Filtering and estimation of noise -contaminated chaotic time series

Filtering noise and estimation of a signal in chaotic time series is possible in some cases. Consider a time series [special characters omitted], where Yi = X i + Zi. Xi = F(Xi −1), where F is an Axiom A diffeomorphism and the Zi\u27s are jointly independent. If the Zi\u27s are uniformly bounded by a constant δ, Lalley (1999) presented a consistent estimate of the signal values X i. We show that this estimate is within [special characters omitted] almost surely for any α ∈ (0, 1/2). Estimates from simulated data from three different dynamical systems are presented and summarized. A second stage smoothing estimate is presented and shown to be consistent. When the noise is unbounded, a kernel density estimate of ∫ u (x)dμ(x) is presented, where μ is the F-invariant SRB measure and u is a bounded [special characters omitted] real-valued function on the invariant set Λ. We conjecture that for a suitably chosen bandwidth, this estimate is consistent. For a time series of the form Yi = F( Yi−1 + Zi −1), it is shown that there is a stationary measure ν governing the Yi\u27s, and a consistent estimate of F−1 is given