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Time series analysis of real-world complex systems [Elektronische Ressource] : climate, finance, proteins, and physiology / vorgelegt von Daniel T. Schmitt

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225 pages
ECOD·ODNEICS·MLUTÄTInstitut für Theoretische PhysikUniversität UlmTime Series Analysis of Real-worldComplex Systems — Climate, Finance,Proteins, and PhysiologyDissertationzur Erlangung des DoktorgradesDr. rer. nat.der Fakultät für Naturwissenschaftender Universität Ulmvorgelegt vonDaniel T. Schmittaus Frankfurt-HöchstUlm 2007ISREVINU·ODNARUC·ODNAmtierender Dekan: Prof. Dr. Klaus-Dieter Spindler1. Gutachter: Prof. Dr. Michael Schulz2. Gutachter: Prof. Dr. Peter Reineker3. Gutachter: Dr. Sc. Plamen Ch. IvanovTag der Promotion: 19.11.2007To everybody who believesin meand my abilitiesiiiCopyright 2007 Daniel T. SchmittAll Rights ReservedivAcknowledgementWe do not live, and even less do research, in a vacuum. This thesis hasbenefittedfrommanypeopledirectlyandindirectly. Itisimpossibletomentionall people by name — I apologize for that in advance.First, I would like to thank to my parents who trust me in my decisions,and always gave me the freedom to follow my dreams.My advisor Prof. Dr. Michael Schulz at the University of Ulm, Germany,was very resourceful providing guidance, as well as, freedom to foster the de-velopment of my ability to conduct independent research.In order to come to Boston University, USA, Prof. H. Eugene Stanley wasvery helpful arranging the formalities and I enjoyed working in the very re-sourceful environment of his research group full of very bright and energizedpeople.
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Institut für Theoretische Physik
Universität Ulm
Time Series Analysis of Real-world
Complex Systems — Climate, Finance,
Proteins, and Physiology
Dissertation
zur Erlangung des Doktorgrades
Dr. rer. nat.
der Fakultät für Naturwissenschaften
der Universität Ulm
vorgelegt von
Daniel T. Schmitt
aus Frankfurt-Höchst
Ulm 2007
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NAmtierender Dekan: Prof. Dr. Klaus-Dieter Spindler
1. Gutachter: Prof. Dr. Michael Schulz
2. Gutachter: Prof. Dr. Peter Reineker
3. Gutachter: Dr. Sc. Plamen Ch. Ivanov
Tag der Promotion: 19.11.2007To everybody who believes
in me
and my abilities
iiiCopyright 2007 Daniel T. Schmitt
All Rights Reserved
ivAcknowledgement
We do not live, and even less do research, in a vacuum. This thesis has
benefittedfrommanypeopledirectlyandindirectly. Itisimpossibletomention
all people by name — I apologize for that in advance.
First, I would like to thank to my parents who trust me in my decisions,
and always gave me the freedom to follow my dreams.
My advisor Prof. Dr. Michael Schulz at the University of Ulm, Germany,
was very resourceful providing guidance, as well as, freedom to foster the de-
velopment of my ability to conduct independent research.
In order to come to Boston University, USA, Prof. H. Eugene Stanley was
very helpful arranging the formalities and I enjoyed working in the very re-
sourceful environment of his research group full of very bright and energized
people.
I have learned a lot during my intense collaboration with Dr. Sc. Plamen
Ch. Ivanov, PhD at Boston University and at Harvard Medical School. He is
an extraordinary person and researcher — I really enjoyed working with him.
Money makes the world go around — and this definitely applies to even the
most detached ivory towers of research, nowadays. Therefore, most researchers
would probably agree that applying for (and worrying about) funding is the
most annoying part of their (research) life and I am no exception. Because of
that I am very thankful to the following funding agencies: The German Aca-
demic Exchange Service (DAAD) supported an initial short visit to Boston
University for 2 month. The Volkswagen Foundation funded my research stay
in Boston for more than one year. I am especially thankful for the unbureau-
cratic extension of their support for additional 3 month which helped me to
finish many started projects.
Finally, I would like to thank everybody who read manuscripts, provided
support, advise, interesting questions, discussions, food, or in general made
my life more enjoyable.
Boston, July 2007 Daniel T. Schmitt
vContents
I Introduction 1
1 Complex Systems 3
1.1 Deterministic and Stochastic Dynamics . . . . . . . . . . . . . 6
1.2 Statistical Physics in Physiology . . . . . . . . . . . . . . . . . 8
II Methods 11
2 Correlations 13
2.1 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Power Spectrum . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Scale-invariant Power-law Correlations . . . . . . . . . . . . . 16
2.3 Rescaled Range Analysis | R/S analysis | Hurst Exponent . . . 17
2.4 Fluctuation Analysis (FA) . . . . . . . . . . . . . . . . . . . . 19
2.5 Detrended Fluctuation Analysis (DFA) . . . . . . . . . . . . . 20
2.6ation (DFA) of Data with Gaps . . 22
2.7 DFA and Power Spectrum Analysis . . . . . . . . . . . . . . . 24
2.8 Magnitude and Sign Decomposition . . . . . . . . . . . . . . . 26
3 Fractal Dimension 29
3.1 The Concept of Dimension . . . . . . . . . . . . . . . . . . . . 29
3.2 Hausdorff-Besicovitch Dimension . . . . . . . . . . . . . . . . 29
3.3 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Fractal Dimension of a Time Series . . . . . . . . . . . . . . . 31
4 Memory-Kernel Analysis (MKA) 33
4.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.1 Surrogate Time Series . . . . . . . . . . . . . . . . . . 38
4.2 Master Technique . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Prediction/Forecasting . . . . . . . . . . . . . . . . . . . . . . 41
5 Control 47
viContents
5.1 Deterministic Control Theory . . . . . . . . . . . . . . . . . . 48
5.2 Control Theory non-local in Time . . . . . . . . . . . . . . . . 49
5.3 Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
III Applications 55
6 Climate Dynamics 57
6.1 Memory of the Wind . . . . . . . . . . . . . . . . . . . . . . . 57
7 Dynamical Measures of Climate Change 65
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 Applications/Results . . . . . . . . . . . . . . . . . . . . . . . 67
7.2.1 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . 67
7.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8 Single-Molecule Spectroscopy 79
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.2 Mori-Zwanzig-Theory . . . . . . . . . . . . . . . . . . . . . . . 82
8.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
8.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 87
9 Financial Markets 91
9.1 Financial Time Series . . . . . . . . . . . . . . . . . . . . . . . 91
9.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
10 Sleep-Stage Dynamics 101
10.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . 101
10.1.1 Healthy Sleep Regulation . . . . . . . . . . . . . . . . . 102
10.1.2 Sleep in Elderly . . . . . . . . . . . . . . . . . . . . . . 103
10.1.3 Sleep Research . . . . . . . . . . . . . . . . . . . . . . 103
10.2 Data and Methods . . . . . . . . . . . . . . . . . . . . . . . . 105
10.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
10.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
10.4.1 Self-organized Criticality (SOC) . . . . . . . . . . . . . 109
10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
11 Age-related Changes in Heart Rate Dynamics 113
11.1 The Electrocardiogram (ECG) . . . . . . . . . . . . . . . . . . 113
11.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
11.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
11.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
viiContents
11.4.1 Detrended Fluctuation Analysis (DFA) . . . . . . . . . 120
11.4.2 Magnitude and Sign Analyzes (MSA) . . . . . . . . . . 122
11.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
11.5.1 Variability in heartbeat intervals and their increments . 123
11.5.2 Fractal Correlations . . . . . . . . . . . . . . . . . . . . 126
11.5.3 Magnitude and sign scaling analysis (MSA) . . . . . . 127
11.5.4 Fractal Dimension Analysis . . . . . . . . . . . . . . . 137
11.6 Interpretation and Modeling . . . . . . . . . . . . . . . . . . . 139
11.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
12 Heart Rate Dynamics during Different Sleep Stages 151
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
12.2 Data and Methods . . . . . . . . . . . . . . . . . . . . . . . . 153
12.2.1 Subjects . . . . . . . . . . . . . . . . . . . . . . . . . . 153
12.2.2 Detrended Fluctuation Analysis (DFA) . . . . . . . . . 154
12.2.3 Magnitude and Sign Analyzes (MSA) . . . . . . . . . . 154
12.2.4 Data processing . . . . . . . . . . . . . . . . . . . . . . 155
12.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
12.3.1 Variability in Heartbeat Intervals and their Increments 158
12.3.2 Fractal Temporal Correlations . . . . . . . . . . . . . . 159
12.3.3 Magnitude and Sign Correlation . . . . . . . . . . . . . 161
12.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
12.4.1 Static measures . . . . . . . . . . . . . . . . . . . . . . 165
12.4.2 Dynamic measures . . . . . . . . . . . . . . . . . . . . 167
13 Conclusion 171
A Mori-Zwanzig Theory 173
B MZ Equation and ARCH-/GARCH 177
B.1 ARCH/GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . 177
B.2 Mori-Zwanzig Theory . . . . . . . . . . . . . . . . . . . . . . . 177
B.2.1 Fluctuaiton-Dissipation Theorem revisited . . . . . . . 178
B.3 ARCH/GARCH - Mori-Zwanzig Equation . . . . . . . . . . . 179
C Fractional Calculus and Memory Kernel of Power-law Form 183
Bibliography 185
viiiList of Figures
1.1 Evolution of the Lyapunov ellipse in a 2-dimensional phase
0 00
space with t<t <t (schematic illustration). . . . . . . . . . 8
2.1 Schematic illustration of the joint probability. . . . . . . . . . 14
2.2 Schematic of the Hurst reservoir. . . . . . . . . . 18
2.3 Sc illustration of the DFA analysis. . . . . . . . . . . 21
2.4 Schematic of the DFA-1 analysis when gaps are
present in the time series. . . . . . . . . . . . . . . . . . . . . 23
2.5 Profile of a fractal signal and its lowpass filtered version. . . . 26
3.1 Fractal dimension and the scaling of the length measure L(k). 32
4.1 Schematic illustration of the Mori-Zwanig equation with one
‘relevant’ observable. . . . . . . . . . . . . . . . . . . . . . . . 35
˜
4.2 Autocorrelation function,hx|xi, Memory kernel,K andK , oft t
the Ornstein-Uhlenbeck process. . . . . . . . . . . . . . . . . 39
4.3 Schematic illustration of the master procedure. . . . . . . . . 40
4.4 Measureoftheforecastqualityfortheexponentiallycorrelated
Orstein-Uhlenbeck process with different restoring forces Ω. . 42
4.5 Measure of the forecast quality for colored noises with different
fractal correlation exponents α. . . . . . . . . . . . . . . . . . 43
4.6 Measure of the forecast quality for Prague temperature fluc-
tuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.7 Measure of the forecast quality for the wind speed in Potsdam,
Germany. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
˜
6.1 Autocorrelation function,hx|xi, Memory kernel,K andK , oft t
the wind strength changes. . . . . . . . . . . . . . . . . . . . 60
6.2 Cumulated memory kernel, Q , of the wind strength changes.t
ThehorizontallinewasfittedtoQ intheintervalt∈ [500, 1000]t
The elementary time unit is 1 h. . . . . . . . . . . . . . . . . 61
˜
6.3 Autocorrelation function,hφ|φi, Memory kernel, K and K ,t t
of the wind direction changes. The elementary time unit is 1 h. 62
ixList of Figures
mastered
6.4 Cumulated mastered Memory kernel, Q , of the windt
direction changes. . . . . . . . . . . . . . . . . . . . . . . . . 63
7.1 Global Surface Air Temperature Anomaly in ℃. . . . . . . . 66
7.2 DFAscalingexponent,α, ofthewindstrengthchangesinPost-
dam, Germany. . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7.3 DFA-2 scaling exponent, α for the detrended hourly tempera-
ture changes at Mt. Washington, USA. . . . . . . . . . . . . . 69
7.4 DFA-2 scaling exponent, α, for the detrended temperatures
changes at the Prague Klementinum, Czech Republic. The
scaling function, F(n) was fitted in the interval n∈ [6, 70]. . . 70
7.5 Fractal dimension, D, of the wind strength in Postdam, Ger-
many. Two year segments of the hourly wind strength data are
analyzed. The scaling function,L(k), was fitted in the interval
k∈ [0, 9] to obtain the fractal dimension, D. . . . . . . . . . 71
7.6 Fractal Dimension, D, for the detrended hourly temperature
at Mt. Washington, USA. The hourly temperatures were de-
trended by subtracting the empirical seasonal trend. Segments
of 2years were analyzed. The scaling function,L(k) was fitted
in the interval k∈ [0, 20]. The error bars indicate the error of
the linear fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.7 Fractal Dimension, D, for the detrended temperatures at the
Prague Klementinum, Czech Republic. The scaling function,
L(k), was fitted in the interval k∈ [0, 16]. . . . . . . . . . . . 73
τP
7.8 Cumulated memory kernel, Q = K , of the wind strengthτ i
i=0
changes in Postdam, Germany. . . . . . . . . . . . . . . . . . 74
7.9 Selected contour lines from figure 7.8 fitted with straight lines. 75
7.10 Cumulated memory kernel, Q , of the detrended hourly tem-t
perature changes at Mt. Washington, USA. . . . . . . . . . . 76
7.11 Cumulated memory kernel, Q , of the detrended temperaturet
changes at the Prague Klementinum, Czech Republic. . . . . 76
7.12 Selected contour lines from Fig. 7.10 fitted with straight lines. 77
7.13 contour lines from Fig. 7.11 fitted witht lines. 77
8.1 Experimental data (Figures courtesy of J. Köhler). . . . . . . 80
8.2 Memory kernels, K, determined for 1552 wavenumbers of the
−1 −1
interval between 12200cm and 13000cm . . . . . . . . . . 86
8.3 Memory kernel, K, for the dark frequency f . The smoothnoise
exponential decay suggests dominating fast processes. . . . . 87
8.4 Memory kernels, K, for the frequencies f , f , f , f . Thenoise 1 2 3
inset shows the data with enlarged K-axis . . . . . . . . . . . 88
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