Sigma Point and Particle Approximations of stochastic differential equations in optimal filtering
Simo Särkkä, Helsinki University of Technology/Nalco Company, Finland
The unscented transform (UT) is a relatively recent
method for approximating non-linear transformations of
random variables. Instead of the classical Taylor series
approximations, it is based on forming a set of sigma
points, which are propagated through the non-linearity.
The unscented Kalman filter (UKF) is an alternative to
the extended Kalman filter (EKF), which utilizes the unscented
transform in the filter computations. However,
in its original form, the UKF is a discrete-time algorithm
and it cannot be directly applied to estimation problems,
where the state dynamics are modeled in continuous-time
as stochastic differential equations.
It has been recently shown that by taking the formal
continuous-time limit of the discrete-time UKF prediction
equations, it is possible to derive sigma-point differential
equations, which can be used for approximating the
mean and covariance of a stochastic differential equation
(SDE). By combining these differential equations with
UKF update equation, we obtain the continuous-discrete
unscented Kalman filter, which can be used for approximate
recursive inference on discretely observed stochastic
differential equations.
The solutions of stochastic differential equations can be
also approximated by simulating random trajectories from
the equation and by forming Monte Carlo or particle approximations
from the simulated trajectories. A commonly
used framework for statistical inference in this context
is sequential importance resampling. In continuoustime
setting the evaluation of importance weights is problematic,
because the exact evaluation would require solving
an instance of Kolmogorov forward partial differential
equation, which is an intractable task in general.
One way of coping with this problem is to use the Girsanov
theorem for evaluation of the likelihood ratios of
stochastic differential equations and in turn importance
weights by numerical simulation. The Girsanov theorem
is a theorem from mathematical probability theory, which
can be used for computing likelihood ratios of stochastic
processes. It states that the likelihood ratio of a stochastic
process and Brownian motion, that is, the Radon-
Nikodym derivative of the measure of the stochastic process
with respect to the measure of Brownian motion, can
be represented as an exponential martingale which is the
solution to a certain stochastic differential equation.
In the talk I will review the Taylor series, sigma-point
(unscented) and particle approximations of stochastic differential
equations in optimal (Bayesian) filtering context
and present some applications of the methods in navigation
systems and in monitoring of chemical processes.
