Motivation
Data
assimilation,
or filtering, refers to the problem of combining noisy observations of a
(typically physical) system together with a model for that system
in
order to infer the state and/or parameters online as data is received. In the Bayesian probabilistic context of a
hidden-Markov model, this leads
to a
recursion of Bayesian updates. The
objective of the filtering problem is then to obtain the posterior distribution
of the unknown as a function of
the
history of observations. One objective
of this research is (i) to
obtain better approximations of this distribution. This is often not possible in practice
even
when it is the objective. One may aim to
obtain an estimator (and some coarse measure of spread) which tracks the truth
rather than aiming
for
the full distribution. Another objective
of this research is (ii) to analyze existing commonly used filtering algorithms
from this perspective to determine
their
accuracy and stability properties. By
the triangle inequality such estimators can be related to the mean of the
filtering distribution (i) in case
this
is stable. This is important both for theoreticians in terms of making sense of
the results of these filters in a rigorous way, and also for the practitioner who may use insight from
these studies to develop better filters.
Methodology
Currently
we
are developing algorithms for (i) which rely on deterministic
parameterizations of the distribution and hence surmount the undesirable convergence
rates
associated with Monte-Carlo sampling [1].
In order to tackle (ii), we identify properties of the underlying system
and leverage these
to
prove rigorous convergence results for filter estimators. In particular, the methodology relies on
observing the unpredictable part of the system, and trusting the observations
sufficiently much [2-5].
Outcomes
We
have
tested and implemented the first of our filters with very promising results
[1]. We have rigorous convergence
results for the most fundamental filter 3DVAR [2,3,4] and are working on
similar results for the more elaborate filters ExKF and EnKF [5].
