## Estimation Algorithms for Non-Gaussian State-Space Models with Application to Positioning

Research output: Book/Report › Doctoral thesis › Collection of Articles

### Details

Original language | English |
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Publisher | Tampere University of Technology |

Number of pages | 71 |

ISBN (Electronic) | 978-952-15-4029-5 |

ISBN (Print) | 978-952-15-4010-3 |

Publication status | Published - 24 Nov 2017 |

Publication type | G5 Doctoral dissertation (article) |

### Publication series

Name | Tampere University of Technology. Publication |
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Volume | 1499 |

ISSN (Print) | 1459-2045 |

### Abstract

State-space models (SSMs) are used to model systems with hidden time-varying state and observable measurement output. In statistical SSMs, the state dynamics is assumed known up to a random term referred to as the process noise, and the measurements contain random measurement noise. Kalman ﬁlter (KF) and Rauch– Tung–Striebel smoother (RTSS) are widely-applied closed-form algorithms that provide the parameters of the exact Bayesian ﬁltering and smoothing distributions for discrete-time linear statistical SSMs where the process and measurement noises follow Gaussian distributions. However, when the SSM involves nonlinear functions and/or non-Gaussian noises, the Bayesian ﬁltering and smoothing distributions cannot in general be solved using closed-form algorithms. This thesis addresses approximate Bayesian time-series inference for two positioning-related problems where the assumption of Gaussian noises cannot capture all useful knowledge of the considered system’s statistical properties: map-assisted indoor positioning and positioning using time-delay measurements.

The motion constraints imposed by the indoor map are typically incorporated in the position estimate using the particle ﬁlter (PF) algorithm. The PF is a Monte Carlo algorithm especially suited for statistical SSMs where the Bayesian posterior distributions are too complicated to be adequately approximated using a well-known distribution family with a low-dimensional parameter space. In mapassisted indoor positioning, the trajectories that cross walls or ﬂoor levels get a low probability in the model. In this thesis, improvements to three different PF algorithms for map-assisted indoor positioning are proposed and compared. In the wall-collision PF, weighted random samples, also known as particles, are moved based on inertial sensor measurements, and the particles that collide with the walls are downweighted. When the inertial sensor measurements are very noisy, map information is used to guide the particles such that fewer particles collide with the walls, which implies that more particles contribute to the estimation. When no inertial sensor information is used, the particles are moved along the links of a graph that is dense enough to approximate the set of expected user paths.

Time-delay based ranging measurements of e.g. ultra-wideband (UWB) and Global Navigation Satellite Systems (GNSSs) contain occasional positive measurement errors that are large relative to the majority of the errors due to multipath effects and denied line of sight. In this thesis, computationally efﬁcient approximate Bayesian ﬁlters and smoothers are proposed for statistical SSMs where the measurement noise follows a skew t -distribution, and the algorithms are applied to positioning using time-delay based ranging measurements. The skew t -distribution is an extension of the Gaussian distribution, which has two additional parameters that affect the heavytailedness and skewness of the distribution. When the measurement noise model is heavy-tailed, the optimal Bayesian algorithm is robust to occasional large measurement errors, and when the model is positively (or negatively) skewed, the algorithms account for the fact that most large errors are known to be positive (or negative). Therefore, the skew t -distribution is more ﬂexible than the Gaussian distribution and captures more statistical features of the error distributions of UWB and GNSS measurements. Furthermore, the skew t -distribution admits a conditionally Gaussian hierarchical form that enables approximating the ﬁltering and smoothing posteriors with Gaussian distributions using variational Bayes (VB) algorithms. The proposed algorithms can thus be computationally efﬁcient compared to Monte Carlo algorithms especially when the state is high-dimensional. It is shown in this thesis that the skew-t ﬁlter improves the accuracy of UWB based indoor positioning and GNSS based outdoor positioning in urban areas compared to the extended KF. The skew-t ﬁlter’s computational burden is higher than that of the extended KF but of the same magnitude.

The motion constraints imposed by the indoor map are typically incorporated in the position estimate using the particle ﬁlter (PF) algorithm. The PF is a Monte Carlo algorithm especially suited for statistical SSMs where the Bayesian posterior distributions are too complicated to be adequately approximated using a well-known distribution family with a low-dimensional parameter space. In mapassisted indoor positioning, the trajectories that cross walls or ﬂoor levels get a low probability in the model. In this thesis, improvements to three different PF algorithms for map-assisted indoor positioning are proposed and compared. In the wall-collision PF, weighted random samples, also known as particles, are moved based on inertial sensor measurements, and the particles that collide with the walls are downweighted. When the inertial sensor measurements are very noisy, map information is used to guide the particles such that fewer particles collide with the walls, which implies that more particles contribute to the estimation. When no inertial sensor information is used, the particles are moved along the links of a graph that is dense enough to approximate the set of expected user paths.

Time-delay based ranging measurements of e.g. ultra-wideband (UWB) and Global Navigation Satellite Systems (GNSSs) contain occasional positive measurement errors that are large relative to the majority of the errors due to multipath effects and denied line of sight. In this thesis, computationally efﬁcient approximate Bayesian ﬁlters and smoothers are proposed for statistical SSMs where the measurement noise follows a skew t -distribution, and the algorithms are applied to positioning using time-delay based ranging measurements. The skew t -distribution is an extension of the Gaussian distribution, which has two additional parameters that affect the heavytailedness and skewness of the distribution. When the measurement noise model is heavy-tailed, the optimal Bayesian algorithm is robust to occasional large measurement errors, and when the model is positively (or negatively) skewed, the algorithms account for the fact that most large errors are known to be positive (or negative). Therefore, the skew t -distribution is more ﬂexible than the Gaussian distribution and captures more statistical features of the error distributions of UWB and GNSS measurements. Furthermore, the skew t -distribution admits a conditionally Gaussian hierarchical form that enables approximating the ﬁltering and smoothing posteriors with Gaussian distributions using variational Bayes (VB) algorithms. The proposed algorithms can thus be computationally efﬁcient compared to Monte Carlo algorithms especially when the state is high-dimensional. It is shown in this thesis that the skew-t ﬁlter improves the accuracy of UWB based indoor positioning and GNSS based outdoor positioning in urban areas compared to the extended KF. The skew-t ﬁlter’s computational burden is higher than that of the extended KF but of the same magnitude.

### Keywords

- algorithm, non-Gaussian distributions, state-space models, nonlinear filtering, positioning

### Field of science, Statistics Finland

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