On the reliability of the surrogate data test for nonlinearity in the analysis of noisy time series

In the analysis of real world data, the surrogate data test is often perfor med in order to investigate nonlinearity in the data. The null hypothesis o f the test is that the original time series is generated from a linear stoc hastic process possibly undergoing a nonlinear static transform. We argue a gainst reported rejection of the null hypothesis and claims of evidence of nonlinearity based on a single nonlinear statistic. In particular, two sche mes for the generation of surrogate data are examined, the amplitude adjust ed Fourier transform (AAFT) and the iterated AAFT (IAFFT) and many nonlinea r discriminating statistics are used for testing, i.e. the fit with the Vol terra series of polynomials and the fit with local average mappings, the mu tual information, the correlation dimension, the false nearest neighbors, t he largest Lyapunov exponent and simple nonlinear averages (the three point autocorrelation and the time reversal asymmetry). The results on simulated data and real data (EEG and exchange rates) suggest that the test depends on the method and its parameters, the algorithm generating the surrogate da ta and the observational data of the examined process.