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# Surrogate data testing

All of the measures of nonlinearity mentioned above share a common property. Their probability distribution on finite data sets is not known analytically - except maybe when strong additional assumptions about the data are made. Some authors have tried to give error bars for measures like predictabilities (e.g. Barahona and Poon [21]) or averages of pointwise dimensions (e.g. Skinner et al. [22]) based on the observation that these quantities are averages (mean values or medians) of many individual terms, in which case the variance (or quartile points) of the individual values yield an error estimate. This reasoning is however only valid if the individual terms are independent, which is usually not the case for time series data. In fact, it is found empirically that nonlinearity measures often do not even follow a Gaussian distribution. Also the standard error given by Roulston [23] for the mutual information is fully correct only for uniformly distributed data. His derivation assumes a smooth rescaling to uniformity. In practice, however, we have to rescale either to exact uniformity or by rank-ordering uniform variates. Both transformations are in general non-smooth and introduce a bias in the joint probabilities. In view of the serious difficulties encountered when deriving confidence limits or probability distributions of nonlinear statistics with analytical methods, it is highly preferable to use a Monte Carlo resampling technique for this purpose.

- Typical vs. constrained realisations
- The null hypothesis: model class vs. properties
- Test design

Next: Typical vs. constrained realisations Up: Surrogate time series Previous: Phase space observables