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We present a model-independent investigation of the Wilkinson Microwave Anisotropy Probe (WMAP) data with respect to scale-independent and scale-dependent non-Gaussianities (NGs). To this end, we employ the method of constrained randomization. For generating so-called surrogate maps a well-specified shuffling scheme is applied to the Fourier phases of the original data, which allows us to test for the presence of higher order correlations (HOCs) also and especially on well-defined scales. Using scaling indices as test statistics for the HOCs in the maps we find highly significant signatures for NGs when considering all scales. We test for NGs in four different l-bands Δl, namely in the bands Δl=[2, 20], [20, 60], [60, 120] and [120, 300]. We find highly significant signatures for both NGs and ecliptic hemispherical asymmetries for the interval Δl=[2, 20] covering the large scales. We also obtain highly significant deviations from Gaussianity for the band Δl=[120, 300]. The result for the full l-range can then easily be interpreted as a superposition of the signatures found in the bands Δl=[2, 20] and [120, 300]. We find remarkably similar results when analysing different ILC-like maps based on the WMAP 3-, 5- and 7-year data. We perform a set of tests to investigate whether and to what extent the detected anomalies can be explained by systematics. While none of these tests 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 cosmic microwave background. 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 would 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 NGs - if it turns out that systematics as a cause for them must be ruled out

We present further investigations of the Wilkinson Microwave Anisotropy Probe (WMAP) data by means of the Minkowski functionals and the scaling index method. In order to test for non-Gaussianities (NGs) with respect to scale-dependencies we use so-called surrogate maps, in which possible phase correlations of the Fourier phases of the original WMAP data and the simulations, respectively, are destroyed by applying a shuffling scheme to the maps. A statistical comparison of the original maps with the surrogate maps then allows to test for the existence of higher order correlations (HOCs) in the original maps, also and especially on well-defined Fourier modes.We calculate the σ-normalised deviation between the Minkowski functionals of original data and 500 surrogates for different hemispheres in the sky and find ecliptic hemispherical asymmetries between the northern and southern ecliptic sky. Using Minkowski functionals as an image analysis technique sensitive to HOCs we find deviations from Gaussianity in the WMAP data with an empirical probability p > 99.8% when considering the low-range with ∆ = [2, 20]. The analysis technique of the scaling indices leads to the same results for this interval with a slightly lower deviation but still at p > 99.8%. Although the underlying foreground reduction methods of the maps differ from each other, we find similar results for the WMAP seven-year ILC map and the WMAP seven-year (needlet-based) NILC map for deviations from Gaussianity in the low-range. Our results point once more to a cosmological nature of the signal. For a higher range with ∆ = [120, 300] the results differ between the two image analysis techniques and between the two maps which makes an intrinsic nature of the signal on this range less likely. When we decrease the size of the analysed sky regions for the low-study, we do not find signatures of NG in the northern ecliptic sky. In the south we find individual spots which show deviations from Gaussianity.In addition, we investigate non-Gaussian CMB simulations that depend on the f NL -parameter of the local type. These simulations with f local NL = [0, ±100, ±1000] cannot account for the detected signatures on the low-range.

Pearson correlation and mutual information-based complex networks of the day-to-day returns of U.S. S&P500 stocks between 1985 and 2015 have been constructed to investigate the mutual dependencies of the stocks and their nature. We show that both networks detect qualitative differences especially during (recent) turbulent market periods, thus indicating strongly fluctuating interconnections between the stocks of different companies in changing economic environments. A measure for the strength of nonlinear dependencies is derived using surrogate data and leads to interesting observations during periods of financial market crises. In contrast to the expectation that dependencies reduce mainly to linear correlations during crises, we show that (at least in the 2008 crisis) nonlinear effects are significantly increasing. It turns out that the concept of centrality within a network could potentially be used as some kind of an early warning indicator for abnormal market behavior as we demonstrate with the example of the 2008 subprime mortgage crisis. Finally, we apply a Markowitz mean variance portfolio optimization and integrate the measure of nonlinear dependencies to scale the investment exposure. This leads to significant outperformance as compared to a fully invested portfolio.