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## Measuring weak nonlinearity

When testing for nonlinearity, we would like to use quantifiers which are optimized for the weak nonlinearity limit, which is not what most time series methods of chaos theory have been designed for. The simple nonlinear prediction scheme (Sec. ) has proven quite useful in this context. If used as a comparative statistic, it should be noted that sometimes seemingly inadequate embeddings or neighborhood sizes may lead to rather big errors which have, however, small fluctuations. The tradeoff between bias and variance may be different from the situation where predictions are desired per se. The same rationale applies to quantities derived from the correlation sum. Neither the small scale limit, genuine scaling, or the Theiler correction are formally necessary in a comparative test. However, any temptation to interpret the results in terms like ``complexity'' or ``dimensionality'' should be resisted, even though ``complexity'' doesn't seem to have an agreed-upon meaning anyway. Apart from average prdiction errors, we have found the stabilities of short periodic orbits (see Sec. ) useful for the detectionof nonlinearity in surrogate data tests. As an alternative to the phase space based methods, more traditional measures of nonlinearity derived from higher order autocorrelation functions ([86]) may also be considered. If a time-reversal asymmetry is present, its statistical confirmation (routine timerev) is a very powerful detector of nonlinearity [87]. Some measures of weak nonlinearity are compared systematically in Ref. [88].

Next: Conclusion and perspectives Up: Testing for nonlinearity Previous: General constrained randomization