discrete time Kalman filter to the continuous time estimate as the
temporal discretization is refined. We derive convergence rate estimates for
different systems, first finite dimensional and then infinite dimensional with
bounded or unbounded observation operators. Finally, we derive the convergence
rate in the case where the system dynamics is governed by an analytic
semigroup. The proofs are based on applying the discrete time Kalman filter on
a dense numerable subset of a certain time interval .Comment: Author's version of the manuscript accepted for publication in
International Journal of Contro
Long-time stability and accuracy of the ensemble Kalman--Bucy filter for fully observed processes and small measurement noise
The ensemble Kalman filter has become a popular data assimilation technique in the geosciences. However, little is known theoretically about its long term stability and accuracy. In this paper, we investigate the behavior of an ensemble Kalman--Bucy filter applied to continuous-time filtering problems. We derive mean field limiting equations as the ensemble size goes to infinity as well as uniform-in-time accuracy and stability results for finite ensemble sizes. The later results require that the process is fully observed and that the measurement noise is small. We also demonstrate that our ensemble Kalman--Bucy filter is consistent with the classic Kalman--Bucy filter for linear systems and Gaussian processes. We finally verify our theoretical findings for the Lorenz-63 system
Testing quantum mechanics: a statistical approach
As experiments continue to push the quantum-classical boundary using
increasingly complex dynamical systems, the interpretation of experimental data
becomes more and more challenging: when the observations are noisy, indirect,
and limited, how can we be sure that we are observing quantum behavior? This
tutorial highlights some of the difficulties in such experimental tests of
quantum mechanics, using optomechanics as the central example, and discusses
how the issues can be resolved using techniques from statistics and insights
from quantum information theory.Comment: v1: 2 pages; v2: invited tutorial for Quantum Measurements and
Quantum Metrology, substantial expansion of v1, 19 pages; v3: accepted; v4:
corrected some errors, publishe
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