Scaled unscented transform Gaussian sum filter: theory and application

Preprint

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) [23], 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 [5].
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 SUTGSF 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 [26]. We will present the details in implementing the SUT-GSF and examine the effects of filter parameters on the performance of the SUT-GSF. 

Authors: Luo, X; Moroz, IM; Hoteit, I

• Publication status: Published
• Peer review status: Not peer reviewed
• Preprint server: arXiv
• Publication date: 2010-05-15
• DOI: 10.48550/arxiv.1005.2665
• EISSN: 2331-8422
• Language: English
• Keywords: data assimilation; Gaussian sum filter; ensemble Kalman filter; scaled unscented Kalman filter