Phase walk analysis of leptokurtic time series

Abstract

A very general definition of nonlinearity in data sets can be obtained from their representation in Fourier space: From the Wiener-Khintchine theorem and the bijectivity of the Fourier transformation it follows that the linear information is entirely represented by the Fourier amplitudes. Hence, all nonlinear information is contained solely in the Fourier phases. Yet, the direct study of the Fourier phases has so far attracted only little attention. Here, we present a novel method to quantify the phase information. In close analogy to random walk analyses, we propose the phase walk statistics as a way to quantify the phase information. We apply it to the analysis of nonlinearities in intermittent, leptokurtic time series like turbulent wind data, the Dow Jones (DJ) day-to-day returns and synthetic leptokurtic data. Testing for nonlinearities by means of surrogates shows that the new method yields strong significances for nonlinear behavior. Due to the drastically decreased computing time as compared to embedding space methods, the number of surrogate realizations can be increased by orders of magnitude [1]. Thereby, the probability distribution of the test statistics can very accurately be derived and parameterized, which allows for much more precise tests on nonlinearities. [1] K. Schreiber et al., Chaos, 28, 063120 (2018)