Correlating phase information with higher order statistics in nonlinear data sets

Kurzfassung

The Wiener-Khinchin theorem establishing an exact mathematical relation between the autocorrelation function of a signal in real space with the power spectrum of the same data derived from the Fourier representation of them is one of the basic pillars of linear data analysis. Yet, all information about nonlinearities and higher order correlations (HOC) being possibly also contained in the signal is coded in the Fourier phases. Calculating the next order correlations and their counterparts in Fourier space allow to quantify some aspects of higher order correlations. Similarly, the estimation static and dynamic complexity measures of a d-dimensional point set being given for spatial structures or being obtained after proper embedding of a time series reveal the presence of nonlinearity with high significance. Another class of measures, namely Minkowski-functionals, quantifying the topology of a point set and can be expressed by a sum of n-point correlation functions thus measuring higher order correlations. Less attention has been paid so far to the explicit analysis of the information contained in the Fourier phases and the correlations among them. There are, however, prominent examples of methods for data analysis where phase properties implicitly play a big role. It is the randomization of the phase information, which is at the heart of algorithms generating so-called surrogate data sets, which were developed to test for weak nonlinearities in a model-independent way. In this contribution we report on establishing for the first time heuristic relations between Fourier phases and higher order statistics for both scalar time series and two-dimensional spatial structures. Using observational data from the XMM-Newton satellite, we demonstrate that by means of surrogate generation algorithms surrogate time series with correlated and anticorrelated phases can be generated. We calculate the nonlinear prediction error (NLPE) and network-based quantities as dynamic and static complexity measures and correlate them with phase statistics. It turns out that both complexity measures yield high linear correlations with phase statistics representing first examples of Wiener-Khinchin-like relations of phases with HOCs. We demonstrate how this result can be understood by generating and analyzing random time series which only mimic the identified phase correlations. Using recent observations of the Cosmic Microwave Background (CMB) we show that known phase correlations at large scales can significantly be diminished when a Bianchi-like template accounting for anisotropies in the radiation field map is subtracted from the CMB-map. By doing so, signatures of anisotropy when measuring the Minkowski functionals and scaling indices in real space are vanishing. Such extensions of the Wiener-Khinchin theorem can open a new road to a better understanding of signatures of non-linearities in complex systems - especially by elucidating the meaning of Fourier phase correlations and their influence on higher-order statistics.