Parametric Bayesian Filters for Nonlinear Stochastic Dynamical Systems: A Survey

Abstract

Nonlinear stochastic dynamical systems are commonly used to model physical processes. For linear and Gaussian systems, the Kalman filter is optimal in minimum mean squared error sense. However, for nonlinear or non-Gaussian systems, the estimation of states or parameters is a challenging problem. Furthermore, it is often required to process data online. Therefore, apart from being accurate, the feasible estimation algorithm also needs to be fast. In this paper, we review Bayesian filters that possess the aforementioned properties. Each filter is presented in an easy way to implement algorithmic form. We focus on parametric methods, among which we distinguish three types of filters: filters based on analytical approximations (extended Kalman filter, iterated extended Kalman filter), filters based on statistical approximations (unscented Kalman filter, central difference filter, Gauss-Hermite filter), and filters based on the Gaussian sum approximation (Gaussian sum filter). We discuss each of these filters, and compare them with illustrative examples.

Effectiveness of Bayesian filters: An information fusion perspective

“…This gives the Bayesian posterior distribution (often called simply "the posterior" [50]. A few statistical parameters such as the 81 mean and variance are sufficient to represent a Gaussian distribution but not a general PDF, for which parameterization is either 82 impossible or will suffer from significant approximation errors.…”

Fast Kalman-Like Filtering for Large-Dimensional Linear and Gaussian State-Space Models

“…However, the approximation (31) that leads to VBPKF conserves dependence between the prediction pdf p(x n |y 0:n−1 ) and the filtering pdf and is thus more consistent with the KF. This can be seen from (48) and (51) in which the covariance changes depending on whether or not the model parameters are time-dependent.…”

Bayesian Nonlinear Filtering via Information Geometric Optimization

“…Bayesian principle provides a general approach for nonlinear filtering, and this approach is called as Bayesian filtering [5]. Bayesian filtering converts the state and measurement from the state-space to probability distribution.…”

“…Along with the spirit of Kalman filter, the approximation methods have been proposed for the nonlinear filtering, such as the Extended Kalman filter (EKF) [1], Unscented Kalman filter (UKF) [2], Gauss-Hermite Kalman filter (GHKF) [3], and Cubature Kalman filter (CKF) [4]. All these methods can be induced by the Bayesian approach with different approximations [5] for nonlinear cases. Apart from the aforementioned methods, the sequential Monte Carlo technique approximating for the Bayesian probability density functions (PDFs) is another feasible approach, for instance, the particle filtering (PF) [6] which uses the particle representations of probability distributions.…”