Beneath the noise, chaos. (English) Zbl 0980.62085

Lalley, Steven P.

Ann. Stat. 27, No. 2, 461-479 (1999).

Summary: 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.

MSC:

62M20	Inference from stochastic processes and prediction
37M10	Time series analysis of dynamical systems
37D20	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Keywords:

Axiom A system; noise reduction; nonlinear filter; hyperbolic attractor

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