This article reports on a new approach to properly analyze time series of dynamical systems which are spoilt by the simultaneous presence of dynamical noise and measurement noise. It is shown that even strong external measurement noise as well as dynamical noise which is an intrinsic part of the dynamical process can be quantified correctly, solely on the basis of measured times series and proper data analysis. Finally real world data sets are presented pointing out the relevance of the new approach.

Atmospheric wind speeds and their fluctuations at different locations (onshore and offshore) are examined. One of the most striking features is the marked intermittency of probability density functions (PDF) of velocity differences -no matter what location is considered. The shape of these PDFs is found to be robust over a wide range of scales which seems to contradict the mathematical concept of stability where a Gaussian distribution should be the limiting one. Motivated by the instationarity of atmospheric winds it is shown that the intermittent distributions can be understood as a superposition of different subsets of isotropic turbulence. Thus we suggest a simple stochastic model to reproduce the measured statistics of wind speed fluctuations.

It is a big challenge in the analysis of experimental data to disentangle the unavoidable measurement noise from the intrinsic dynamical noise. Here we present a general operational method to extract measurement noise from stochastic time series even in the case when the amplitudes of measurement noise and uncontaminated signal are of the same order of magnitude. Our approach is based on a recently developed method for a nonparametric reconstruction of Langevin processes. Minimizing a proper non-negative function, the procedure is able to correctly extract strong measurement noise and to estimate drift and diffusion coefficients in the Langevin equation describing the evolution of the original uncorrupted signal. As input, the algorithm uses only the two first conditional moments extracted directly from the stochastic series and is therefore suitable for a broad panoply of different signals. To demonstrate the power of the method, we apply the algorithm to synthetic as well as climatological measurement data, namely, the daily North Atlantic Oscillation index, shedding light on the discussion of the nature of its underlying physical processes.

No abstract

The meaning of non-Gaussian statistics of disordered systems is discussed. The risks of heavy tailed probability distributions of wind gusts are presented together with the discussion of an eventual power law behavior of such heavy tailed distributions. We summarize the argumentation connecting such probability distributions with Levy processes and the meaning of non-existing moments. Based on turbulence and financial market data, stochastic cascade processes are presented, which lead to anomalous statistics with heavy tails, too. Finally we show how for disordered complex systems with an underlying hierarchical structure, like the cascade structure in turbulence, a stochastic equation can be estimated directly from the data. The numerical solution of this reconstructed stochastic process gives back the anomalous statistics.