On Fourier phases and their relevance for nonlinear time series analysis

Many processes in nature are governed by nonlinear mechanisms that are frequently superposed by pronounced linear components. To characterize such complex dynamics it is desirable to disentangle linear and nonlinear features in empirical data. Quantitative nonlinear measures are also influenced by linear properties of the signals and, for the univariate case, Fourier transform surrogates do not represent properly the null hypothesis of zero nonlinear features. Here we elaborate on the application of a recently published method in Fourier space that does not suffer from comparison with inappropriate surrogate data and that reveals with high sensitivity correlations between Fourier phases and amplitudes. In addition, we propose a simple pre-processing procedure that avoids the mixing of linear and nonlinear features when applying conventional nonlinear measures, which can drastically increase the sensitivity of the statistical evaluation and avoids numerical problems associated with surrogate data. We test our proposal on data derived from numerical models and we analyze electroencephalographic recordings from epilepsy patients as well as heart rate signals from healthy subjects before and during meditation.