Modelling the variability of complex systems by means of Langevin processes [Elektronische Ressource] : on the application of a dynamical approach to experimental data / von Julia Gottschall

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Publié par	carl_von_ossietzky_universitat_oldenburg
Publié le	01 janvier 2009
Nombre de lectures	5
Langue	English
Poids de l'ouvrage	2 Mo

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Modelling the variability of complex
systems by means of Langevin
processes
On the application of a dynamical approach
to experimental data
Julia Gottschall
Von der Fakult at fur Mathematik und Naturwissenschaften
der Carl von Ossietzky Universit at Oldenburg
zur Erlangung des Grades und Titels eines
Doktors der Naturwissenschaften
Dr. rer. nat.
angenommene Dissertation
von Frau Julia Gottschall
geboren am 14.03.1982 in G ottingenGutachter: Prof. Dr. Joachim Peinke
Zweitgutachterin: Prof. Dr. Ulrike Feudel
Tag der Disputation: 20.01.2009Contents
Abstract { Zusammenfassung vii
1 Introduction 1
1.1 Modelling variability of complex systems . . . . . . . . . . . . . . . . 1
1.2 Reconstructing the dynamics of Langevin processes . . . . . . . . . . 3
1.3 Stochastic modelling of experimental data . . . . . . . . . . . . . . . 5
1.4 Extension to Langevin-like processes . . . . . . . . . . . . . . . . . . 7
1.5 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 De nition and handling of di erent drift and di usion estimates 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Finite sampling rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Presence of measurement noise . . . . . . . . . . . . . . . . . . . . . 19
2.4 Combination of low sampling rates and measurement noise . . . . . 22
2.5 Reconstruction through optimization . . . . . . . . . . . . . . . . . . 25
2.6 Handling the estimates and a note on the robustness of xed points 28
2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.8 Appendix: Derivation of third-order terms . . . . . . . . . . . . . . . 31
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 A phenomenological power performance model 35
3.1 Basic considerations on power p testing . . . . . . . . . . 36
3.1.1 Power performance and e ective control . . . . . . . . . . . . 36
3.1.2 Standard procedure for power performance testing (IEC
61400-12-1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.3 Averaged power performance versus short-term dynamics . . 39
3.2 Turbulent structures in wind speed and power output . . . . . . . . 41
3.2.1 Characterization of short-term structures in wind speed time
series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2 A note on gusts . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.3 Transfer of turbulent structures to the power output . . . . . 50
3.3 De nition and reconstruction of a dynamical power characteristic . . 55
3.3.1 Stochastic relaxation model . . . . . . . . . . . . . . . . . . . 55
3.3.2 Reconstruction of dynamical power characteristic . . . . . . . 57
3.4 Discussion of the model . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.1 Self-consistency . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.2 Time scale of description . . . . . . . . . . . . . . . . . . . . 65
3.4.3 Representativity of considered wind speed . . . . . . . . . . . 67
3.4.4 Relevance of the dynamical power characteristic . . . . . . . 70
3.5 General potential of a stochastic xed-point analysis . . . . . . . . . 72
3.6 Appendix A (Publication in Environ. Res. Lett.) . . . . . . . . . . . 75iv CONTENTS
3.7 Appendix B: Data description . . . . . . . . . . . . . . . . . . . . . . 84
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4 Stochastic modelling of human postural control 91
4.1 Exploring the dynamics of balance data . . . . . . . . . . . . . . . . 92
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1.2 Experimental setup and data analysis . . . . . . . . . . . . . 93
4.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 Distinguishing between task and time e ects . . . . . . . . . . . . . 103
4.2.1 Outline of the study . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 104
4.3 Appendix: One-way ANOVA . . . . . . . . . . . . . . . . . . . . . . 106
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5 Summary 111
Danksagung 115
Lebenslauf 117
Erklarung 119
Publikationsliste 120List of Figures
2.1 Time series of an exemplary stochastic process . . . . . . . . . . . . 15
(1)
2.2 D (x; ) for a process with cubic drift and constant di usion . . . 180E;
(1)
2.3 The estimate D (x; ) . . . . . . . . . . . . . . . . . . . . . . . . . 190E;
(1)
2.4 The D (y) . . . . . . . . . . . . . . . . . . . . . . . . . . 21E;
(n) (n)
2.5 Empirical and theoretical estimates D and D (n = 1; 2) . . . . 23E; E;
(1)
2.6 The estimate D (y; ) . . . . . . . . . . . . . . . . . . . . . . . . 240E;
(1) (1)
2.7 Convergence behaviour of the estimates D and D . . . . . . . 25E; E;
(n)
2.8 Empirical and theoretical estimates D and extrapolated o sets . 27E;
2.9 Theoretical drift estimates for four exemplary processes . . . . . . . 29
03.1 Typical time series for u(t), uctuations u (t) and increments u (t) . 42
03.2 Turbulence intensity, skewness and kurtosis for u(t) and u (t), resp. . 43
3.3 Pdfs of high-frequency data u conditioned on 10 min periods . . . . 44
03.4 Entire pdfs of the uctuations u with respect to di erent values of T 45
3.5 Pdfs of the wind speed increments u for di erent time increments 46
23.6 Standard deviation and form parameter for pdfs of u versus 47
3.7 Entire pdf p(u ) and conditioned pdfs p(u ju) . . . . . . . . . . . . . 48
3.8 Pdfs of the wind directions increments for di erent . . . . . . . 49
3.9 Pdfs for the vertical wind speed component u and its increments u 50z z;
03.10 Turbulence intensity, skewness and kurtosis for P (t) and P (t), resp. 51
3.11 Pdfs of high-frequency data P conditioned on 10 min periods . . . . 52
3.12 Pdfs of the power increments P for di erent . . . . . . . . . . . . 53
3.13 Pdfs for the incrementsP and P for = 20 s . . . . . . . . . . . 53 p:c:;
23.14 and for the pdfs of P and P versus . . . . . . . . . . . 54P p:c:;
3.15 Illustration of relaxation model for power performance . . . . . . . . 56
3.16 Reconstructed power curves for numerical data with/without relaxation 57
3.17 Estimation of the drift coe cient from rst conditional moment . . . 59
3.18 Illustration of xed-point analysis . . . . . . . . . . . . . . . . . . . . 60
3.19 Dynamical power characteristic for two di erent power binnings . . . 61
3.20 Reconstructed xed points for numerical power output data . . . . . 63
3.21 Pdfs of reconstr. noise processes (t) for di erent wind speed bins . 64i
3.22 Pdfs of empirical and reconstructed increments P (t) for di erent . 65
3.23 Dynamical power characteristic for averaged wind speed u~ . . . . . 66T
3.24 Correlation between wind speed and power output time series . . . . 68
3.25 Reconstructed drift coe cients with/without additional delay . . . . 69
3.26 dynamical power curves with/without additional delay 69
3.27 Power characteristics for a turbine with multistable dynamics . . . . 72
3.28 Convergence behaviour of mean values and xed points . . . . . . . 74
3.29 Reconstructed power curves for simulated data for di erent values of I 78
3.30 Simulated data with non-ideal noise . . . . . . . . . . . . . . . . . . 80vi LIST OF FIGURES
3.31 Reconstructed power curves for simulated data with non-ideal noise 80
3.32 Schematic comparison of di erent types of power curves . . . . . . . 81
3.33 Histograms of power output data for di erent bins . . . . . . . . . . 82
3.34 Site layout and location of the meteorological mast . . . . . . . . . . 85
3.35 Distribution of wind directions for the considered site . . . . . . . . 86
4.1 Experimental setup (balance experiment) . . . . . . . . . . . . . . . 94
4.2 Angular displacement and velocity in medial-lateral direction . . . . 95
4.3 Reconstruction of drift and di usion coe cient with binning . . . . . 96
4.4 Direct reconstruction of drift and di usion coe cient without binning 97
4.5 Self-consistency test { empirical and reconstructed pdfs . . . . . . . 98
4.6 Performance of balance with and without supra-postural tasks . . . 101
4.7 Experimental setup for sitting condition . . . . . . . . . . . . . . . . 103Abstract – Zusammenfassung vii
Abstract
Central topic of this thesis is the modelling of complex systems as stochastic
processes. In this regard, the underlying dynamics of the considered process is
described by means of e ective Langevin equations. The functions for drift and
di usion determining these equations are derived directly form a set of measured
data, based on a procedure proposed by Friedrich and Peinke.
The focus of this thesis is on the application of this procedure to experimental
data. In particular, respective disturbances are discussed that complicate a reliable
estimation of the drift and di usion coe cients. After an introduction to the theory
forming the basis of the considered method and a critical overview over previous
examples of application, two speci c applications are discussed in more detail.
The rst ap