Multiscale Analysis of Complex Time Series Integration of Chaos and Random Fractal Theory, and Beyond
, by Gao, Jianbo; Cao, Yinhe; Tung, Wen-wen; Hu, Jing- ISBN: 9780471654704 | 0471654701
- Cover: Hardcover
- Copyright: 9/17/2007
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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