TY - JOUR

T1 - On concentration properties of partially observed chaotic systems

AU - Paulin, Daniel

AU - Jasra, Ajay

AU - Crisan, Dan

AU - Beskos, Alexandros

N1 - Generated from Scopus record by KAUST IRTS on 2019-11-20

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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.

AB - 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.

UR - https://www.cambridge.org/core/product/identifier/S0001867818000216/type/journal_article

UR - http://www.scopus.com/inward/record.url?scp=85050666641&partnerID=8YFLogxK

U2 - 10.1017/apr.2018.21

DO - 10.1017/apr.2018.21

M3 - Article

SN - 0001-8678

VL - 50

JO - Advances in Applied Probability

JF - Advances in Applied Probability

IS - 2

ER -