Testing for nonlinearity: the role of surrogate data

Statistical testing for nonlinearity involves the use of surrogate time series which mimic given features of the original time series but are random otherwise. Using the framework of constrained randomization by Schreiber [Phys. Rev. Lett. 80 (1998) 2105] the required structures are imposed on the random sequences by an optimization technique. As a result, the surrogate data fulfil given constraints, specified by a null hypothesis, with some error. In our approach to testing for nonlinearity we require that measures of significance for rejecting the null hypothesis must be independent of this error. This criterion turns out to be useful in the investigation of typical examples – even for weakly nonstationary time series. Furthermore, it is shown that testing for unstable periodic orbits (UPOs) is a robust measure for nonlinearity with respect to different types of surrogate data.