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On concentration properties of partially observed chaotic systems

Published online by Cambridge University Press:  26 July 2018

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
Corresponding
E-mail address:

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
Original Article
Copyright
Copyright © Applied Probability Trust 2018 

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