Multiscale Analysis of Complex Time Series: Integration of Chaos and Random Fractal Theory, and Beyond Book

Preface     xiii
Introduction     1
Examples of multiscale phenomena     4
Examples of challenging problems to be pursued     9
Outline of the book     12
Bibliographic notes     14
Overview of fractal and chaos theories     15
Prelude to fractal geometry     15
Prelude to chaos theory     18
Bibliographic notes     23
Warmup exercises     23
Basics of probability theory and stochastic processes     25
Basic elements of probability theory     25
Probability system     25
Random variables     27
Expectation     30
Characteristic function, moment generating function, Laplace transform, and probability generating function     32
Commonly used distributions     34
Stochastic processes     41
Basic definitions     41
Markov processes     43
Special topic: How to find relevant information for a new field quickly     49
Bibliographic notes     51
Exercises     51
Fourier analysis and wavelet multiresolution analysis     53
Fourier analysis     54
Continuous-time (CT) signals     54
Discrete-time (DT) signals     55
Sampling theorem     57
Discrete Fourier transform     58
Fourier analysis of real data     58
Wavelet multiresolution analysis     62
Bibliographic notes     67
Exercises     67
Basics of fractal geometry     69
The notion of dimension     69
Geometrical fractals     71
Cantor sets     71
Von Koch curves     74
Power law and perception of self-similarity     75
Bibliographic notes     76
Exercises     76
Self-similar stochastic processes     79
General definition     79
Brownian motion (Bm)     81
Fractional Brownian motion (fBm)     84
Dimensions of Bm and fBm processes     87
Wavelet representation of fBm processes     89
Synthesis of fBm processes     90
Applications     93
Network traffic modeling     93
Modeling of rough surfaces     97
Bibliographic notes     97
Exercises     98
Stable laws and Levy motions     99
Stable distributions      100
Summation of strictly stable random variables     103
Tail probabilities and extreme events     104
Generalized central limit theorem     107
Levy motions     108
Simulation of stable random variables     109
Bibliographic notes     111
Exercises     112
Long memory processes and structure-function-based multifractal analysis     115
Long memory: basic definitions     115
Estimation of the Hurst parameter     118
Random walk representation and structure-function-based multifractal analysis     119
Random walk representation     119
Structure-funciion-based multifractal analysis     120
Understanding the Hurst parameter through multifractal analysis     121
Other random walk-based scaling parameter estimation     124
Other formulations of multifractal analysis     124
The notion of finite scaling and consistency of H estimators     126
Correlation structure of ON/OFF intermittency and Levy motions     130
Correlation structure of ON/OFF intermittency     130
Correlation structure of Levy motions     131
Dimension reduction of fractal processes using principal component analysis     132
Broad applications      137
Detection of low observable targets within sea clutter     137
Deciphering the causal relation between neural inputs and movements by analyzing neuronal firings     139
Protein coding region identification     147
Bibliographic notes     149
Exercises     151
Multiplicative multifractals     153
Definition     153
Construction of multiplicative multifractals     154
Properties of multiplicative multifractals     157
Intermittency in fully developed turbulence     163
Extended self-similarity     165
The log-normal model     167
The log-stable model     168
The[beta]-model     168
The random[beta]-model     168
The p model     169
The SL model and log-Poisson statistics of turbulence     169
Applications     171
Target detection within sea clutter     173
Modeling and discrimination of human neuronal activity     173
Analysis and modeling of network traffic     176
Bibliographic notes     178
Exercises     179
Stage-dependent multiplicative processes     181
Description of the model      181
Cascade representation of 1/f[subscript beta] processes     184
Application: Modeling heterogeneous Internet traffic     189
General considerations     189
An example     191
Bibliographic notes     193
Exercises     193
Models of power-law-type behavior     195
Models for heavy-tailed distribution     195
Power law through queuing     195
Power law through approximation by log-normal distribution     196
Power law through transformation of exponential distribution     197
Power law through maximization of Tsallis nonextensive entropy     200
Power law through optimization     202
Models for 1/f[superscript beta] processes     203
1/f[superscript beta] processes from superposition of relaxation processes     203
1/f[superscript beta] processes modeled by ON/OFF trains     205
1/f[superscript beta] processes modeled by self-organized criticality     206
Applications     207
Mechanism for long-range-dependent network traffic     207
Distributional analysis of sea clutter     209
Bibliographic notes     210
Exercises     211
Bifurcation theory      213
Bifurcations from a steady solution in continuous time systems     213
General considerations     214
Saddle-node bifurcation     215
Transcritical bifurcation     215
Pitchfork bifurcation     215
Bifurcations from a steady solution in discrete maps     217
Bifurcations in high-dimensional space     218
Bifurcations and fundamental error bounds for fault-tolerant computations     218
Error threshold values for arbitrary K-input NAND gates     219
Noisy majority gate     222
Analysis of von Neumann's multiplexing system     226
Bibliographic notes     233
Exercises     233
Chaotic time series analysis     235
Phase space reconstruction by time delay embedding     236
General considerations     236
Defending against network intrusions and worms     237
Optimal embedding     240
Characterization of chaotic attractors     243
Dimension     244
Lyapunov exponents     246
Entropy     251
Test for low-dimensional chaos     254
The importance of the concept of scale     258
Bibliographic notes      258
Exercises     259
Power-law sensitivity to initial conditions (PSIC)     261
Extending exponential sensitivity to initial conditions to PSIC     262
Characterizing random fractals by PSIC     263
Characterizing 1/f[superscript beta] processes by PSIC     264
Characterizing Levy processes by PSIC     265
Characterizing the edge of chaos by PSIC     266
Bibliographic notes     268
Multiscale analysis by the scale-dependent Lyapunov exponent (SDLE)     271
Basic theory     271
Classification of complex motions     274
Chaos, noisy chaos, and noise-induced chaos     274
1/f [superscript beta] processes     276
Levy flights     277
SDLE for processes defined by PSIC     279
Stochastic oscillations     279
Complex motions with multiple scaling behaviors     280
Distinguishing chaos from noise     283
General considerations     283
A practical solution     284
Characterizing hidden frequencies     286
Coping with nonstationarity     290
Relation between SDLE and other complexity measures     291
Broad applications     297
EEG analysis      297
HRV analysis     298
Economic time series analysis     300
Sea clutter modeling     303
Bibliographic notes     304
Description of data     307
Network traffic data     307
Sea clutter data     308
Neuronal firing data     309
Other data and program listings     309
Principal Component Analysis (PCA), Singular Value Decomposition (SVD), and Karhunen-Loeve (KL) expansion     311
Complexity measures     313
FSLE     314
LZ complexity     315
PE     317
References     319
Index     347