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Universität Karlsruhe (TH)
Institut für Industrielle Informationstechnik

Parameterschätzung
und Chaostheorie
-
digitale
Verarbeitung
biologischer Signale

cand. el. Torsten Ziegler

Betreuer:	Dipl.-Ing. Stephan Schulz
	Dipl.-Ing. Andreas Schwarzhaupt

Universität Karlsruhe (TH)
Institut für Industrielle Informationstechnik

Parameterschätzung und Chaostheorie
-
digitale Verarbeitung
biologischer Signale

Torsten Ziegler

geb. am 27. März 1971 in Karlsruhe

Mtr.Nr.:	0756787
Adresse:	Frauenalberstr. 33
	76199 Karlsruhe

Januar - Juli 1998, Karlsruhe

Zusammenfassung:

In dieser Diplomarbeit zum Thema ,,Parameterschätzung und Chaostheorie - digitale Verarbeitung biologischer Signale``werden verschiedene nichtlineare Signalverarbeitungsmethoden dargestellt und ihre praktische Anwendbarkeit in der Analyse von Signalen aus physiologischen Versuchen untersucht. Es handelt sich dabei vornämlich um Methoden und Maße die aus der Chaostheorie bekannt sind. Ziel der Analyse ist die Systemidentifikation und das Erkennen verschiedener physiologischer Zustände.

Die Maße der Chaostheorie, sind die Dimensionalität und die Lyapunov Exponenten eines Systemes. Diese Maße sind vorallem deshalb interessant, da sie es erlauben ohne vorherige Modellbildung für ein System Parameter zu bestimmen, falls das System deterministisches Chaos aufzeigt. Zum Test, ob diese Voraussetzung erfüllt ist, wird die Methode des Surrogate Data benutzt. Sie stellt einen statistischen Nullhypothesentest dar, der dazu benutzt wird die tatsächliche Herkunft der Signale zu bestimmen.

Zur Erzeugung von Datensätzen, die bestimmten Nullhypothesen entsprechen wird ein Verfahren vorgestellt, das auf der Fouriertransformation basiert und numerisch sehr stabil ist.

Neben den Methoden der Chaostheorie werden als weitere Maße die Recurrence Plots dargestellt, die sich als besonders nützlich in der Analyse physiologischer Signale erwiesen haben und keinerlei Voraussetzungen an das zu analysierende Signal stellen.

Diese Arbeit beschäftigt sich zuerst mit der Chaostheorie, mit dem Begriff deterministisches Chaos und den Maßen der Dimensionalität und der Sensitivität. Dann werden die Recurrence Plots mit ihren Maßen eingeführt. Die Anwendung dieser Parameter in der Systemidentifikation wird besprochen und auf ihre Probleme Bezug genommen. Der theoretische Teil schließt mit der Vorstellung von Testmethoden, die eine Validierung von Analyseergebnissen ermöglichen.

Nach der Darstellung einer Reihe von bekannten und neuen Algorithmen zur Ermittlung der zuvor besprochenen Parameter wird ein Computerprogramm vorgestellt, das zur Analyse von Daten mit den vorgestellten Methoden dient. Neben der Vorstellung der graphischen Benutzeroberfläche wird anhand einer Beispielsitzung die Handhabung und die Interpretation der Ergebnisse erläutert.

Schließlich werden Ergebnisse dargestellt, die sich bei der Analyse von Meßdaten aus Tierversuchen unter Benutzung des vorgestellen Programmes ergaben.

Im praktischen Teil dieser Arbeit konnte gezeigt werden, daß Signale aus physiologischem Ursprung, im besonderen Fall Signale aus dem cardiovasculären System von Versuchstieren, nichtlinear und der Analyse mit Methoden der Chaostheorie zugänglich sind. Als sehr positiv hat sich die Anwendung der Recurrence Plots zur dynamischen Analyse der Signale gezeigt, hier konnten interessante Ergebnisse erzielt werden, die einen Ausblick auf einen zukünftigen Einsatz dieser Analysemethode im klinischen Bereich versprechen.

Zusammenfassung:

This diploma thesis is about the ``Estimation of Parameters and the Chaos Theory - Digital Processing of Biological Signals``.

In this work methods derived from the chaos theory like the estimation of dimensions or lyapunov exponents, as well as some other methods like recurrence plot strategies are discussed and evaluated upon their use in analysing signals from physiological systems. The main interest in this is systemidentification and the possibility to detect changing conditions in biological systems.

For a long time linear analysis techniques have been used in observing biological systems. The analysis was based on linearity and stationarity. This although it is known that a lot of physiological regulation is based on nonlinearities. The different states that can be observed in these systems, like beeing asleep or awake, with their different organic functionalities and foremost the instabilities causing the transitions between these states are a sign of their nonlinear nature. A linear system would reach some singular point if it were to change its state but nonlinear systems can reach another quasi stationar point after a transition. These instabilities are due to the combination of high gains and long reaction times in regulation loops.

Analyses of measured data from animal experiments showed, that the analytic thechniques derived from the chaos theory can show these changing conditions although the analysis is rather sensitive to the quality of the measured signals. On the other hand analysis using the recurrence plot startegies showed good results not only in determining changing states but also in tracking these changes themselfes.

The main concern in chaos theory is the spatial correlation of points in the phase space. A phase space is constructed either by the indipendent variables of a system or by a phase space reconstruction. This reconstruction can be achieved through assigning a scalar observable and time delayed versions of this observable to the coordinate axes of the phase space. As the reconstruction theorem by Takens et al. prooves, such a reconstructed space is identical to the original one except for a smooth coordinate transformation, if the dimensionality of the reconstructed phase space is at least twice that of the original dynamics.

Given such a phase space the system shows a trajectory as it evolves in time. In case of a chaotic system this is called an attractor.

The term dimensionality describes measures that address the distribution of attractor points in small local volumes of the phase space. While for non chaotic systems the distribution of points is kind of uniform in higher dimensions, for chaotic systems the amount of points in a certain volume is restricted and the dimension measures converge to an upper limit. This is due to the fractal structure of strange attractors, as we observe self similarity and forbidden zones in the attractor.

A second measure of the chaos theory are the Lyapunov exponents. As chaotic systems are nonlinear any small initial difference between some starting conditions will be magnified exponentially in time. This behaviour of instability is measured with the Lyapunov exponents. And the existance of at least one positive exponent is said to be a sure sign for chaotic behaviour.

The advantage of these measures is that they can be calculated without much knowledge of the signal and its underlying system. No modell building is needed, the only demand is that the signal emerges from a chaotic system. On the other hand the problem with measures of chaos theory is that their calculation will give some results no matter if the use of the calculation is justified or not. Therefor, to check the validity of a calculation, the method of surrogate data is used. This method constructs data with similar properties as the measured signal but with the difference that the newly constructed surrogate data corresponds to some null hypothesis. By using a special null hypothesis, like that the data emerged from a set of independent and identically distributed random variables, it is possible to compute the significance of the original analysis. For this the measures of chaos theory are calculated on both, the measured signal as well as on the created surrogate data. The comparison of the results and error bars derived from the calculations on several surrogate data sets lead to evidence for the analysis.

Recurrence plot strategies offer some measures that are not restricted to chaotic systems. These plots analyse a combination of the spatial and the time correlation of points in the phase space. Recurrence plots are constructed by plotting a matrix of points where the position of each point is determined by the time indizes of orbit points that are near to each other. This reveals a typical structure of points for a given system. Recurrence plots provide an analysis for stationarity by just looking at the plot. And second, the analysis of the structure of the plot quantifies measures as the determinism of the system or the complexity of this deterministic part.

Part of this diploma thesis has been the development of a computer program named Chaos Analyse Tool that provides an easy access to the above mentioned analysis techniques.

This program and the methods of the chaos theory have been applied to experimental data. The data are measurements in the cardiovascular system of pigs, i.e. the left ventricular preasure, the aortic preasure and the aortic flow. This data has been recorded during rest and during an occlusion of the descending aorta. The analysis of these measurements showed a rising dimensionality during the occlusion. This points to more active regulation mechanisms than in rest. Parameters derived from recurrence plots that measure the spatial periodicity and the detaerminism of the signal showed a very transient behaviour. The organism's reaction onto the occlusion led to a period of transition that is dominated by huge oscillations, the parameters show overshoot and under swing.

The analysis of experimental data showed that the methods of chaos theory can determine changing conditions in physilogical systems. Recurrence plot strategies can also track these changes and give usefull insight in the reaction of an organism to changing conditions. Next is to qualify these changing parameters with physiological meanings to use them in the clinical situation.