Phase correlations in nonlinear time series analysis

Abstract

Nonlinear times series properties are represented by the Fourier phases of the time series and for a comprehensive description of nonlinear data it is necessary to include the quantification of Fourier phase information. We introduce a novel method that traces nonlinearities by detecting correlations among the Fourier phases. The basic idea of the so-called phase walk statistics (PWS) is to measure - in close analogy to random walks - the deviation from randomness of the unwrapped differences of phases. The method compares distributions of phases from experimental and simulated time series with distributions that follow the null hypothesis of having totally random and uniformly distributed steps. It is shown that the obtained signifcances of nonlinear influence are comparable to those of well established measures. However, the results are obtained with orders of magnitude less computational effort, enabling much refined statistical analyses. We point out that the method can even identify specific scales on which nonlinear effects occur - a feature, not yet reached by any conventional method. A surrogate assisted analysis further shows that PWS is able to disentangle different classes of nonlinearities which have indistinguishable PDFs. The presented algorithms can be used to quantify nonlinear influence on time series of any physical, biological or financial system. We demonstrate the performance of the new approach via examples from Econophysics (Dow Jones time series), Plasmaphysics (trajectories of dust particles in vortices) and Aerodynamics (wind data with intermittencies).