On Concentration Properties of Partially Observed Chaotic Systems

TL;DR

This paper investigates the concentration behavior of filtering and smoothing distributions in partially observed chaotic systems, revealing that these distributions do not necessarily concentrate around the true state as observations increase, due to inherent chaos.

Contribution

It demonstrates that in chaotic systems like Lorenz models, the support of filtering and smoothing distributions remains bounded away from the true state, challenging assumptions of convergence with more data.

Findings

Support diameter remains lower bounded by noise level

Support diameter is upper bounded by noise level under certain conditions

Applications to Lorenz 63 and Lorenz 96 models are discussed

Abstract

This article presents 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 infinity. 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 times 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 times 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.