Removing the Noise from Chaos Plus Noise

Abstract

The problem of extracting a “signal” x_n generated by a dynamical system from a time series y_n = x_n + e_n, where e_n is an observational error, is considered. It is shown that consistent signal extraction is impossible when the errors are distributed according to a density with unbounded support, and the underlying dynamical system admits bomoclinic pairs. It is also shown that consistent signal extraction is possible when the errors are uniformly bounded by a suitable constant and the underlying dynamical system has the “weak orbit separation property”. Simple algorithms for signal recovery are described in the latter case.