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) (Julier and Uhlmann (2004) [16]), 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 (Arulampalam et al. (2002) [7]). In
applications, a potential problem of a Gaussian sum filter is that the number of Gaussian
distributions may increase very rapidly. To this end, we also propose
an auxiliary algorithm to conduct pdf re-approximation so that the
number of Gaussian
distributions can be reduced. With the auxiliary algorithm, in principle
the SUT-GSF can achieve almost the same computational speed as the SUKF
if the SUT-GSF is implemented in parallel. As an example, we will use
the SUT-GSF to assimilate a 40-dimensional system due to Lorenz and
Emanuel (1998) [27]. We will present the details of implementing the
SUT-GSF and examine the effects of filter parameters on the performance of the SUT-GSF.

DOI: 10.1016/j.physd.2010.01.022