Christoph Raeth | Searcharxiv

arXiv2008

Surrogates with random Fourier Phases

The method of surrogates is widely used in the field of nonlinear data analysis for testing for weak nonlinearities. The two most commonly used algorithms for generating surrogates are the amplitude adjusted Fourier transform (AAFT) and the iterated amplitude adjusted Fourier transfom (IAAFT) algorithm. Both the AAFT and IAAFT algorithm conserve the amplitude distribution in real space and reproduce the power spectrum (PS) of the original data set very accurately. The basic assumption in both algorithms is that higher-order correlations can be wiped out using a Fourier phase randomization procedure. In both cases, however, the randomness of the Fourier phases is only imposed before the (first) Fourier back tranformation. Until now, it has not been studied how the subsequent remapping and iteration steps may affect the randomness of the phases. Using the Lorenz system as an example, we show that both algorithms may create surrogate realizations containing Fourier phase correlations. We present two new iterative surrogate data generating methods being able to control the randomization of Fourier phases at every iteration step. The resulting surrogate realizations which are truly linear by construction display all properties needed for surrogate data.

physics.data-an↗

arXiv2004

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 $M= \{(ϕ_{\vec{k}}, ϕ_{\vec{k} + \vecΔ})\}$, where $ ϕ_{\vec{k}}$ is the phase of the $k$-mode of the Fourier transform of the data set and $\vecΔ$ 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 $\vecΔ$. 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.

physics.data-an↗