Fixed-Interval Kalman Smoothing Algorithms in Singular State–Space Systems

Abstract

Fixed-interval Bayesian smoothing in statespace systems has been addressed for a long time. However, as far as the measurement noise is concerned, only two cases have been addressed so far : the regular case, i.e., with positive definite covariance matrix; and the perfect measurement case, i.e., with zero measurement noise. In this paper we address the smoothing problem in the intermediate case where the measurement noise covariance is positive semi definite with arbitrary rank. We exploit the singularity of the model in order to transform the original state-space system into a pairwise Markov model with reduced state dimension. Finally, the a posteriori Markovianity of the reduced state enables us to propose a family of fixed-interval smoothing algorithms.

Ensemble Kalman filtering with coloured observation noise

“…These require statistical models to be established beforehand to describe the observational noise time‐correlations. It is often suggested that coloured noise can be efficiently described by a first‐order autoregressive (AR) model driven by white Gaussian noise (e.g., Bryson and Johansen, 1965; Bryson and Henrikson, 1968; Ait‐El‐Fquih and Desbouvries, 2011; Wang et al ., 2012; Chang, 2014). In this context, two main approaches have been introduced to derive KF variants for coloured observational noise.…”

“…The first approach is based on state augmentation in which the state is augmented by the observational noise (e.g., Bryson and Johansen, 1965; Bryson and Henrikson, 1968; Wang et al ., 2012). On top of increasing the dimension of the state vector, this approach results in a state‐space system with perfect observations that is prone to singularities in the involved matrix inversions (e.g., Gelb, 1974; Chen, 1992; Sun and Deng, 2004; Simon, 2006; Ait‐El‐Fquih and Desbouvries, 2011). Remedies to this problem were proposed by Bryson and Johansen (1965) and Ait‐El‐Fquih and Desbouvries (2011) through order reduction, and by Wang et al .…”

Optimal Filter Approximations in Conditionally Gaussian Pairwise Markov Switching Models

“…In this section we introduce a new model, which is a particular case of the CGPMSM given by (6), (7) and in which the exact optimal filter can be computed with linear complexity in time. Called "Conditionally Gaussian Observed Markov Switching Model" (CGOMSM), this new model is a CGPMSM such that A 3 n+1 (R n+1 n ) = 0.…”

“…Due to their popularity in many different fields and, in particular, in different "tracking" problems [1], [2], speech processing problems [3], or biomedical engineering ones [4], these models are known under different names as "switching linear dynamic systems" [4], "jump Markov linear systems" [5], "switching linear state-space models" [6], "conditional linear Gaussian models" [7], or still "conditionally Gaussian linear state-space models" [8].…”

Off-line estimation of trajectory in discrete state space using the minimal-covariance adaptive FIR smoothing with extended output vector

“…In [19] the fixed-interval smoothing problem in state space systems with singular measurement noise (i.e., in the case where the covariance matrix of the measurement noise is either null or singular with arbitrary rank) is solved using the twofilter approach.…”