From phenomenological modelling of anomalous diffusion through continuous-time random walks and fractional calculus to correlation analysis of complex systems [Elektronische Ressource] / vorgelegt von Daniel Fulger

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From phenomenological modellingof anomalous di usion throughcontinuous-time random walks and fractional calculusto correlation analysis of complex systemsDissertationzur Erlangung des Doktorgrades der Naturwisschaften(Dr. rer. nat.)dem Fachbereich Chemieder Philipps-Universit at Marburgvorgelegt vonDaniel Fulgeraus VasluiMarburg/Lahn 2009Vom Fachbereich Chemie der Philipps-Universit at Marburg als Dissertation am............angenommen.Erstgutachter: Prof. Dr. Guido GermanoZweitgutachter: Prof. Dr. Enrico ScalasTag der mundlic hen Prufu ng: 20. M arz 2009’But I don’t want to go among mad people,’ said Alice. ’Oh, you can’thelp that,’ said the cat. ’We’re all mad here.’ . . .Charles Lutwidge Dodgson alias Lewis CarrollDodgson condensation is a method of computing the determinants of square matrices,named after its inventor.Contents1 Overview 21.1 Complex system eclecticism . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Papers and logistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Continuous-time random walks and anomalous di usion { two birds withone stone 92.1 From random walks to the macroscopic di usion equation . . . . . . . . . . 92.2 Continuous-time random walks . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 The time and space fractional di usion equation . . . . . . . . . . . . . . . 172.4 Monte Carlo solution of the fractional di usion equation . . . . . . . . . . . 202.

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From phenomenological modelling
of anomalous di usion through
continuous-time random walks and fractional calculus
to correlation analysis of complex systems
Dissertation
zur Erlangung des Doktorgrades der Naturwisschaften
(Dr. rer. nat.)
dem Fachbereich Chemie
der Philipps-Universit at Marburg
vorgelegt von
Daniel Fulger
aus Vaslui
Marburg/Lahn 2009Vom Fachbereich Chemie der Philipps-Universit at Marburg als Dissertation am
............
angenommen.
Erstgutachter: Prof. Dr. Guido Germano
Zweitgutachter: Prof. Dr. Enrico Scalas
Tag der mundlic hen Prufu ng: 20. M arz 2009’But I don’t want to go among mad people,’ said Alice. ’Oh, you can’t
help that,’ said the cat. ’We’re all mad here.’ . . .
Charles Lutwidge Dodgson alias Lewis Carroll
Dodgson condensation is a method of computing the determinants of square matrices,
named after its inventor.Contents
1 Overview 2
1.1 Complex system eclecticism . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Papers and logistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Continuous-time random walks and anomalous di usion { two birds with
one stone 9
2.1 From random walks to the macroscopic di usion equation . . . . . . . . . . 9
2.2 Continuous-time random walks . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 The time and space fractional di usion equation . . . . . . . . . . . . . . . 17
2.4 Monte Carlo solution of the fractional di usion equation . . . . . . . . . . . 20
2.5 Isotropic random walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Transformation formulas for non-uniform random numbers . . . . . . . . . . 24
2.6.1 Symmetric Levy -stable probability distribution . . . . . . . . . . . 24
2.6.2 One-parameter Mittag-Le ery . . . . . . . . 24
2.7 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Spectral densities of Wishart-Levy free stable random matrices 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Free stable random variables and the Wishart-Levy ensemble . . . . . . . . 38
3.4 The analytic spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 Monte Carlo validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.7 Computer codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Random numbers 48
4.1 Non-uniform variates for arbitrary densities with nite support . . . . . . . 49
4.1.1 Introduction and background . . . . . . . . . . . . . . . . . . . . . . 49
4.1.2 The tiling and numerical considerations . . . . . . . . . . . . . . . . 53
4.1.3 Discontinuous probability densities . . . . . . . . . . . . . . . . . . . 57
4.1.4 Measurements and comparisons . . . . . . . . . . . . . . . . . . . . . 59
4.2 Random numbers from the distribution tails using the transformation and
the tiling methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.2 The Levy -stable probability density and its transform map . . . . 66
4.2.3 Sampling method and example application . . . . . . . . . . . . . . 71
4.2.4 The Mittag-Le er probability distribution . . . . . . . . . . . . . . 74
4.3 Fast generation of rotationally invariant random matrices . . . . . . . . . . 74
4.4 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
ii5 Comparison of the Fourier and Pearson correlation estimators 79
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 The context of correlation matrices . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Preliminary numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4 Matlab code for the Fourier correlation estimator . . . . . . . . . . . . . . . 89
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6 Correlation matrices of arti cial continuous time-random walks and em-
pirical data 91
6.1 Eigenvectors of correlation matrices . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Toy example: arti cial stock market data . . . . . . . . . . . . . . . . . . . 94
7 On the relevance of the sampling of continuous-time random walks in
correlation matrix analysis 100
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.2 The distribution of sampled increments x . . . . . . . . . . . . . . . . . . 101
7.3 The distri of correlation coe cients with missing data . . . . . . . . . 103
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8 Spectral properties of correlation matrices { towards enhanced spectral
clustering 108
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2 Scenario 1 { Correlated noise with many variables and many measurements
per variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.2.1 One correlated cluster . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.2.2 Two clusters . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.3 On the distribution of eigenvector elements . . . . . . . . . . . . . . . . . . 124
8.4 Improved spectral clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.5 Scenario 2 { Un-correlated noise with more variables than measurements
per variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.6 Scenario 3 { Correlated noise with more variables than measurements per
variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.7 Intermediate discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.8 Genetic pro le scenario of microarray data on di erential expressions . . . . 138
8.9 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
9 Networks of synchronisation in electroencephalographic activity 145
9.1 De nition and measure of synchronisation . . . . . . . . . . . . . . . . . . . 146
9.2 Non-mathematical comments . . . . . . . . . . . . . . . . . . . . . . . . . . 148
9.3 Procedure and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.4 Some remarks on eigenvalues of coe cient matrices . . . . . . . . . . . . . . 156
9.5 Patterns in data and remarks on the signi cance . . . . . . . . . . . . . . . 157
9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
10 Some quintessence 159
A Numerical calculation of the Levy probability density 163
B Serious research 166
C Future research, seriously 168
Bibliography 168
Acknowledgments 169
iiiList of Tables
2.1 Average number n of jumps per run and total CPU time t in secondsCPU
7for 10 runs with t2 [0; 2] on a 2.2 GHz AMD Athlon 64 X2 Dual-Core
with Fedora Core 4 Linux, using the ran1 uniform random number gen-
erator [148] and the Intel C++ compiler version 9.1 with the -O3 -static
optimization options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1 Rejection rate and number of tiles for a uni-modal PDF. . . . . . . . . . . . 60
4.2 rate and number of tiles for a multi-modal PDF. . . . . . . . . . . 61
6.1 Parameters of the Gaussian distributionsN( ; ;m;M) truncated atm; M
used to generate the parameters ; ; ; of the arti cial market. . . . . 94x t
8.1 With the transition T > N to T N the rank of the sample correlation
matrix drops earlier and persistently by one than the rank of the Wishart
matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.1 Typical classi cation of sleep stages. The non-REM stages are not clearly
distinct and continuous in transition as opposed to REM and wake which
in turn are similar with respect to typical classi cation criteria. . . . . . . . 150
9.2 Classi cation of brain waves into frequency bands (in this order). There are
di erent de nitions in neurology and the number of bands di ers between
4 to 7 with exible ranges or overlap. . . . . . . . . . . . . . . . . . . . . . . 150
ivList of Figures
1.1 Where the work was done. On a logarithmic scale the font size is an estimate
of the respective amount. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Brownian motion (left) and a Levy ight (right) of a particle with = 1:5
whose position for each time index is connected to the previous with a
straight line. The realisations are shown for 1000, 5000, 1000 and 20000
jumps. The di erence between A and B is small. During the 5000 jumps
the walker spent its time in the upper cluster and behaved Brownian. This
alteration of domains is typical for Levy ights. . . . . . . . . . . . . . . . 15
2.2 Schematic picture of a continuous-time random walk. Waiting-times and
jumps are distributed according to the densities () and (x) functions. . 17
=
2.3 Sample paths of CTRWs with scale parameters = 0:001; = andt x t
di erent choices of and. With smaller the jumps become larger; with
smaller