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On concentration properties of partially observed chaotic systems

Part of: Dynamical systems with hyperbolic behavior Parametric inference Applications - Dynamical systems and ergodic theory

Published online by Cambridge University Press:  26 July 2018

Daniel Paulin ,
Ajay Jasra ,
Dan Crisan  and
Alexandros Beskos
Daniel Paulin*
Affiliation:
National University of Singapore
Ajay Jasra*
Affiliation:
National University of Singapore
Dan Crisan*
Affiliation:
Imperial College London
Alexandros Beskos*
Affiliation:
University College London
*
* Postal address: Department of Statistics & Applied Probability, National University of Singapore, 117546, Singapore.
* Postal address: Department of Statistics & Applied Probability, National University of Singapore, 117546, Singapore.
*** Postal address: Department of Mathematics, Imperial College London, London SW7 2AZ, UK.
**** Postal address: Department of Statistical Science, University College London, London WC1E 6BT, UK.
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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.

Keywords

Dynamical systemchaosfilteringsmoothingLorenz equation

MSC classification

Primary: 37N10: Dynamical systems in fluid mechanics, oceanography and meteorology 37D45: Strange attractors, chaotic dynamics
Secondary: 62F15: Bayesian inference
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 440 - 479
Copyright
Copyright © Applied Probability Trust 2018 

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