Improved Calibration of Numerical Integration Error in Sigma-Point Filters

Abstract

The sigma-point filters, such as the unscented Kalman filter, are popular alternatives to the ubiquitous extended Kalman filter. The classical quadrature rules used in the sigmapoint filters are motivated via polynomial approximation of the integrand; however, in the applied context, these assumptions cannot always be justified. As a result, a quadrature error can introduce bias into estimated moments, for which there is no compensatory mechanism in the classical sigma-point filters. This can lead in turn to estimates and predictions that are poorly calibrated. In this article, we investigate the Bayes-Sard quadrature method in the context of sigma-point filters, which enables uncertainty due to quadrature error to be formalized within a probabilistic model. Our first contribution is to derive the well-known classical quadratures as special cases of the Bayes-Sard quadrature method. Based on this, a general-purpose moment transform is developed and utilized in the design of a novel sigma-point filter, which explicitly accounts for the additional uncertainty due to quadrature error.

A Probabilistic Taylor Expansion with Applications in Filtering and Differential Equations

“…5) for online state estimation in non-linear, partially observed systems. Unlike conventional non-linear filters, the proposed filter is capable of incorporating the error from model linearisations into filtering error covariances, and as a result is both more robust and significantly easier to tune than other recently developed GP based filters (Deisenroth et al, 2012;Prüher and Straka, 2018;Prüher et al, 2017Prüher et al, , 2021. The filter is shown to outperform both the EKF and the popular derivative-free unscented Kalman filter (UKF) in a challenging target tracking problem where both of these alternatives tend to diverge.…”

Testing whether a Learning Procedure is Calibrated

“…The nci, which is also called the inclusion indicator in Li and Zhao [2006], takes values close to 0 if the filtering algorithm is calibrated and is quite widely used [e.g. Prüher et al, 2020]. Further discussion can be found in Li et al [2011].…”

A modern retrospective on probabilistic numerics

“…The most successful area of research to date has been on the development of Bayesian methods for global optimisation [Snoek et al, 2012], which have become standard to the point of being embedded into commercial software [The MathWorks Inc.] and deployed in realistic [Acerbi, 2018, Paul et al, 2018 and indeed high-profile [Chen et al, 2018] applications. Other numerical tasks have yet to experience the same level of practical interest, though we note applications of probabilistic methods for cubature in computer graphics [Marques et al, 2013] and tracking [Prüher et al, 2018], as well as applications of probabilistic numerical methods in medical tractography [Hauberg et al, 2015] and nonlinear state estimation in an industrial context.…”