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

Bayesian Estimation and Quality Monitoring for Personal Positioning Systems

Personal positioning is a dynamic estimation problem where the ability to assess the quality of the positioning service is as important as obtaining accurate position estimates. When estimating the position of a person, as opposed to e.g. an airplane, the type of motion can change at any time as a pedestrian can board a bus, or a cyclist can board a train. Also the changing surroundings in urban navigation influence the observation noise as tall buildings blocking the line of sight to satellites are full of reflecting surfaces. First we investigate classic robust estimation methods applied to the positioning problem, but then we focus on the Bayesian framework, as its generality allows us to take into account the abrupt changes in the state-space system. Gaussian mixture distributions and Markov chain indicator processes are used to model the changing systems. We evaluate the resulting systems mainly with sequential Monte Carlo methods, as this approach gives us an approximative joint posterior distribution of the errors and the state. We propose a general framework for the Bayesian receiver autonomous integrity monitoring in urban navigation based on the posterior probabilities. We also use the Bayesian framework to solve the explicit effect of the sensor errors in a nominal system that estimates the state with the assumption of no changes in the models. We use the estimated cumulated effect of the errors in the time series to determine whether error is present in the system at any time. Finally, a variational Bayes algorithm is developed for detecting changes in the system noise covariances

Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorial

On top of machine learning models, uncertainty quantification (UQ) functions as an essential layer of safety assurance that could lead to more principled decision making by enabling sound risk assessment and management. The safety and reliability improvement of ML models empowered by UQ has the potential to significantly facilitate the broad adoption of ML solutions in high-stakes decision settings, such as healthcare, manufacturing, and aviation, to name a few. In this tutorial, we aim to provide a holistic lens on emerging UQ methods for ML models with a particular focus on neural networks and the applications of these UQ methods in tackling engineering design as well as prognostics and health management problems. Toward this goal, we start with a comprehensive classification of uncertainty types, sources, and causes pertaining to UQ of ML models. Next, we provide a tutorial-style description of several state-of-the-art UQ methods: Gaussian process regression, Bayesian neural network, neural network ensemble, and deterministic UQ methods focusing on spectral-normalized neural Gaussian process. Established upon the mathematical formulations, we subsequently examine the soundness of these UQ methods quantitatively and qualitatively (by a toy regression example) to examine their strengths and shortcomings from different dimensions. Then, we review quantitative metrics commonly used to assess the quality of predictive uncertainty in classification and regression problems. Afterward, we discuss the increasingly important role of UQ of ML models in solving challenging problems in engineering design and health prognostics. Two case studies with source codes available on GitHub are used to demonstrate these UQ methods and compare their performance in the life prediction of lithium-ion batteries at the early stage and the remaining useful life prediction of turbofan engines

Interactive multiple model filtering for robotic navigation and tracking applications

The work contained in this thesis focuses on two main objectives. The first objective is to evaluate the Interactive Multiple Model (IMM) filter method for robotic applications including inertial navigation systems (INS) and computer vision tracking. The second objective is to design an experimental testbed for multi-model mobile robot state estimation research in the Intelligent Systems Laboratory (ISLAB) at Memorial University. An IMM estimator uses multiple filters that run simultaneously to produce a combined weighted estimation of an observed system’s states. The weights are functions of the likelihood of how well each individual filter matches the current behaviour exhibited by the system. The performance of IMM filtering is evaluated using two different strategies for augmenting the system’s filter banks. The first method uses multiple kinematic models (constant velocity and constant acceleration models) in a mean-shift-based computer vision tracking application. The results of this experiment indicate that the IMM improves tracking performance due to its ability to adapt to the continuously changing motion characteristics of 2D blobs in videos. The second approach uses the same kinematics for each filter; however, the process and sensor noise parameters are tuned differently for each model. This method is tested in INS applications for both an automobile and a skid-steer mobile robot (Seekur Jr). Results show that the method improves INS tracking over single model Extended Kalman Filter (EKF) designs. Furthermore, an augmented state-space model containing skid-steer instantaneous center of rotation (ICR) kinematics is presented for future testing on the Seekur Jr INS. The experimental testbed designed in this thesis work is an operational data acquisition system developed for use with the Seekur Jr robot. The Seekur Jr platform has been Robot Operating System (ROS) enabled with access to data streams from 2D Lidar, 3D nodding Lidar, inertial measurement unit, digital compass, wheel encoder, onboard Global Positioning System (GPS), real-time kinematic (RTK) differential global positioning system (DGPS) ground truth, and vision sensors. The physical setup and data networking aspects of the testbed have been used for validation of an IMM filter presented in this thesis and is fully configured for future multi-model localization experiments of the ISLAB

Robust Approaches to Nonlinear Filtering with Applications to Navigation

Linear estimators, like the extended Kalman filter (EKF), find continual use (especially in the field of navigation) mostly for their familiarity and computational efficiency. Often, these estimations must be safeguarded from the realistic elements of physical systems, such as nonlinearities, non-Gaussian noises, and unmodeled effects. To this end, existing linear estimators are frequently outfitted with procedure-first robustness techniques—ad hoc mechanisms designed specifically to prevent filter failure—such as measurement editing, gain underweighting, filter resets, and more. As an alternative, this dissertation elects a model-first ethos, proposing nonlinear Gaussian mixture (GM) filters that are derived from first principles to be robust. These inherently robust algorithms are split into two approaches—1) non-Bayesian filters and 2) fault-cognizant filters—the end result being a collection of filters that challenge the status quo of current practical estimation; instead of reusing preexisting filter frameworks for the sake of ease, customized filters can be designed specifically for the system at hand. 1) Bayes’ rule, while the archetypal basis for measurement fusion, relies on a fundamental assumption; all specified models, such as prior distributions and measurement likelihoods, are presumed to exactly reflect reality. In practice, this is rarely the case, warranting an investigation into non-Bayesian alternatives to traditional measurement updates. Fortunately, generalized variational inference (GVI) provides an established foundation for such updates and is used in this work to prototype several robust non-Bayesian filters. As closed-form filters are usually preferred, an iterative confidence-based update is derived, which, through Monte Carlo analyses, is shown to be selectively conservative, such that a desired level of robustness can be user-appointed. 2) Whereas traditional filtering screens out undesirable, or faulty, measurements, fault-cognizant filtering attempts to directly model these erroneous measurements, yielding estimators inherently capable of processing returns that conflict with the conventional model of a sensor. As the nature of both valid and faulty measurements can differ significantly between systems, several different fault-cognizant updates (FCUs) are derived, each purposed for a specific application. Subsequent analyses illustrate the robustness of the FCU to faulty measurements, both known and unknown