Topology-based denoising of chaos

Abstract

In this article, we propose a denoising algorithm to denoise a time series y_i = x_i + e_i, where {x_i} is a time series obtained from a time-T map of a uniformly hyperbolic or Anosov flow, and lcubeircub a uniformly bounded sequence of independent and identically distributed (i.i.d.) random variables. Making use of observations up to time n, we create an estimate of x_i for i is < n. We show under typical limiting behaviours of the orbit and the recurrence properties of x_i, the estimation error converges to zero as n tends to infinity with probability 1.