Joachim Peinke

ANALYSIS OF DATA SETS OF STOCHASTIC SYSTEMS

Abstract This paper deals with the analysis of data sets of stochastic systems which can be described by a Langevin equation. By the method presented in this paper drift and diffusion terms of the corresponding Fokker-Planck equation can be extracted from the noisy data sets, and deterministic laws and fluctuating forces of the dynamics can be identified. The method is validated by the application to simulated one- and two-dimensional noisy data sets.

How to quantify deterministic and random influences on the statistics of the foreign exchange market

It is shown that price changes of the U.S. dollar-German mark exchange rates upon different delay times can be regarded as a stochastic Marcovian process. Furthermore, we show how Kramers-Moyal coefficients can be estimated from the empirical data. Finally, we present an explicit Fokker-Planck equation which models very precisely the empirical probability distributions, in particular, their non-Gaussian heavy tails.

Extracting model equations from experimental data

This letter wants to present a general data-driven method for formulating suitable model equations for nonlinear complex systems. The method is validated in a quantitative way by its application to experimentally found data of a chaotic electric circuit. Furthermore, the results of an analysis of tremor data from patients suffering from Parkinsons disease, from essential tremor, and from normal subjects with physiological tremor are presented, discussed and compared. They allow a distinction between the different forms of tremor.

On the statistics of wind gusts

Wind measurementsnear the North Seaborder of Northern Germany andvelocity measurements under localisotropic conditions of a turbulent wakebehind a cylinder are compared. It is shownthat wind gusts, measured by means ofvelocity increments, do show similar statisticsto the laboratory data if they are conditionedon an averaged wind speed value.Clear differences between the laboratory dataand the atmospheric measurements arefound for the waiting time statistics betweensuccessive gusts above acertain threshold.

Statistical properties of a turbulent cascade

Abstract Statistical properties of a turbulent cascade are evaluated by considering the joint probability distribution p(ν1, L1; ν2, L2) for two velocity increments ν1, ν2 of different length scales L1, L2. We present experimental evidence that the conditional probability distribution p(ν2, L2|ν1, L1) obeys a Chapman-Kolmogorov equation. We evaluate the Kramers-Moyal coefficient and show evidence that higher-order coefficients vanish except for the drift and diffusion coefficient. As a result the joint probability distributions obeys a Fokker-Planck equation. We calculate drift and diffusion coefficients and discuss their relationship to universal behaviour in the scaling region and to intermittency of the turbulent cascade.

On a quantitative method to analyze dynamical and measurement noise

This letter reports on a new method of analysing experimentally gained time series with respect to different types of noise involved, namely, we show that it is possible to differentiate between dynamical and measurement noise. This method does not depend on previous knowledge of model equations. For the complicated case of a chaotic dynamics spoiled at the same time by dynamical and measurement noise, we even show how to extract from data the magnitude of both types of noise. As a further result, we present a new criterion to verify the correct embedding for chaotic dynamics with dynamical noise.

How to improve the estimation of power curves for wind turbines

We introduce a dynamical approach for the determination of power curves for wind turbines and compare it with two common methods—among them the standard procedure due to IEC 61400-12-1, i.e. the international standard prepared and published by the International Electrotechnical Commission. The main idea of the new method is to separate the dynamics of a wind turbines power output into a deterministic and a stochastic part, corresponding to the actual behaviour of the wind turbine and external influences such as the turbulence of the wind, respectively. In particular, the governing coefficients are reconstructed from the data, and the power characteristic is extracted as the stationary states of the deterministic behaviour. Our results prove that a dynamical approach enables one to grasp the actual conversion dynamics of a wind turbine and to gain most accurate results for the power curve, independent of site-specific influences.

Evidence of Markov properties of high frequency exchange rate data

We present a stochastic analysis of a data set consisting of 106 quotes of the US Dollar–German Mark exchange rate. Evidence is given that the price changes x(τ) upon different delay times τ can be described as a Markov process evolving in τ. Thus, the τ-dependence of the probability density function (pdf) p(x,τ) on the delay time τ can be described by a Fokker–Planck equation, a generalized diffusion equation for p(x,τ). This equation is completely determined by two coefficients D1(x,τ) and D2(x,τ) (drift- and diffusion coefficient, respectively). We demonstrate how these coefficients can be estimated directly from the data without using any assumptions or models for the underlying stochastic process. Furthermore, it is shown that the solutions of the resulting Fokker–Planck equation describe the empirical pdfs correctly, including the pronounced tails.

Mapping stochastic processes onto complex networks

We introduce a method by which stochastic processes are mapped onto complex networks. As examples, we construct the networks for such time series as those for free-jet and low-temperature helium turbulence, the German stock market index (the DAX), and white noise. The networks are further studied by contrasting their geometrical properties, such as the mean length, diameter, clustering, and average number of connections per node. By comparing the network properties of the original time series investigated with those for the shuffled and surrogate series, we are able to quantify the effect of the long-range correlations and the fatness of the probability distribution functions of the series on the networks constructed. Most importantly, we demonstrate that the time series can be reconstructed with high precision by means of a simple random walk on their corresponding networks.

Comment on “Indispensable Finite Time Corrections for Fokker-Planck Equations from Time Series Data”

A Comment on the Letter by Mario Ragwitz and Holger Kantz, Phys. Rev. Lett. 87, 254501 (2001). The authors of the Letter offer a Reply.