This book introduces a number of key concepts from random fractal theory by emphasizing their usage in signal processing. A number of applications will be discussed in depth. These include DNA sequence analysis, network traffic modelling, analysis of neuron inter-spike interval data, and study of heart rate variability and ambiguous visual perception.

Jianbo Gao is an Assistant Professor of the Department of Electrical and Computer Engineering at the University of Florida.

Yinhe Cao is the CEO of BioSieve.

Wen-wen Tung is an Assistant Professor of the Department of Earth and Atmospheric Sciences at Purdue University, West Lafayette, Indiana.

Jing Hu is a Research Engineer of the Department of Electrical and Computer Engineering at the University of Florida.

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

9780471654704

0471654701

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