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April 1999 Beneath the noise, chaos
Steven P. Lalley
Ann. Statist. 27(2): 461-479 (April 1999). DOI: 10.1214/aos/1018031203


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.


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Steven P. Lalley. "Beneath the noise, chaos." Ann. Statist. 27 (2) 461 - 479, April 1999.


Published: April 1999
First available in Project Euclid: 5 April 2002

zbMATH: 0980.62085
MathSciNet: MR1714721
Digital Object Identifier: 10.1214/aos/1018031203

Primary: 62M20
Secondary: 58F15

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 2 • April 1999
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