Nonlinearity test for complex time series without surrogates

The linear or nonlinear nature of a dynamical system is often evaluated by applying nonlinearity tests to an experimental time series that is typically the only observable output of the system. The most widely used nonlinearity test is the method of surrogates: a set of linear time series that replicate the properties of the experimental time series, specifically the marginal distribution and the autocorrelation function. This set of surrogates represents the null hypothesis of linearity, and can be created by several techniques. However, these techniques typically require some degree of manipulation in the frequency domain that often produces correlations in the Fourier phases leading to undesired nonlinear correlations in the surrogates. Here, we present a nonlinearity test that does not rely on generating surrogates, therefore avoiding the problem of spurious nonlinearities in the null hypothesis. Our approach uses the autocorrelation function of the time series under assessment to statistically determine whether the observed correlations could arise from a linear Gaussian time series that has been reversibly transformed into the experimental time series. If so, the experimental time series is considered linear; if not, it is considered nonlinear. We have applied the test to several well known models of linear and nonlinear time series, obtaining excellent results in both cases.