Unbiased reconstruction of the dynamics underlying a noisy chaotic time series

Refined methods for the construction of a deterministic dynamical system which can consistently reproduce observed aperiodic data are discussed. The determination of the dynamics underlying a noisy chaotic time series suffers strongly from two systematic errors: One is a consequence of the so-called "error-in-variables problem." Standard least-squares fits implicitly assume that the independent variables are noise free and that the dependent variable is noisy. We show that due to the violation of this assumption one receives considerably wrong results for moderate noise levels. A straightforward modification of the cost function solves this problem. The second problem consists in a mutual inconsistency between the images of a point under the model dynamics and the corresponding observed values. For an improved fit we therefore introduce a multistep prediction error which exploits the information stored in the time series in a better way. The performance is demonstrated by several examples, including experimental data. (c) 1996 American Institute of Physics.