Surrogate Data

Surrogate data are nondeterministic, artificially generated data that mimic certain features (although not all features) of a measured time series but are otherwise stochastic. (Stochastic means having no regularities in behavior, and thus characterizable only by statistical properties.) Surrogate data, for instance, might have the same mean, variance, Fourier power spectrum, autocorrelation, and so on as the measured time series. The idea is to use such artificial, nondeterministic data as a control and compare them to the measured time series. Theiler et al. (1992) and Prichard & Price (1993) suggest adaptations of this long-established idea to nonlinear dynamics; Theiler et al. (1993) review other applications in nonlinear dynamics and chaos theory.
The object is to see whether the observed time series, similar in many respects to the specially designed nondeterministic data, is likely to have essentially the same value of some chosen nonlinear measure (embedding dimension, for instance). If it has the same value (to within limits that you specify), then the real time series doesn't differ from the artificial (nondeterministic) data in regard to that measure. For example, if we were testing some value of embedding dimension, we'd conclude that there's no advantage to that embedding. That, in turn, would mean that the next lower (and hence easier to deal with) embedding might be just as good for analyzing the actual time series. Or, testing for determinism, we might conclude that the observed data probably aren't deterministic. On the other hand, if the observed time series has a significantly different value of the chosen measure, then there's something special about that time series (e.g. the embedding is an improvement, the time series might be deterministic, etc.). The general steps in the surrogate data technique are:
1. Generate many surrogate datasets, each similar to the original time series. ''Similar" means the artificial datasets are of the same length as the measured time series, and the artificial data are statistically indistinguishable from the observed time series in regard to characteristics that you specify (e.g. mean, standard deviation, etc.). The purpose of creating a group of surrogate datasets rather than one set is so that the test can be repeated on many qualifying surrogate datasets. That gives a distribution of results and lets you evaluate the comparison in terms of statistical significance. The number of surrogate datasets in practice seems to range from about 5 to 100.
2. Compute a discriminating statistic for the measured time series and for each of the surrogate datasets. The discriminating statistic is any number that quantifies some aspect of the time series. Typical examples in nonlinear dynamics are the correlation dimension, largest Lyapunov exponent, and forecasting error.
3. Hypothesize that there's no difference between the discriminating statistic of the measured time series, on the one hand, and the same statistic for the members of the ensemble of plausible alternatives (the surrogate datasets), on the other. (Such a hypothesis of no difference is called a null hypothesis.)
4. Ask whether the discriminating statistic computed for the original data differs significantly from those computed for each of the surrogate datasets (see, for instance, Kachigan 1986: 177). If it does, then reject the hypothesis of no difference. That is, conclude that there is a discernible difference.