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