The method of surrogates is one of the key concepts of nonlinear data analysis. Here, we demonstrate that commonly used algorithms for generating surrogates often fail to generate truly linear time series. Rather, they create surrogate realizations w
ith Fourier phase correlations leading to non-detections of nonlinearities. We argue that reliable surrogates can only be generated, if one tests separately for static and dynamic nonlinearities.
We present a model-independent investigation of the WMAP data with respect to scale- dependent non-Gaussianities (NGs) by employing the method of constrained randomization. For generating so-called surrogate maps a shuffling scheme is applied to the
Fourier phases of the original data, which allows to test for the presence of higher order correlations (HOCs) on well-defined scales. Using scaling indices as test statistics we find highly significant signatures for non-Gaussianities when considering all scales. We test for NGs in the bands l = [2,20], l = [20,60], l = [60,120] and l = [120,300]. We find highly significant signatures for non-Gaussianities and ecliptic hemispherical asymmetries for l = [2, 20]. We also obtain highly significant deviations from Gaussianity for the band l = [120,300]. The result for the full l-range can be interpreted as a superposition of the signatures found in the bands l = [2, 20] and l = [120, 300]. We find remarkably similar results when analyzing different ILC-like maps. We perform a set of tests to investigate if the detected anomalies can be explained by systematics. While no test can convincingly rule out the intrinsic nature of the anomalies for the low l case, the ILC map making procedure and/or residual noise in the maps can also lead to NGs at small scales. Our investigations prove that there are phase correlations in the WMAP data of the CMB. In the absence of an explanation in terms of Galactic foregrounds or known systematic artefacts, the signatures at low l must so far be taken to be cosmological at high significance. These findings strongly disagree with predictions of isotropic cosmologies with single field slow roll inflation. The task is now to elucidate the origin of the phase correlations and to understand the physical processes leading to these scale-dependent non-Gaussianities - if systematics as cause for them must be ruled out.
Probing Gaussianity represents one of the key questions in modern cosmology, because it allows to discriminate between different models of inflation. We test for large-scale non-Gaussianities in the cosmic microwave background (CMB) in a model-indepe
ndent way. To this end, so-called first and second order surrogates are generated by first shuffling the Fourier phases belonging to the scales not of interest and then shuffling the remaining phases for the length scales under study. Using scaling indices as test statistics we find highly significant signatures for both non-Gaussianities and asymmetries on large scales for the WMAP data of the CMB. We find remarkably similar results when analyzing different ILC-maps based on the WMAP five and seven year data. Such features being independent from the map-making procedure would disfavor the fundamental principle of isotropy as well as canonical single-field slow-roll inflation - unless there is some undiscovered systematic error in the collection or reduction of the CMB data or yet unknown foreground contributions.
We present a model-independent method to test for scale-dependent non-Gaussianities in combination with scaling indices as test statistics. Therefore, surrogate data sets are generated, in which the power spectrum of the original data is preserved, w
hile the higher order correlations are partly randomised by applying a scale-dependent shuffling procedure to the Fourier phases. We apply this method to the WMAP data of the cosmic microwave background (CMB) and find signatures for non-Gaussianities on large scales. Further tests are required to elucidate the origin of the detected anomalies.
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.
