The Annals of Statistics

Beneath the noise, chaos

Steven P. Lalley

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The problem of extracting a signal $x_{n}$ from a noise-corrupted time series $y_{n} = x_{n}+e_{n}$ is considered. The signal $x_{n}$ is assumed to be generated by a discrete-time, deterministic, chaotic dynamical system $F$, in particular, $x_{n} = F^{n}(x_{0})$, where the initial point $x_{0}$ is assumed to lie in a compact hyperbolic $F$-invariant set. It is shown that (1) if the noise sequence $e_{n}$ is Gaussian then it is impossible to consistently recover the signal $x_{n}$ , but (2) if the noise sequence consists of i.i.d. random vectors uniformly bounded by a constant $\delta > 0$, then it is possible to recover the signal $x_{n}$ provided $\delta < 5\Delta$, where $\Delta$ is a separation threshold for $F$. A filtering algorithm for the latter situation is presented.

Article information

Ann. Statist., Volume 27, Number 2 (1999), 461-479.

First available in Project Euclid: 5 April 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11]
Secondary: 58F15

noise reduction nonlinear filter Axiom A hyperbolic attractor.


Lalley, Steven P. Beneath the noise, chaos. Ann. Statist. 27 (1999), no. 2, 461--479. doi:10.1214/aos/1018031203.

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