Improved Calibration of Numerical Integration Error in Sigma-Point Filters

Jakub Prüher, Toni Karvonen, Chris J. Oates, Ondrej Straka, Simo Särkkä

Research output: Contribution to journalArticleScientificpeer-review

3 Citations (Scopus)


The sigma-point filters, such as the UKF, are popular alternatives to the ubiquitous EKF. The classical quadrature rules used in the sigma-point filters are motivated via polynomial approximation of the integrand, however in the applied context these assumptions cannot always be justified. As a result, 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 formalised 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 utilised in the design of novel sigma-point filter, which explicitly accounts for the additional uncertainty due to quadrature error.
Original languageEnglish
Pages (from-to)1286-1292
Number of pages8
JournalIEEE Transactions on Automatic Control
Issue number3
Publication statusPublished - 6 May 2020
MoE publication typeA1 Journal article-refereed


  • Kalman filters
  • Bayesian quadrature
  • Quantification of uncertainty
  • Sigma-points
  • Gaussian processes


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