Lack of Robustness of the Method of Surrogate Data: Examples from Physiology and Finance
Author: Dimitris Kugiumtzis
The direct application of nonlinear tools (such as the estimation of correlation
dimension or the Lyapunov exponents) to real time series, seems to give rarely convincing
evidence on the existence of nonlinear dynamics.
On the other hand, the indirect approach, attempting to exclude that the data are
''linear'', has gained much interest in the last years. This is typically done by using
the method of surrogate data. The most general null hypothesis in this respect is that the
data are generated by a Gaussian (linear) process undergoing a possibly nonlinear static
transform.
The null hypothesis is tested by comparing the value of an estimate of a nonlinear
characteristic derived from the original data with the corresponding values computed from
a set of surrogate time series representing the null hypothesis.
The generated surrogate data do not always represent completely this null hypothesis.
Often the original linear correlations may not be preserved in the surrogate data, which
can effect on the result of the test. The rejection of the null hypothesis may also depend
on the applied nonlinear method and the choice of the method parameters. Here, we argue
against reported success of the test on physiological and financial data and claims of
evidence of nonlinearity based on a single nonlinear statistic.
In particular, two schemes for the generation of surrogate data are examined, the
amplitude adjusted Fourier transform (AAFT) and the iterated AAFT (IAFFT) and many
nonlinear discriminating statistics are used for testing, i.e. the fit with the Volterra
series of polynomials and the fit with local average mappings, the mutual information, the
correlation dimension, the false nearest neighbors, the largest Lyapunov exponent and
simple nonlinear averages (the three point autocorrelation and the time reversal
asymmetry). The results on arbitrarily selected EEG data and stock index data suggest that
rejections of the null hypothesis vary with the method and its parameters and depend also
on the algorithm generating the surrogate data.
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http://www.mpipks-dresden.mpg.de/~dimitris
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