A novel progressive Gaussian approximate filter with variable step size based on a variational Bayesian approach

The selection of step sizes in the progressive Gaussian ap- proximate filter (PGAF) is important, and it is difficult to se- lect optimal values in practical applications. Furthermore, in the PGAF, significant integral approximation errors are gener- ated by the repeated approximate calculations of the Gaussian weighted integrals, which results in an inaccurate measure- ment noise covariance matrix (MNCM). To solve these prob- lems, in this paper, the step sizes and the MNCM are jointly estimated based on the variational Bayesian (VB) approach. By incorporating the adaptive estimates of step sizes and the MNCM into the PGAF framework, a novel PGAF with vari- able step size is proposed. Simulation results illustrate that the proposed filter has higher estimation accuracy than exist- ing state-of-the-art nonlinear Gaussian approximate filters

Joint State Estimation and Noise Identification Based on Variational Optimization

In this article, the state estimation problems with unknown process noise and measurement noise covariances for both linear and nonlinear systems are considered. By formulating the joint estimation of system state and noise parameters into an optimization problem, a novel adaptive Kalman filter method based on conjugate-computation variational inference, referred to as CVIAKF, is proposed to approximate the joint posterior probability density function of the latent variables. Unlike the existing adaptive Kalman filter methods utilizing variational inference in natural-parameter space, CVIAKF performs optimization in expectation-parameter space, resulting in a faster and simpler solution. Meanwhile, CVIAKF divides optimization objectives into conjugate and non-conjugate parts of nonlinear dynamical models, whereas conjugate computations and stochastic mirror-descent are applied, respectively. Remarkably, the reparameterization trick is used to reduce the variance of stochastic gradients of the non-conjugate parts. The effectiveness of CVIAKF is validated through synthetic and real-world datasets of maneuvering target tracking.Comment: 13 page

Robust Rauch-Tung-Striebel smoothing framework for heavy-tailed and/or skew noises

A novel robust Rauch-Tung-Striebel smoothing framework is proposed based on a generalized Gaussian scale mixture (GGScM) distribution for a linear state-space model with heavy-tailed and/or skew noises. The state trajectory, mixing parameters and unknown distribution parameters are jointly inferred using the variational Bayesian approach. As such, a major contribution of this work is unifying results within the GGScM distribution framework. Simulation and experimental results demonstrate that the proposed smoother has better accuracy than existing smoothers

Approximate Bayesian inference methods for stochastic state space models

This thesis collects together research results obtained during my doctoral studies related to approximate Bayesian inference in stochastic state-space models. The published research spans a variety of topics including 1) application of Gaussian filtering in satellite orbit prediction, 2) outlier robust linear regression using variational Bayes (VB) approximation, 3) filtering and smoothing in continuous-discrete Gaussian models using VB approximation and 4) parameter estimation using twisted particle filters. The main goal of the introductory part of the thesis is to connect the results to the general framework of estimation of state and model parameters and present them in a unified manner.Bayesian inference for non-linear state space models generally requires use of approximations, since the exact posterior distribution is readily available only for a few special cases. The approximation methods can be roughly classified into to groups: deterministic methods, where the intractable posterior distribution is approximated from a family of more tractable distributions (e.g. Gaussian and VB approximations), and stochastic sampling based methods (e.g. particle filters). Gaussian approximation refers to directly approximating the posterior with a Gaussian distribution, and can be readily applied for models with Gaussian process and measurement noise. Well known examples are the extended Kalman filter and sigma-point based unscented Kalman filter. The VB method is based on minimizing the Kullback-Leibler divergence of the true posterior with respect to the approximate distribution, chosen from a family of more tractable simpler distributions.The first main contribution of the thesis is the development of a VB approximation for linear regression problems with outlier robust measurement distributions. A broad family of outlier robust distributions can be presented as an infinite mixture of Gaussians, called Gaussian scale mixture models, and include e.g. the t-distribution, the Laplace distribution and the contaminated normal distribution. The VB approximation for the regression problem can be readily extended to the estimation of state space models and is presented in the introductory part.VB approximations can be also used for approximate inference in continuous-discrete Gaussian models, where the dynamics are modeled with stochastic differential equations and measurements are obtained at discrete time instants. The second main contribution is the presentation of a VB approximation for these models and the explanation of how the resulting algorithm connects to the Gaussian filtering and smoothing framework.The third contribution of the thesis is the development of parameter estimation using particle Markov Chain Monte Carlo (PMCMC) method and twisted particle filters. Twisted particle filters are obtained from standard particle filters by applying a special weighting to the sampling law of the filter. The weighting is chosen to minimize the variance of the marginal likelihood estimate, and the resulting particle filter is more efficient than conventional PMCMC algorithms. The exact optimal weighting is generally not available, but can be approximated using the Gaussian filtering and smoothing framework

Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond

Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov–Bayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed ‘Gaussian conjugacy’ in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity

Practice and Innovations in Sustainable Transport

The book continues with an experimental analysis conducted to obtain accurate and complete information about electric vehicles in different traffic situations and road conditions. For the experimental analysis in this study, three different electric vehicles from the Edinburgh College leasing program were equipped and tracked to obtain over 50 GPS and energy consumption data for short distance journeys in the Edinburgh area and long-range tests between Edinburgh and Bristol. In the following section, an adaptive and robust square root cubature Kalman filter based on variational Bayesian approximation and Huber’s M-estimation is proposed to accurately estimate state of charge (SOC), which is vital for safe operation and efficient management of lithium-ion batteries. A coupled-inductor DC-DC converter with a high voltage gain is proposed in the following section to match the voltage of a fuel cell stack to a DC link bus. Finally, the book presents a review of the different approaches that have been proposed by various authors to mitigate the impact of electric buses and electric taxis on the future smart grid

Poisson Multi-Bernoulli Mixtures for Multiple Object Tracking

Multi-object tracking (MOT) refers to the process of estimating object trajectories of interest based on sequences of noisy sensor measurements obtained from multiple sources. Nowadays, MOT has found applications in numerous areas, including, e.g., air traffic control, maritime navigation, remote sensing, intelligent video surveillance, and more recently environmental perception, which is a key enabling technology in automated vehicles. This thesis studies Poisson multi-Bernoulli mixture (PMBM) conjugate priors for MOT. Finite Set Statistics provides an elegant Bayesian formulation of MOT based on random finite sets (RFSs), and a significant trend in RFSs-based MOT is the development of conjugate distributions in Bayesian probability theory, such as the PMBM distributions. Multi-object conjugate priors are of great interest as they provide families of distributions that are suitable to work with when seeking accurate approximations to the true posterior distributions. Many RFS-based MOT approaches are only concerned with multi-object filtering without attempting to estimate object trajectories. An appealing approach to building trajectories is by computing the multi-object densities on sets of trajectories. This leads to the development of many multi-object filters based on sets of trajectories, e.g., the trajectory PMBM filters. In this thesis, [Paper A] and [Paper B] consider the problem of point object tracking where an object generates at most one measurement per time scan. In [Paper A], a multi-scan implementation of trajectory PMBM filters via dual decomposition is presented. In [Paper B], a multi-trajectory particle smoother using backward simulation is presented for computing the multi-object posterior for sets of trajectories using a sequence of multi-object filtering densities and a multi-object dynamic model. [Paper C] and [Paper D] consider the problem of extended object tracking where an object may generate multiple measurements per time scan. In [Paper C], an extended object Poisson multi-Bernoulli (PMB) filter is presented, where the PMBM posterior density after the update step is approximated as a PMB. In [Paper D], a trajectory PMB filter for extended object tracking using belief propagation is presented, where the efficient PMB approximation is enabled by leveraging the PMBM conjugacy and the factor graph formulation

Uncertainty-aware variational inference for target tracking

In the low Earth orbit, target tracking with ground based assets in the context of situational awareness is particularly difficult. Because of the nonlinear state propagation between the moments of measurement arrivals, the inevitably accumulated errors will make the target state prediction and the measurement likelihood inaccurate and uncertain. In this paper, optimizable models with learned parameters are constructed to model the state and measurement prediction uncertainties. A closed-loop variational iterative framework is proposed to jointly achieve parameter inference and state estimation, which comprises an uncertainty-aware variational filter (UnAVF). The theoretical expression of the evidence lower bound and the maximization of the variational lower bound are derived without the need for the true states, which reflect the awareness and reduction of uncertainties. The evidence lower bound can also evaluate the estimation performance of other Gaussian density filters, not only the UnAVF. Moreover, two rules, estimation consistency and lower bound consistency, are proposed to conduct the initialization of hyperparameters. Finally, the superior performance of UnAVF is demonstrated over an orbit state estimation problem