Large sample asymptotics for the ensemble Kalman filter.
François Le Gland (INRIA / IRISA Rennes)
(joint work with Valérie Monbet (UBS) and Vu-Duc Tran (UBS))
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The ensemble Kalman filter (EnKF) has been proposed in sequential
data assimilation, where state vectors of huge dimension (e.g.
resulting from the discretization of pressure and velocity fields
over a continent, as considered in meteorology) should be estimated
from noisy measurements (e.g. collected at sparse in-situ stations).
Even if the state and measurement equations are linear with additive
Gaussian white noise, computing and storing the error covariance
matrices involved in the Kalman filter is practically impossible,
and it has been proposed to represent the filtering distribution
with a sample (ensemble) of a few elements and to think of the
corresponding empirical covariance matrix as an approximation of
the intractable error covariance matrix. Extensions to nonlinear
state equations have also been proposed.
Surprisingly, very little is known about the asymptotic behaviour
of the EnKF, whereas on the other hand, the asymptotic behaviour of
many different classes of particle filters is well understood, as
the number of particles goes to infinity. Interpreting the ensemble
elements as particles with mean-field interactions (and not only as
an instrumental device producing the ensemble mean value as a point
estimator of the hidden state), we prove the convergence of the EnKF,
with the classical rate 1/sqrt(N), as the number N of ensemble
elements increases to infinity. In the linear case, the limit of
the empirical distribution of the ensemble elements is the usual
(Gaussian distribution associated with the) Kalman filter, but in
the more general case of a nonlinear state equation with linear
observations, this limit differs from the usual Bayesian filter,
and is characterized here. To get the correct limit in this case,
the mechanism that generates the elements in the EnKF should be
interpreted as a proposal importance distribution, and appropriate
importance weights should be assigned to the ensemble elements.
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