[Paper Review] Nonlinear Bayesian Estimation: From Kalman Filtering to a Broader Horizon

Huazhen Fang, Ning Tian+3 more|arXiv (Cornell University)|Dec 4, 2017

Target Tracking and Data Fusion in Sensor Networks39 references3 citations

TL;DR

This paper presents a comprehensive tutorial on nonlinear Bayesian estimation, systematically reviewing Kalman filtering variants—EKF, UKF, and EnKF—and extending to advanced methods like particle filtering and moving horizon estimation. It establishes a unified Bayesian framework for state estimation in nonlinear systems, emphasizing probabilistic modeling and uncertainty quantification.

ABSTRACT

This article presents an up-to-date tutorial review of nonlinear Bayesian estimation. State estimation for nonlinear systems has been a challenge encountered in a wide range of engineering fields, attracting decades of research effort. To date, one of the most promising and popular approaches is to view and address the problem from a Bayesian probabilistic perspective, which enables estimation of the unknown state variables by tracking their probabilistic distribution or statistics (e.g., mean and covariance) conditioned on the system's measurement data. This article offers a systematic introduction of the Bayesian state estimation framework and reviews various Kalman filtering (KF) techniques, progressively from the standard KF for linear systems to extended KF, unscented KF and ensemble KF for nonlinear systems. It also overviews other prominent or emerging Bayesian estimation methods including the Gaussian filtering, Gaussian-sum filtering, particle filtering and moving horizon estimation and extends the discussion of state estimation forward to more complicated problems such as simultaneous state and parameter/input estimation.

Motivation & Objective

• To provide a systematic, bottom-up introduction to Bayesian state estimation for nonlinear systems.
• To review and compare major Kalman filtering techniques (EKF, UKF, EnKF) within a Bayesian probabilistic framework.
• To extend the discussion to advanced Bayesian estimators such as Gaussian filtering, Gaussian-sum filtering, particle filtering, and moving horizon estimation.
• To address complex estimation challenges, including simultaneous state and parameter estimation.
• To identify and discuss future research directions in nonlinear Bayesian estimation for cyber-physical and data-intensive systems.

Proposed method

• Formulates nonlinear state estimation as a Bayesian inference problem, modeling states as random vectors with time-varying probability distributions.
• Applies Bayes' theorem to recursively update the posterior distribution of the state conditioned on measurement data.
• Uses linear and nonlinear approximations (e.g., Jacobian and unscented transforms) to propagate Gaussian distributions through nonlinear functions in EKF and UKF.
• Employs Monte Carlo sampling via ensembles to approximate the posterior distribution in ensemble Kalman filtering (EnKF).
• Introduces Gaussian filtering (GF) and Gaussian-sum filtering (GSF) to handle non-Gaussian and multi-modal posterior distributions.
• Applies particle filtering (PF) for non-Gaussian, nonlinear systems using sequential Monte Carlo methods with weighted particles.
• Extends the framework to moving horizon estimation (MHE), which uses finite-horizon optimization to improve estimation accuracy.

Experimental results

Research questions

• RQ1 How can the Bayesian framework be systematically applied to nonlinear state estimation problems?
• RQ2 What are the strengths and limitations of EKF, UKF, and EnKF in approximating nonlinear transformations of probability distributions?
• RQ3 How do advanced methods like particle filtering and moving horizon estimation improve upon traditional Kalman filters in complex scenarios?
• RQ4 What are the key challenges in accurately characterizing the result of nonlinear transformations on probability distributions?
• RQ5 How can Bayesian estimation be extended to handle simultaneous state and parameter estimation in dynamic systems?

Key findings

• The Bayesian framework provides a unified and intuitive approach to state estimation, enabling probabilistic modeling of uncertainty in nonlinear systems.
• Extended Kalman Filtering (EKF) achieves reasonable performance for mildly nonlinear systems but suffers from linearization errors due to Jacobian approximations.
• Unscented Kalman Filtering (UKF) improves accuracy by using sigma points to capture the mean and covariance of nonlinearly transformed distributions more effectively than EKF.
• Ensemble Kalman Filtering (EnKF) offers a robust, sampling-based alternative suitable for high-dimensional systems, particularly in data assimilation contexts.
• Particle filtering enables estimation in non-Gaussian and highly nonlinear systems, though at the cost of computational complexity and potential particle degeneracy.
• Moving horizon estimation (MHE) enhances estimation accuracy by solving a finite-horizon optimization problem, making it suitable for systems with constraints and noisy measurements.