## The Annals of Statistics

### Beneath the noise, chaos

Steven P. Lalley

#### Abstract

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

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

Dates
First available in Project Euclid: 5 April 2002

https://projecteuclid.org/euclid.aos/1018031203

Digital Object Identifier
doi:10.1214/aos/1018031203

Mathematical Reviews number (MathSciNet)
MR1714721

Zentralblatt MATH identifier
0980.62085

Subjects
Secondary: 58F15

#### Citation

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

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• WEST LAFAYETTE, INDIANA 47907 E-MAIL: lalley@stat.purdue.edu