Topics in Nonlinear Time Series Analysis: With Implications for EEG Analysis

360 Pages·2000·27 MB·English

by  Andreas Galka

A D V A N C ED S E R I ES IN
N O N L I N E AR D Y N A M I CS
V O L U ME 14
il
OPICS IN
NONLINEAR T ME
SERIES ANALYSIS
With Implications for EEG Analysis
Andreas Galka
World Scientific
TOPICS IN
NONLINEAR TIME
SERIES ANALYSIS
ADVANCED SERIES IN NONLINEAR DYNAMICS
Editor-in-Chief: R. S. MacKay (Cambridge)
Published
Vol. 1 Dynamical Systems
ed. Ya G. Sinai
Vol. 2 Statistical Mechanics
ed. Ya G. Sinai
Vol. 3 Topics in Bifurcation Theory & Application
G. looss & M. Adelmeyer
Vol. 4 Hamiltonian Systems & Celestial Mechanics
eds. J. Llibre & E. A. Lacomba
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L. Alseda at al.
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R. S. MacKay
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ed. A. Mielke et al.
Vol. 8 New Trends for Hamiltonian Systems & Celestial Mechanics
eds. J. Llibre & E. Lacomba
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S. Benkadda, F. Doveil & Y. Elskens
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Y.-P. Jiang
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A. V. Getting
Vol. 12 Localization and Solitary Waves in Solid Mechanics
A. R. Champneys, G. W. Hunt & J. M. T. Thompson
Vol. 13 Time Reversibility, Computer Simulation, and Chaos
W. G. Hoover
Vol. 14 Topics in Nonlinear Time Series Analysis - With Implications for EEG Analysis
A. Galka
Forthcoming
Symplectic Twist Maps
C. Gole
Wave Collapse
E. A. Kuznetsov & V. E. Zakharov
Positive Transfer Operators and Decay of Correlations
V. Baladi
Methods in Equivariant Bifurcation and Dynamical Systems, with Applications
P. Chossat & R. Lauterbach
Combinatorial Dynamics and Entropy in Dimension One - 2nd Edition
L. Alsedb, J. Llibre & M. Misiurewicz
ADVANCED SERIES I N
NONLINEAR D YNAMIC S
V 0 L U M E 1 4
TOPICS IN
NONLINEAR TIME
SERIES ANALYSIS
With Implications for EEG Analysis
Andreas Galka
Christian-Albrechts-University of Kiel, Germany
`A World Scientific
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TOPICS IN NONLINEAR TIME SERIES ANALYSIS
With Implications for EEG Analysis
Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd.
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This page is intentionally left blank
Preface
This book is the result of my interest in the question, whether the reports
of low-dimensional deterministic dynamics in human electroencephalogram
(EEG) recordings which were published in increasing numbers since 1985
were indicating a serious new approach for medical diagnosis and brain
research. It is for this reason that the book has a twofold intention:
First, I have tried to give a review of a number of (more or less) new
techniques for nonlinear time series analysis which were developed in the
wake of the breakthrough of Nonlinear Dynamics in the Seventies of the
20th century, and secondly I have analysed (and, to some extent, advanced)
some of these techniques with a particular view to EEG time series analysis.
So I hope that the book will be useful both for the reader who is look-
ing for a general introduction into theory and application of some recently
introduced techniques for the analysis of complex dynamical data, and also
for the reader who is interested especially in the application of these tech-
niques to EEG time series analysis.
I acknowledge with gratitude the help that I have received from many
friends, colleagues and relatives in writing this book. In particular, I would
like to thank Stephan Wolff for many helpful discussions, Thorsten MaaB
for good cooperation and for the permission to include some of the results
of his diploma thesis [1381 into this book, Gunther Fritzer for providing the
EEG time series analysed in this study, Gerd Pfister and Ulrich Stephani for
putting forward the whole project, Daniel Kaplan for interesting discussions
and Oliver Malki for proofreading.
vii
viii Preface
How to read this book
A detailed outline of the organisation of this book will be given in section
1.4; here I would like to give some further advice to the reader.
Each chapter builds more or less on the preceding chapters, and readers
who have sufficient time at their disposal should simply follow the given
order; but it is also possible to confine the reading to certain chapters or
groups of chapters which provide reviews of specific subjects:
• Chapter 2 provides a basic introduction to dynamical systems and
time series.
• Chapters 4 and 5 provide a review of the important subject of state
space reconstruction, both from the theoretical and the applied
point of view. This review covers most of the relevant literature
published until 1998.
• Chapters 6, 8 and 9 deal with correlation dimension estimation,
again both from the theoretical and the applied point of view. Prac-
tical and theoretical aspects of this subject have been addressed in
virtually hundreds of papers by now, and it was my aim to condense
the results of this work into a useful review. Nevertheless I will al-
most certainly have missed some important contributions, I apolo-
gise to the authors (the same remark applies to the other reviews).
• Chapter 11 reviews the technique of surrogate data testing.
• The first part of chapter 13 summarises a number of techniques for
the detection of determinism in time series data.
I expect these reviews to be also useful for the reader who is not particularly
interested in EEG time series analysis. A review (including critical assess-
ment) of some recent approaches to EEG time series analysis can be found
in chapter 12. But there will also be occasional references to published
work on EEG time series in other chapters.
The remaining chapters contain additional material and some original
contributions, such as the Monte Carlo analysis in chapter 10 and the inter-
spike interval approach in chapter 13; some of these results and proposals
are to be regarded as rather tentative and much less well established than
the material presented in the review chapters. They are still subject of
ongoing work, and I would like to ask the reader to read and rate them
with indulgence.
A.Q.
Contents
Preface vii
Chapter 1 Introduction 1
1.1 Linearity and the beginning of time series analysis . . . . . . . 1
1.2 Irregular time series and determinism . . . . . . . . . . . . . . 3
1.3 The objective of nonlinear time series analysis . . . . . . . . . . 4
1.4 Outline of the organisation of the present study . . . . . . . . . 5
Chapter 2 Dynamical systems, time series and attractors 9
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Dynamical systems and state spaces . . . . . . . . . . . . . . . 9
2.3 Measurements and time series . . . . . . . . . . . . . . . . . . . 10
2.4 Deterministic dynamical systems . . . . . . . . . . . . . . . . . 12
2.4.1 Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.2 Linear systems . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.3 Invariant measures . . . . . . . . . . . . . . . . . . . . . 15
2.4.4 Sensitive dependence on initial conditions . . . . . . . . 18
2.4.5 Maps and discretised flows . . . . . . . . . . . . . . . . 19
2.4.6 Some important maps . . . . . . . . . . . . . . . . . . . 21
2.4.7 Some important flows . . . . . . . . . . . . . . . . . . . 24
2.5 Stochastic dynamical systems . . . . . . . . . . . . . . . . . . . 28
2.5.1 Pure noise time series . . . . . . . . . . . . . . . . . . . 29
2.5.2 Noise in dynamical systems . . . . . .. . . . . . . . .. 30
2.5.3 Linear stochastic systems . . . . . . . . . . . . . . . . . 31
2.6 Nonstationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
ix