R. Friedrich

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.

A theoretical model of phase transitions in the human brain

An experiment using a multisensor SQUID (superconducting quantum interference device) array was performed by Kelso and colleagues (1992) which combined information from three different sources: perception, motor response, and brain signals. When an acoustic stimulus frequency is changed systematically, a spontaneous transition in coordination occurs at a critical frequency in both motor behavior and brain signals. Qualitatively analogous transitions are known for physical and biological systems such as changes in the coordination of human hand movements (Kelso 1981, 1984). In this paper we develop a theoretical model based on methods from the interdisciplinary field of synergetics (Haken 1983, 1987) and nonlinear oscillator theory that reproduces the main experimental features very well and suggests a formulation of a fundamental biophysical coupling.

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.

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.

On the velocity distribution in homogeneous isotropic turbulence: correlations and deviations from Gaussianity

We investigate the single-point probability density function of the velocity in threedimensional stationary and decaying homogeneous isotropic turbulence. To this end, we apply the statistical framework of the Lundgren–Monin–Novikov hierarchy combined with conditional averaging, identifying the quantities that determine the shape of the probability density function. In this framework, the conditional averages of the rate of energy dissipation, the velocity diffusion and the pressure gradient with respect to velocity play a key role. Direct numerical simulations of the Navier–Stokes equation are used to complement the theoretical results and assess deviations from Gaussianity.

Reconstruction of the spatio-temporal dynamics of a human magnetoencephalogram

Abstract We reconstruct the entire experimentally observed spatio-temporal signal of a human magnetoencephalogram (MEG) observed in a sensori-motor-coordination experiment by Kelso et al. In this experiment, when an acoustic stimulus frequency is changed systematically, a spontaneous transition in coordination occurs at a critical frequency in both the motor behavior and brain signals. Here we present a stepwise approach for the reconstruction of the spatio-temporal signal: First, we identify the order parameters and recall a theoretical model by the present authors and Kelso which reproduces the temporal dynamics of the order parameters. Second, we use the variational method by Uhl et al. in order to determine the spatial modes of the order parameters. Third, we present a variational method for the reconstruction of the remaining spatio-temporal signal and determine the spatial modes and temporal dynamics of the enslaved variables and possible order parameter modifications. The obtained set of spatial modes proves to be a fixed spatial base system of the observed temporal dynamics in the brain.

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.