Approximate Bayesian Smoothing with Unknown Process and Measurement Noise Covariances

Abstract

We present an adaptive smoother for linear statespace models with unknown process and measurement noise covariances. The proposed method utilizes the variational Bayes technique to perform approximate inference. The resulting smoother is computationally efficient, easy to implement, and can be applied to high dimensional linear systems. The performance of the algorithm is illustrated on a target tracking example.

Roundabout Entry Capacity Calculation—A Case Study Based on Roundabouts in Tokyo, Japan, and Tokyo Surroundings

“…The position of vehicles was extracted in 0.5 s intervals, and then, the position coordinates were converted into a global coordinate system by the projective transformation. The vehicle trajectories transformed were smoothened by the Kalman smoothing method [47][48][49].…”

A Novel Adaptive Kalman Filter With Inaccurate Process and Measurement Noise Covariance Matrices

“…In this simulation scenario, the target moves according to the continuous white noise acceleration motion model in two dimensional Cartesian coordinates, and the target's positions are collected by a sensor. The state is defined as x k [x k y kẋkẏk ], where x k , y k ,ẋ k andẏ k denote the cartesian coordinates and corresponding velocities [13], [20]. The state transition matrix F k−1 and observation matrix H k are respectively given by…”

“…However, the performance of the existing VBAKF will degrade for an inaccurate process noise covariance matrix (PNCM) since it assumes accurate PNCM. Although the VB based Rauch-Tung-Striebel smoother can estimate unknown PNCM and MNCM simultaneously [13], [14], it can only estimate unknown and constant noise covariance matrices off-line. To the best of the knowledge of the authors, it is always a challenge to design a VBAKF for linear Gaussian state-space models with inaccurate PNCM and MNCM since the PNCM is difficult to be estimated directly with a rather small window of data.…”

A Robust Gaussian Approximate Fixed-Interval Smoother for Nonlinear Systems With Heavy-Tailed Process and Measurement Noises

“…In this simulation, the standard cubature Kalman smoother (CKS) [2], outlier robust CKS [4], CKS with unknown noise covariances (CKSWUNC) [13], [14], the proposed robust CKS with fixed noise parameters (the proposed CKS-fixed), the proposed robust CKS with estimated Q and R and fixed ω and ν (the proposed CKS-QR), the proposed robust CKS with estimated ω and ν and fixed Q and R (the proposed CKS-ων), and the proposed robust CKS with estimated Q, R, ω and ν (the proposed CKS-QRων) are tested. Note that CKSWUNC is obtained by using the Rauch-Tung-Striebel smoother in [13] combined with the third degree spherical radial cubature rule [3] based statistical linearization of the nonlinear system. The scale matrix and dof parameter of the existing outlier robust CKS are set as Σ v and 5.…”

“…However, this filter requires the growth of the degree of freedom (dof) parameters to be prevented and thereby maintain the assumption that the estimated state and process/measurement noise are jointly Student's t with a common dof parameter in the filter recursion [12]. An adaptive smoother based on a variational Bayesian (VB) approach for a linear state space model with Gaussian noises and unknown noise covariances was proposed in [13], [14], but it is sensitive to heavy-tailed process and measurement noises, as will be confirmed in Section IV. An approach to estimate the unknown parameters of a Student's t distribution for an autoregressive model was proposed in [15], however, this approach is not suitable for the state space model in this work.…”