Abstract
In this paper, the fixed-lag smoothing problem for conditionally linear Gaussian state-space models is investigated from a factor graph perspective. More specifically, after formulating Bayesian smoothing for an arbitrary state-space model as a forward–backward message passing over a factor graph, we focus on the above-mentioned class of models and derive two novel particle smoothers for it. Both the proposed techniques are based on the well-known two-filter smoothing approach and employ marginalized particle filtering in their forward pass. However, on the one hand, the first smoothing technique can only be employed to improve the accuracy of state estimates with respect to that achieved by forward filtering. On the other hand, the second method, that belongs to the class of Rao–Blackwellized particle smoothers, also provides a point mass approximation of the so-called joint smoothing distribution. Finally, our smoothing algorithms are compared, in terms of estimation accuracy and computational requirements, with a Rao–Blackwellized particle smoother recently proposed by Lindsten et al. (“Rao–Blackwellized particle smoothers for conditionally linear Gaussian models,” IEEE J. Sel. Topics Signal Process. , vol. 10, no. 2, pp. 353–365, 2016).
Highlights
Bayesian filtering and Bayesian smoothing for state space models (SSMs) are two interrelated problems that have received significant attention for a number of years [1]
Approximate solutions are available for general nonlinear models; these are based on sequential Monte Carlo (SMC) techniques which represent a powerful tool for numerical approximations [3]-[5]
In developing our first particle smoothing algorithm the following fundamental requirements have been set to limit its computational complexity as much as possible: 1) MPF is employed in its forward pass4; 2) the statistical information generated by the BIF technique adopted in its backward pass can be combined with that provided by MPF to generate the marginal smoothed densities of interest; c) the estimates of both the linear component and the nonlinear component are available at the end of the backward pass and each of them is represented by a single trajectory
Summary
Bayesian filtering and Bayesian smoothing for state space models (SSMs) are two interrelated problems that have received significant attention for a number of years [1]. The proposed solutions are represented by two novel particle smoothing methods, the first one dubbed serial particle smoothing (SPS), the second one Rao-Blackwellized serial smoothing (RBSS) These methods share the following relevant features: a) they are based on the two-filter smoothing formula and employ MPF in their forward pass; b) they can be derived applying the well known sum-product algorithm (SPA) [23], [24], together with a specific scheduling procedure, to the same FG developed in [21] and [22] for a CLG SSM; c) unlike the RBPS methods devised in [13] and [14], they can be employed for a SSM in which both the linear and. Notations: The probability density function (pdf) of a random vector R evaluated at point r is denoted f (r); N (r; ηr, Cr) represents the pdf of a Gaussian random vector R characterized by the mean ηr and covariance matrix Cr evaluated at point r; the precision (or weight) matrix associated with the covariance matrix Cr is denoted Wr, whereas the transformed mean vector Wrηr is denoted wr
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