Removing the Noise from Chaos Plus Noise

The problem of extracting a “signal” xn generated by a dynamical system from a times series yn = xn + en, where en 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 homoclinic pairs. It is also shown that consistent signal extraction is possible when the errors are uniformly bounded by a suitable constant and when the underlying dynamical system has the “weak orbit separation property”. Simple algorithms for signal recovery are described in the latter case.