Estimation of the direction of the coupling by conditional probabilities of recurrence

Identifying hidden common causes from bivariate time series: a method using recurrence plots

We propose a method for inferring the existence of hidden common causes from observations of bivariate time series. We detect related time series by excessive simultaneous recurrences in the corresponding recurrence plots. We also use a noncoverage property of a recurrence plot by the other to deny the existence of a directional coupling. We apply the proposed method to real wind data.

Estimating causal dependencies in networks of nonlinear stochastic dynamical systems

The inference of causal interaction structures in multivariate systems enables a deeper understanding of the investigated network. Analyzing nonlinear systems using partial directed coherence requires high model orders of the underlying vector-autoregressive process. We present a method to overcome the drawbacks caused by the high model orders. We calculate the corresponding statistics and provide a significance level. The performance is illustrated by means of model systems and in an application to neurological data.

Physics of brain dynamics: Fokker-Planck analysis reveals changes in EEG delta and theta activity during anaesthesia

We use drift and diffusion coefficients to reveal interactions between different oscillatory processes underlying a complex signal and apply the method to EEG delta and theta frequencies in the brain. By analysis of data recorded from rats during anaesthesia we consider the stability and basins of attraction of fixed points in the phase portrait of the deterministic part of the retrieved stochastic process. We show that different classes of dynamics are associated with deep and light anaesthesia, and we demonstrate that the predominant directionality of the interaction is such that theta drives delta.

Reliable detection of directional couplings using rank statistics

To detect directional couplings from time series various measures based on distances in reconstructed state spaces were introduced. These measures can, however, be biased by asymmetries in the dynamics' structure, noise color, or noise level, which are ubiquitous in experimental signals. Using theoretical reasoning and results from model systems we identify the various sources of bias and show that most of them can be eliminated by an appropriate normalization. We furthermore diminish the remaining biases by introducing a measure based on ranks of distances. This rank-based measure outperforms existing distance-based measures concerning both sensitivity and specificity for directional couplings. Therefore, our findings are relevant for a reliable detection of directional couplings from experimental signals.

From Granger causality to long-term causality: application to climatic data

Quantitative characterization of interaction between processes from time series is often required in different fields of natural science including geophysics and biophysics. Typically, one estimates "short-term" influences, e.g., the widely used Granger causality is defined via one-step-ahead predictions. Such an approach does not reveal how strongly the "long-term" behavior of one process under study is affected by the others. To overcome this problem, we introduce the concept of long-term causality, which extends the concept of Granger causality. The long-term causality is estimated from data via empirical modeling and analysis of model dynamics under different conditions. Apart from mathematical examples, we apply both approaches to find out how strongly the global surface temperature (GST) is affected by variations in carbon dioxide atmospheric content, solar activity, and volcanic activity during the last 150 years. Influences of all the three factors on GST are detected with the Granger causality. However, the long-term causality shows that the rise in GST during the last decades can be explained only if the anthropogenic factor (CO2) is taken into account in a model.

Inferring direct directed-information flow from multivariate nonlinear time series

Estimating the functional topology of a network from multivariate observations is an important task in nonlinear dynamics. We introduce the nonparametric partial directed coherence that allows disentanglement of direct and indirect connections and their directions. We illustrate the performance of the nonparametric partial directed coherence by means of a simulation with data from synchronized nonlinear oscillators and apply it to real-world data from a patient suffering from essential tremor.

Detection of couplings in ensembles of stochastic oscillators

The problem of detection and quantitative characterization of directional couplings in an ensemble of noisy oscillators from a time series is addressed. We suggest estimators for the strengths of couplings which are based on modeling the observed oscillations with a set of stochastic phase oscillators and easily interpreted from a physical viewpoint. Moreover, we present an analytic formula for a statistical significance level allowing to reveal an architecture of couplings reliably from a relatively short time series. The technique applies to weakly coupled nonsynchronized oscillators. It is introduced for oscillators with close basic frequencies but can be readily generalized to the case of arbitrary frequencies. Efficiency of the technique is demonstrated in numerical experiments.

Hypothesis test for synchronization: twin surrogates revisited

The method of twin surrogates has been introduced to test for phase synchronization of complex systems in the case of passive experiments. In this paper we derive new analytical expressions for the number of twins depending on the size of the neighborhood, as well as on the length of the trajectory. This allows us to determine the optimal parameters for the generation of twin surrogates. Furthermore, we determine the quality of the twin surrogates with respect to several linear and nonlinear statistics depending on the parameters of the method. In the second part of the paper we perform a hypothesis test for phase synchronization in the case of experimental data from fixational eye movements. These miniature eye movements have been shown to play a central role in neural information processing underlying the perception of static visual scenes. The high number of data sets (21 subjects and 30 trials per person) allows us to compare the generated twin surrogates with the "natural" surrogates that correspond to the different trials. We show that the generated twin surrogates reproduce very well all linear and nonlinear characteristics of the underlying experimental system. The synchronization analysis of fixational eye movements by means of twin surrogates reveals that the synchronization between the left and right eye is significant, indicating that either the centers in the brain stem generating fixational eye movements are closely linked, or, alternatively that there is only one center controlling both eyes.

Epilepsy and nonlinear dynamics

This overview summarizes findings obtained from analyzing electroencephalographic (EEG) recordings from epilepsy patients with methods from the theory of nonlinear dynamical systems. The last two decades have shown that nonlinear time series analysis techniques allow an improved characterization of epileptic brain states and help to gain deeper insights into the spatial and temporal dynamics of the epileptic process. Nonlinear EEG analyses can help to improve the evaluation of patients prior to neurosurgery, and with an unequivocal identification of precursors of seizures, they can be of great value in the development of seizure warning and prevention techniques.

Symbolic transfer entropy

We propose to estimate transfer entropy using a technique of symbolization. We demonstrate numerically that symbolic transfer entropy is a robust and computationally fast method to quantify the dominating direction of information flow between time series from structurally identical and nonidentical coupled systems. Analyzing multiday, multichannel electroencephalographic recordings from 15 epilepsy patients our approach allowed us to reliably identify the hemisphere containing the epileptic focus without observing actual seizure activity.

Measuring interdependences in dissipative dynamical systems with estimated Fokker-Planck coefficients

We propose a data-driven approach to measure interdependences between dissipative dynamical systems under the influence of noise. We estimate drift and diffusion coefficients of a Fokker-Planck equation and derive measures that allow one to quantify the asymmetry in coupling in a fully automated and computationally inexpensive and simple way. Our approach makes it possible to discriminate between interdependences in the deterministic and stochastic parts of the dynamics. We report results of numerical studies of exemplary time series from coupled stochastic and deterministic model systems and of an application to electroencephalographic recordings from epilepsy patients.

Distinguishing a driver from a response system

This study presents an approach for distinguishing a driver from a response system. The proposed method can be applied to both unidirectional or bidirectional interactions, and to identical or structurally different systems. Compared with most previously proposed schemes, the present method is so "simple" that the driver-response relationships can generally be detected using a direct graphic way. On the other hand, quantitative estimation is also developed using the idea of the correlation dimension.