4,528 research outputs found
A Gaussian mixture ensemble transform filter
We generalize the popular ensemble Kalman filter to an ensemble transform
filter where the prior distribution can take the form of a Gaussian mixture or
a Gaussian kernel density estimator. The design of the filter is based on a
continuous formulation of the Bayesian filter analysis step. We call the new
filter algorithm the ensemble Gaussian mixture filter (EGMF). The EGMF is
implemented for three simple test problems (Brownian dynamics in one dimension,
Langevin dynamics in two dimensions, and the three dimensional Lorenz-63
model). It is demonstrated that the EGMF is capable to track systems with
non-Gaussian uni- and multimodal ensemble distributions
Particle Kalman Filtering: A Nonlinear Bayesian Framework for Ensemble Kalman Filters
This paper investigates an approximation scheme of the optimal nonlinear
Bayesian filter based on the Gaussian mixture representation of the state
probability distribution function. The resulting filter is similar to the
particle filter, but is different from it in that, the standard weight-type
correction in the particle filter is complemented by the Kalman-type correction
with the associated covariance matrices in the Gaussian mixture. We show that
this filter is an algorithm in between the Kalman filter and the particle
filter, and therefore is referred to as the particle Kalman filter (PKF). In
the PKF, the solution of a nonlinear filtering problem is expressed as the
weighted average of an "ensemble of Kalman filters" operating in parallel.
Running an ensemble of Kalman filters is, however, computationally prohibitive
for realistic atmospheric and oceanic data assimilation problems. For this
reason, we consider the construction of the PKF through an "ensemble" of
ensemble Kalman filters (EnKFs) instead, and call the implementation the
particle EnKF (PEnKF). We show that different types of the EnKFs can be
considered as special cases of the PEnKF. Similar to the situation in the
particle filter, we also introduce a re-sampling step to the PEnKF in order to
reduce the risk of weights collapse and improve the performance of the filter.
Numerical experiments with the strongly nonlinear Lorenz-96 model are presented
and discussed.Comment: Accepted manuscript, to appear in Monthly Weather Revie
Bridging the ensemble Kalman and particle filter
In many applications of Monte Carlo nonlinear filtering, the propagation step
is computationally expensive, and hence, the sample size is limited. With small
sample sizes, the update step becomes crucial. Particle filtering suffers from
the well-known problem of sample degeneracy. Ensemble Kalman filtering avoids
this, at the expense of treating non-Gaussian features of the forecast
distribution incorrectly. Here we introduce a procedure which makes a
continuous transition indexed by gamma in [0,1] between the ensemble and the
particle filter update. We propose automatic choices of the parameter gamma
such that the update stays as close as possible to the particle filter update
subject to avoiding degeneracy. In various examples, we show that this
procedure leads to updates which are able to handle non-Gaussian features of
the prediction sample even in high-dimensional situations
Ensemble Kalman methods for high-dimensional hierarchical dynamic space-time models
We propose a new class of filtering and smoothing methods for inference in
high-dimensional, nonlinear, non-Gaussian, spatio-temporal state-space models.
The main idea is to combine the ensemble Kalman filter and smoother, developed
in the geophysics literature, with state-space algorithms from the statistics
literature. Our algorithms address a variety of estimation scenarios, including
on-line and off-line state and parameter estimation. We take a Bayesian
perspective, for which the goal is to generate samples from the joint posterior
distribution of states and parameters. The key benefit of our approach is the
use of ensemble Kalman methods for dimension reduction, which allows inference
for high-dimensional state vectors. We compare our methods to existing ones,
including ensemble Kalman filters, particle filters, and particle MCMC. Using a
real data example of cloud motion and data simulated under a number of
nonlinear and non-Gaussian scenarios, we show that our approaches outperform
these existing methods
Scaled unscented transform Gaussian sum filter: theory and application
In this work we consider the state estimation problem in
nonlinear/non-Gaussian systems. We introduce a framework, called the scaled
unscented transform Gaussian sum filter (SUT-GSF), which combines two ideas:
the scaled unscented Kalman filter (SUKF) based on the concept of scaled
unscented transform (SUT), and the Gaussian mixture model (GMM). The SUT is
used to approximate the mean and covariance of a Gaussian random variable which
is transformed by a nonlinear function, while the GMM is adopted to approximate
the probability density function (pdf) of a random variable through a set of
Gaussian distributions. With these two tools, a framework can be set up to
assimilate nonlinear systems in a recursive way. Within this framework, one can
treat a nonlinear stochastic system as a mixture model of a set of sub-systems,
each of which takes the form of a nonlinear system driven by a known Gaussian
random process. Then, for each sub-system, one applies the SUKF to estimate the
mean and covariance of the underlying Gaussian random variable transformed by
the nonlinear governing equations of the sub-system. Incorporating the
estimations of the sub-systems into the GMM gives an explicit (approximate)
form of the pdf, which can be regarded as a "complete" solution to the state
estimation problem, as all of the statistical information of interest can be
obtained from the explicit form of the pdf ...
This work is on the construction of the Gaussian sum filter based on the
scaled unscented transform
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