Henning U. Voss

NONLINEAR DYNAMICAL SYSTEM IDENTIFICATION FROM UNCERTAIN AND INDIRECT MEASUREMENTS

We review the problem of estimating parameters and unobserved trajectory components from noisy time series measurements of continuous nonlinear dynamical systems. It is first shown that in parameter estimation techniques that do not take the measurement errors explicitly into account, like regression approaches, noisy measurements can produce inaccurate parameter estimates. Another problem is that for chaotic systems the cost functions that have to be minimized to estimate states and parameters are so complex that common optimization routines may fail. We show that the inclusion of information about the time-continuous nature of the underlying trajectories can improve parameter estimation considerably. Two approaches, which take into account both the errors-in-variables problem and the problem of complex cost functions, are described in detail: shooting approaches and recursive estimation techniques. Both are demonstrated on numerical examples.

The seizure prediction characteristic: a general framework to assess and compare seizure prediction methods.

The unpredictability of seizures is a central problem for all patients suffering from uncontrolled epilepsy. Recently, numerous methods have been suggested that claim to predict from the EEG the onset of epileptic seizures. In parallel, new therapeutic devices are in development that could control upcoming seizures provided that their onset is known in advance. A reliable clinical application controlling seizures, consisting of a seizure prediction method and an intervention system, would improve patient quality of life. The question therefore arises as to whether the performance of the seizure prediction methods is already sufficient for clinical applications. The answer requires assessment criteria to judge and compare these methods, but recognized criteria still do not exist. Based on clinical, behavioral, and statistical considerations, we suggest the seizure prediction characteristic to evaluate seizure prediction methods. Results of this approach are exemplified by its application to the dynamical similarity index seizure prediction method using 582 hours of intracranial EEG data, including 88 seizures.

Nonlinear analysis of complex phenomena in cardiological data

Zusammenfassung Das Hauptanliegen dieses Beitrages ist es, verschiedene Ansätze in der Herzfrequenz- und Blutdruckvariabilität zu diskutieren und damit das Verständnis der kardiovaskulären Regulation zu verbessern. Wir betrachten Komplexitätsmaße basierend auf der symbolischen Dynamik, die renormierte Entropie und die ,finite-time’ Wachstumsraten. Weiterhin werden die duale Sequenzmethode zur Bestimmung der Baroreflexsensitivität sowie die Maximalkorrelationsmethode zur Schätzung der nichtlinearen Kopplung in bivariaten Daten vorgestellt. Letztere stellt eine geeignete Methode zur Bestimmung der Kopplungsstärke und –richtung dar. Herzfrequenz- und Blutdruckvariabilitätsdaten einer klinischen Pilotstudie und einer großangelegten klinischen Studie werden analysiert. Wir demonstrieren in diesem Beitrag, dass Methoden der nichtlinearen Dynamik nützlich sind für die Risikostratifizierung nach Herzinfarkt, für die Vorhersage von lebensbedrohlichen Rhythmusstörungen sowie für die Modellierung der Herzfrequenz- und Blutdruckregulation. Diese Ergebnisse könnten in der klinischen Diagnostik sowie für therapeutische und präventive Zwecke von implantierbaren Defibrillatoren der nächsten Generation von Bedeutung sein.Summary The main intention of this contribution is to discuss different nonlinear approaches to heart rate and blood pressure variability analysis for a better understanding of the cardiovascular regulation. We investigate measures of complexity which are based on symbolic dynamics, renormalised entropy and the finite time growth rates. The dual sequence method to estimate the baroreflex sensitivity and the maximal correlation method to estimate the nonlinear coupling between time series are employed for analysing bivariate data. The latter appears to be a suitable method to estimate the strength of the nonlinear coupling and the coupling direction. Heart rate and blood pressure data from clinical pilot studies and from very large clinical studies are analysed. We demonstrate that parameters from nonlinear dynamics are useful for risk stratification after myocardial infarction, for the prediction of life-threatening cardiac events even in short time series, and for modelling the relationship between heart rate and blood pressure regulation. These findings could be of importance for clinical diagnostics, in algorithms for risk stratification, and for therapeutic and preventive tools of next generation implantable cardioverter defibrillators.

Parameter estimation in nonlinear delayed feedback systems from noisy data

Abstract We propose a method for the estimation of parameters of nonlinear delayed feedback systems from a time series. Being based on the multiple shooting approach it is fairly robust against high levels of observation noise and yields precise parameter estimates. We evaluate its performance using simulated data of the Mackey–Glass equation and present an application to observed time series of an electronic circuit with time delay.

REAL-TIME ANTICIPATION OF CHAOTIC STATES OF AN ELECTRONIC CIRCUIT

This Letter presents an experimental realization of a recently proposed method to anticipate future states of nonlinear time-delayed feedback systems. The electronic circuit allows for a real-time anticipation of even strongly irregular signals. It is found that synchronization of the driven circuit with chaotic future states of the driving circuit is insensitive to signal and system perturbations.

Linear and Nonlinear Time Series Analysis of the Black Hole Candidate Cygnus X-1

We analyze the variability in the x-ray lightcurves of the black hole candidate Cygnus X-1 by linear and nonlinear time series analysis methods. While a linear model describes the overall second order properties of the observed data well, surrogate data analysis reveals a significant deviation from linearity. We discuss the relation between shot noise models usually applied to analyze these data and linear stochastic autoregressive models. We debate statistical and interpretational issues of surrogate data testing for the present context. Finally, we suggest a combination of tools from linear and nonlinear time series analysis methods as a procedure to test the predictions of astrophysical models on observed data.

A backward time shift filter for nonlinear delayed-feedback systems

Abstract Using a simple nonlinear filter, it is possible to shift arbitrarily complex wave forms produced by systems with a delayed feedback backwards in time. Physically, this corresponds to a seemingly noncausal transmission of signals. Filter chains allow for signals travelling against the coupling direction of the chain. These apparent paradoxes are resolved, and possible physical implications are discussed.

Non-parametric identification of non-linear oscillating systems

The problem of system identification from a time series of measurements is solved by using non-parametric additive models. Having only few structural information about the system, a non-parametric approach may be more appropriate than a parametric one for which detailed prior knowledge is needed. Based on non-parametric regression, the functions in the additive models are estimated by a penalized least-squares approach using backfitting. The optimal smoothing parameters are determined via generalized cross-validation, making this approach completely adaptive to the data. The procedure is applied to identify the non-linear restoring force of vibrationally excited helical wire rope isolators.

Reconstruction of grand minima of solar activity from Δ14C data: Linear and nonlinear signal analysis

Using a special technique of data analysis, we have found 34 grand minima of solar activity in a 7700-year-long Δ14C record. The method used rests on a proper filtering of the Δ14C record and the extrapolation of verifiable results for the later history back in time. Additionally, we have applied a method of nonlinear dynamics, the recurrence rate, to back up the results. Our findings are not contradictory to the record of grand minima by Eddy [1977] but constitute a considerable extension. Hence it has become possible to look closer at the validity of models. This way, we have tested the model of Barnes et al. [1980]. There are hints for that the grand minima might solely be driven by the 209-year period found in the Δ14C record.

Classical Möbius-ring resonators exhibit fermion-boson rotational symmetry.

The behavior of coupled harmonic oscillators in systems with specified boundary conditions is typically characterized by resonances whose frequency spectra represent harmonics according to properties of the individual oscillators, the interactions between them, and the overall symmetry of the system. Here it is demonstrated that classical one- and two-dimensional radio frequency resonators constrained to a Möbius topology are the formal partners of cylindrical ring resonators, and specifically give rise to half-integral harmonic excitations that are orthogonal to the integral excitations of a ring. In particular, the half-integral harmonics are formally invariant under rotations at a minimum of 4pi rather than 2pi rad, in analogy to the rotational symmetry of fermions in quantum mechanics. The results offer a pathway for discovery in other physical systems as well as the design of novel materials and electronic instrumentation.