部分可观测混沌系统的集中性
On Concentration Properties of Partially Observed Chaotic Systems
摘要：
本文给出了部分可观测混沌动力系统的平滑分布和滤波分布的集中性结果。令人惊讶的是，我们发现，对于洛伦兹方程的几何模型，以及其他一些混沌动力系统，平滑和滤波分布不集中在信号的真实位置，因为观测的数量趋于无穷大。相反，在各种假设下的观测噪声，我们表明，直径的平滑和过滤分布的支持的期望值保持下界由一个常数倍的标准差的噪声，独立的观测数。相反，在相当一般的条件下，平滑和滤波分布的支撑直径的上限是常数乘以噪声的标准差。在一定程度上，考虑了对三维Lorenz 63'模型和任意大维数Lorenz 96'模型的应用。
英文摘要：
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.
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