Density-ratio robustness in dynamic state estimation

Abstract

The filtering problem is addressed by taking into account imprecision in the knowledge about the probabilistic relationships involved. Imprecision is modelled in this paper by a particular closed convex set of probabilities that is known with the name of density ratio class or constant odds-ratio (COR) model. The contributions of this paper are the following. First, we shall define an optimality criterion based on the squared-loss function for the estimates derived from a general closed convex set of distributions. Second, after revising the properties of the density ratio class in the context of parametric estimation, we shall extend these properties to state estimation accounting for system dynamics. Furthermore, for the case in which the nominal density of the COR model is a multivariate Gaussian, we shall derive closed-form solutions for the set of optimal estimates and for the credible region. Third, we discuss how to perform Monte Carlo integrations to compute lower and upper expectations from a COR set of densities. Then we shall derive a procedure that, employing Monte Carlo sampling techniques, allows us to propagate in time both the lower and upper state expectation functionals and, thus, to derive an efficient solution of the filtering problem. Finally, we empirically compare the proposed estimator with the Kalman filter. This shows that our solution is more robust to the presence of modelling errors in the system and that, hence, appears to be a more realistic approach than the Kalman filter in such a case.

On the median in imprecise ordinal problems

“…Similar results have been obtained for the L 2 loss and the interval bounded by lower and upper expectations (Benavoli and Zaffalon 2013).…”

“…5 suggest that symmetry and strict convexity are necessary conditions. Also, in the case of maximality, results from Benavoli and Zaffalon (2013) linking expected value and L 2 loss, and Example 3 indicate that retrieving the median interval with other symmetric losses than the L 1 loss may be difficult; -More generally, it would be interesting to know what are the conditions to impose on losses for the prediction to be a closed interval, as this means that one only needs to compute the bounds of such intervals. Again, this paper and (Benavoli and Zaffalon 2013) suggest that strict convexity of the loss function is a necessary condition; -Study to which extent the presented results can be used in other applications involving ordinal variables, in particular the field of multi-criteria decision making, in which some methods are closely related to ordinal regression problems (Angilella et al 2015).…”

“…(10). Yet using other V-shaped loss functions such as the L 2 loss will lead to other predictions (Benavoli and Zaffalon 2013).…”

Unifying neighbourhood and distortion models: part I – new results on old models

“…It was given a behavioural interpretation by Peter Walley in [32, Sec. 2.9.4], and studied in [1,2,25,29]. 6.1.…”

A probabilistic interpretation of set-membership filtering: Application to polynomial systems through polytopic bounding

“…Proof: LetS be a polytope belonging to the set of feasibility of problem (42) (i.e., M ⊆S) which does not minimize (42). This means that there exists a polytopeS such that M ⊆S ⊆S and a pointp given as input of Algorithm 2…”