1,509 research outputs found
Controlling overestimation of error covariance in ensemble Kalman filters with sparse observations: A variance limiting Kalman filter
We consider the problem of an ensemble Kalman filter when only partial
observations are available. In particular we consider the situation where the
observational space consists of variables which are directly observable with
known observational error, and of variables of which only their climatic
variance and mean are given. To limit the variance of the latter poorly
resolved variables we derive a variance limiting Kalman filter (VLKF) in a
variational setting. We analyze the variance limiting Kalman filter for a
simple linear toy model and determine its range of optimal performance. We
explore the variance limiting Kalman filter in an ensemble transform setting
for the Lorenz-96 system, and show that incorporating the information of the
variance of some un-observable variables can improve the skill and also
increase the stability of the data assimilation procedure.Comment: 32 pages, 11 figure
Linear theory for filtering nonlinear multiscale systems with model error
We study filtering of multiscale dynamical systems with model error arising
from unresolved smaller scale processes. The analysis assumes continuous-time
noisy observations of all components of the slow variables alone. For a linear
model with Gaussian noise, we prove existence of a unique choice of parameters
in a linear reduced model for the slow variables. The linear theory extends to
to a non-Gaussian, nonlinear test problem, where we assume we know the optimal
stochastic parameterization and the correct observation model. We show that
when the parameterization is inappropriate, parameters chosen for good filter
performance may give poor equilibrium statistical estimates and vice versa.
Given the correct parameterization, it is imperative to estimate the parameters
simultaneously and to account for the nonlinear feedback of the stochastic
parameters into the reduced filter estimates. In numerical experiments on the
two-layer Lorenz-96 model, we find that parameters estimated online, as part of
a filtering procedure, produce accurate filtering and equilibrium statistical
prediction. In contrast, a linear regression based offline method, which fits
the parameters to a given training data set independently from the filter,
yields filter estimates which are worse than the observations or even divergent
when the slow variables are not fully observed
Kalman-Takens filtering in the presence of dynamical noise
The use of data assimilation for the merging of observed data with dynamical
models is becoming standard in modern physics. If a parametric model is known,
methods such as Kalman filtering have been developed for this purpose. If no
model is known, a hybrid Kalman-Takens method has been recently introduced, in
order to exploit the advantages of optimal filtering in a nonparametric
setting. This procedure replaces the parametric model with dynamics
reconstructed from delay coordinates, while using the Kalman update formulation
to assimilate new observations. We find that this hybrid approach results in
comparable efficiency to parametric methods in identifying underlying dynamics,
even in the presence of dynamical noise. By combining the Kalman-Takens method
with an adaptive filtering procedure we are able to estimate the statistics of
the observational and dynamical noise. This solves a long standing problem of
separating dynamical and observational noise in time series data, which is
especially challenging when no dynamical model is specified
A Nonparametric Adaptive Nonlinear Statistical Filter
We use statistical learning methods to construct an adaptive state estimator
for nonlinear stochastic systems. Optimal state estimation, in the form of a
Kalman filter, requires knowledge of the system's process and measurement
uncertainty. We propose that these uncertainties can be estimated from
(conditioned on) past observed data, and without making any assumptions of the
system's prior distribution. The system's prior distribution at each time step
is constructed from an ensemble of least-squares estimates on sub-sampled sets
of the data via jackknife sampling. As new data is acquired, the state
estimates, process uncertainty, and measurement uncertainty are updated
accordingly, as described in this manuscript.Comment: Accepted at the 2014 IEEE Conference on Decision and Contro
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