Phase correlations in leptokurtic time series

Kurzfassung

Many time series of nonlinear origin, like turbulence or financial markets not only exhibit lep- tokurtic (also called heavy-tailed or fat-tailed) probability distribution functions (PDF) but are also classified or characterized by them. In the Fourier representation of these time series these non- linearities must be contained in the phase information. Here, we demonstrate how to exploit this information to both synthesize and analyze leptokurtic time series. We first show that empirical data of market indices show linear phase correlations. Based on these findings we impose a set of linear phase correlations on white Gaussian noise and demonstrate that the so-obtained time series can reproduce the scaling properties of the PDF and volatility remark- ably well [1]. Next we introduce a novel nonlinear statistic called phase walk statistics (PWS), which measures – in close analogy to random walk analyses – the deviation from randomness of the phases. Applying this statistics to a number of empirical and synthetic leptokurtic time series reveals that nonlinear- ities can be detected unprecedented significance. A surrogate-assisted analysis further shows that PWS is able to disentangle different classes of nonlinearities, which have indistinguishable PDFs [2]. In summary, the study of Fourier phases offers new and refined insights into nonlinearities in time series. Thus, this approach opens new ways for better understanding and modelling the underlying nonlinear processes.