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On concentration properties of partially observed chaotic systems

Paulin, D; Jasra, A; Crisan, D; Beskos, A; (2018) On concentration properties of partially observed chaotic systems. Advances in Applied Probability , 50 (2) pp. 440-479. 10.1017/apr.2018.21. Green open access

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Abstract

In this paper we present results on the concentration properties of the smoothing and filtering distributions of some partially observed chaotic dynamical systems. We show that, rather surprisingly, for the geometric model of the Lorenz equations, as well as some other chaotic dynamical systems, the smoothing and filtering distributions do not concentrate around the true position of the signal, as the number of observations tends to ∞. Instead, under various assumptions on the observation noise, we show that the expected value of the diameter of the support of the smoothing and filtering distributions remains lower bounded by a constant multiplied by the standard deviation of the noise, independently of the number of observations. Conversely, under rather general conditions, the diameter of the support of the smoothing and filtering distributions are upper bounded by a constant multiplied by the standard deviation of the noise. To some extent, applications to the three-dimensional Lorenz 63 model and to the Lorenz 96 model of arbitrarily large dimension are considered.

Type: Article
Title: On concentration properties of partially observed chaotic systems
Open access status: An open access version is available from UCL Discovery
DOI: 10.1017/apr.2018.21
Publisher version: https://doi.org/10.1017/apr.2018.21
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Dynamical system; chaos; filtering; smoothing; Lorenz equation
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
URI: https://discovery.ucl.ac.uk/id/eprint/1518523
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