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## Higher order statistics

Traditional measures of nonlinearity are derived from generalisations of the two-point auto-covariance function or the power spectrum. The use of higher order cumulants as well as bi- and multi-spectra is discussed for example in Ref. [10]. One particularly useful third order quantity is
since it measures the asymmetry of a series under time reversal. (Remember that the statistics of linear stochastic processes is always symmetric under time reversal. This can be most easily seen when the statistical properties are given by the power spectrum which contains no information about the direction of time.) Time reversibility as a criterion for discriminating time series is discussed in detail in Ref. [11], where, however, a different statistic is used to quantify it. The concept itself is quite folklore and has been used for example in Refs. [6, 12].

Time irreversibility can be a strong signature of nonlinearity. Let us point out, however, that it does not imply a dynamical origin of the nonlinearity. We will later (Sec. 7.1) give an example of time asymmetry generated by a measurement function involving a nonlinear time average.

Next: Phase space observables Up: Detecting weak nonlinearity Previous: Detecting weak nonlinearity