Detecting non-linearities in data sets. Characterization of Fourier phase maps using the Weighted Scaling Indices

We present a methodology for detecting non-linearities in data sets based on the characterization of the structural features of the Fourier phase maps. A Fourier phase map is a 2D set of points $M= \{(\phi_{\vec{k}}, \phi_{\vec{k} + \vec{\Delta}})\}$, where $ \phi_{\vec{k}}$ is the phase of the $k$-mode of the Fourier transform of the data set and $\vec{\Delta}$ a phase shift. The information thus rendered on this space is analyzed using the spectrum of weighted scaling indices to detect phase coupling at any scale $\vec{\Delta}$. We propose a statistical test of significance based on the comparison of the properties of phase maps created from both the original data and surrogate realizations. We have applied our method to the Lorenz system and the logarithmic stock returns of the Dow Jones index. Applications to higher dimensional data are straightforward. The results indicate that both the Lorenz system and the Dow Jones time series exhibit significant signatures of non-linear behavior.