Stability and dimensionality reduction in nonlinear filtering

Thesis

The focus of this thesis is the analysis of the stability and robustness of continuous-time, finite state-space nonlinear filters, in order to provide new and practically relevant quantitative error bounds for a general class of approximate filters. This analysis is carried out through the use of the Hilbert projective metric.

We begin by providing a self-contained introduction to the Hilbert metric and its fundamental properties, with a particular focus on the space of probability measures. We then derive and study various dual formulations, and exploit these to obtain a contraction result for linear operators on convex cones with respect to a new distance, the hyperbolic tangent of the Hilbert metric. This general observation directs us naturally towards a range of new results on stability and robustness in nonlinear filtering.

Specifically, we turn to the problem of estimating the state of a continuous-time Markov chain from noisy observations. As regards stability, our key contribution is a proof that the corresponding optimal filter, called the Wonham filter, is contracting pathwise in the aforementioned distance given by the hyperbolic tangent of the Hilbert metric. Moreover, we give explicit deterministic and pathwise rates of convergence. By utilising these results, we are able to take an alternative approach to the study of the robustness of the Wonham filter, thereby improving on known error estimates and deriving rigorous, computable error bounds of theoretical and practical relevance as concerns the analysis and implementation of approximate filters.

Finally, we consider the problem of reducing the dimensionality of the Wonham filter via geometric projections, with a view towards defining an optimal projection filter. Building on the intuition provided by our error bounds, we find a natural submanifold for the Wonham filter such that the error of the projection filter is minimized.

Authors: Fausti, E

• DOI: 10.5287/ora-9oypoyozm
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: projection filters; probability simplex; stability; Hilbert projective metric; nonlinear filtering
• Subjects: Applied mathematics; Stochastic analysis