# Scaled unscented transform Gaussian sum filter: Theory and application

- Handle URI:
- http://hdl.handle.net/10754/561455
- Title:
- Scaled unscented transform Gaussian sum filter: Theory and application
- Authors:
- Abstract:
- 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. © 2010 Elsevier B.V. All rights reserved.
- KAUST Department:
- Publisher:
- Journal:
- Issue Date:
- May-2010
- DOI:
- 10.1016/j.physd.2010.01.022
- Type:
- Article
- ISSN:
- 01672789
- Additional Links:
- http://arxiv.org/abs/arXiv:1005.2665v1

- Appears in Collections:
- Articles; Environmental Science and Engineering Program; Physical Sciences and Engineering (PSE) Division

# Full metadata record

DC Field | Value | Language |
---|---|---|

dc.contributor.author | Luo, Xiaodong | en |

dc.contributor.author | Moroz, Irene M. | en |

dc.contributor.author | Hoteit, Ibrahim | en |

dc.date.accessioned | 2015-08-02T09:11:48Z | en |

dc.date.available | 2015-08-02T09:11:48Z | en |

dc.date.issued | 2010-05 | en |

dc.identifier.issn | 01672789 | en |

dc.identifier.doi | 10.1016/j.physd.2010.01.022 | en |

dc.identifier.uri | http://hdl.handle.net/10754/561455 | en |

dc.description.abstract | 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. © 2010 Elsevier B.V. All rights reserved. | en |

dc.publisher | Elsevier | en |

dc.relation.url | http://arxiv.org/abs/arXiv:1005.2665v1 | en |

dc.subject | Data assimilation | en |

dc.subject | Ensemble Kalman filter | en |

dc.subject | Gaussian sum filter | en |

dc.subject | Scaled unscented Kalman filter | en |

dc.title | Scaled unscented transform Gaussian sum filter: Theory and application | en |

dc.type | Article | en |

dc.contributor.department | Physical Sciences and Engineering (PSE) Division | en |

dc.contributor.department | Environmental Science and Engineering Program | en |

dc.contributor.department | Earth Fluid Modeling and Prediction Group | en |

dc.identifier.journal | Physica D: Nonlinear Phenomena | en |

dc.contributor.institution | Math Inst, Oxford OX1 3LB, England | en |

dc.contributor.institution | Univ Oxford, Oxford Man Inst, Oxford OX2 6ED, England | en |

dc.identifier.arxivid | arXiv:1005.2665 | en |

kaust.author | Luo, Xiaodong | en |

kaust.author | Hoteit, Ibrahim | en |

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