Fractal-Based Point Processes

Abstract

Fractals, Point Processes, Fractal-Based Point Processes, Problems

Full text

Fractal-Based
Point Processes
Steven Bradley Lowen
Harvard Medical School
McLean Hospital
Malvin Carl Teich
Boston University
Columbia University
A JOHN WILEY & SONS, INC., PUBLICATION
lowen-fm.qxd 6/1/2005 9:18 AM Page iii

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Library of Congress Cataloging-in-Publication Data:
Lowen, Steven Bradley, 1962–
Fractal-based point processes / Steven Bradley Lowen, Malvin Carl Teich.
p. cm.
Includes bibliographical references and index.
ISBN-13 978-0-471-38376-5 (acid-free paper)
ISBN-10 0-471-38376-7 (acid-free paper)
1. Point processes. 2. Fractals. I. Teich, Malvin Carl. II. Title.
QA274.42.L69 2005
519.2'3—dc22 2005048977
Printed in the United States of America.
10987654321
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Preface
Fractals and Point Processes
Fractals are objects that possess a form of self-scaling; a part of the whole can be
made to recreate the whole by shifting and stretching. Many objects are self-scaling
only in a statistical sense, meaning that a part of the whole can be made to recreate the
whole in the likeness of their probability distributions, rather than as exact replicas.
Examples of random fractals includethe length of a segment of coastline, the variation
of water flow in the river Nile, and the human heart rate.
Point processes are mathematical representations of random phenomena whose
individual events are largely identical and occur principally at discrete times and
locations. Examples include the arrival of cars at a tollbooth, the release of neuro-
transmitter molecules at a biological synapse, and the sequence of human heartbeats.
Fractals began to find their way into the scientific literature some 50 years ago.
For point processes this took place perhaps 100 years ago, although both concepts
developed far earlier. These two fields of study have grown side-by-side, reflecting
their increasing importance in the natural and technological worlds. However, the
domains in which point processes and fractals both play a role have received scant
attention. It is the intersection of these two fields that forms the topic of this treatise.
Fractal-based point processes exhibit both the scaling properties of fractals and
the discrete character of random point processes. These constructs are useful for
representing a wide variety of diverse phenomena in the physical and biological
sciences, from information-packet arrivals on a computer network to action-potential
occurrences in a neural preparation.
v

vi PREFACE
Scope
The presentation begins with several concrete examples of fractals and point pro-
cesses, without devoting undue attention to mathematical detail (Chapter 1). A brief
introduction to fractals and chaos follows (Chapter 2). We then define point processes
and consider a collection of measures useful in characterizing them (Chapter 3). This
is followed by a number of salient examples of point processes (Chapter 4). With the
concepts of fractals and point processes in hand, we proceed to integrate them (Chap-
ter 5). Mathematical formulations for several important fractal-based point-process
families are then set forth (Chapters 6–10). An exposition detailing how various
operations modify such processes follows (Chapter 11). We then proceed to examine
analysis and estimation techniques suitable for these processes (Chapter 12). Finally,
we examine computer network traffic (Chapter 13), an important application used to
illustrate the various approaches and models set forth in earlier chapters.
To facilitate the smooth flow of material, lengthy Derivations are relegated to
Appendix A.Problem Solutions appear in Appendix B. For convenience, Appendix C
contains a List of Symbols. A comprehensive Bibliography is provided.
Approach
We have been inspired by Feller’s venerable and enduring Introduction to Probability
Theory and Its Applications (1968; 1971) and Cox and Isham’s concise but superb
Point Processes (1980).
We provide an integrated exposition of fractal-based point processes, from defi-
nitions and measures to analysis and estimation. The material is set forth in a self-
contained manner. We approach the topic from a practical and informal perspective
— and with a distinct engineering bent. Chapters 3, 4, and 11 can serve as a compre-
hensive stand-alone introduction to point processes.
A number of important applications are examined in detail with the help of a
canonical set of point processes drawn from biological signals and computer network
traffic. This set includes action-potential sequences recorded from the retina, lateral
geniculate nucleus, striate cortex, descending contralateral movement detector, and
cochlea; as well as vesicular exocytosis and human-heartbeat sequences. We revisit
these data sets throughout our presentation.
Other applications are drawn from a diverse collection of topics, including 1/f
noise events in electronic devices and systems, trapping in amorphous semiconduc-
tors, semiconductor high-energy particle detectors, diffusion processes, error clus-
tering in telephone networks, digital generation of 1/f αnoise, photon statistics of
ˇ
Cerenkov radiation, power-law mass distributions, molecular evolution, and the statis-
tics of earthquake occurrences.
Audience
Our exposition is addressed principally to students and researchers in the mathe-
matical, physical, biological, psychological, social, and medical sciences who seek

PREFACE vii
to understand, explain, and make use of the ever-growing roster of phenomena that
are found to exhibit fractal and point-process characteristics. The reader is assumed
to have a strong mathematical background and a solid grasp of probability theory.
While not required, a rudimentary knowledge of fractals and a familiarity with point
processes will prove useful.
This book will likely find use as a text for graduate-level courses in fields as
diverse as statistics, electrical engineering, neuroscience, computer science, physics,
and psychology. An extensive set of solved problems accompanies each chapter.
Website and Supplementary Material
Supplementary materials related to the practical aspects of data analysis and simula-
tion are linked from the book’s website. Errata are posted and readers are encouraged
to contribute to the compilation. Kindly visit http://www.wiley.com/statistics/ and
scroll down to the icon labeled “Download Software and Supplements for Wiley
Math & Statistics Titles.” Then find the entry “Lowen and Teich.” Alternatively, you
may directly access the authors’ websites at http://cordelia.mclean.org/∼lowen/ and
http://people.bu.edu/teich/.
Photo Credits
We express our appreciation to the many organizations that have provided assistance
in connection with our efforts to assemble the photographs used at the beginnings of
each chapter: Penck (courtesy of Bildarchiv der ¨
Osterreichischen Nationalbibliothek,
Vienna); Richardson (courtesy of Olaf K. F. Richardson); Cantor and Poincar´
e (cour-
tesy of the Aldebaran Group for Astrophysics, Prague); Poisson, Yule, Pareto, Hurst,
and Erlang [from Heyde & Seneta (2001), courtesy of Chris Heyde, Eugene Seneta,
and Springer-Verlag]; Lapicque (courtesy of the National Library of Medicine); Cox
(courtesy of Sir David R. Cox); Fourier (courtesy of John Wiley & Sons); Haar (cour-
tesy of Akad´
emiai Kiad´
o, Budapest); Kolmogorov (courtesy of A. N. Shiryaev); Van
Ness (courtesy ofJohn W. Van Ness); Mandelbrot(courtesy of BenoitB. Mandelbrot);
Gauss (S. Bendixen portrait, 1828); L´
evy and Feller [from Reid (1982), courtesy of
Ingram Olkin, Constance Reid, and Springer-Verlag]; Schottky (from the Schottky
family album); Rice (courtesy of the IEEE History Center, Rutgers University); Ney-
man [from Reid (1982), courtesy of Constance Reid and Springer-Verlag]; Bartlett
(courtesy of Walter Bird and Godfrey Argent); Bernoulli [frontispiece from Fleck-
enstein (1969), courtesy of Birkh¨
auser-Verlag]; Allan (courtesy of David W. Allan);
Palm (courtesy of Jan Karlqvist, from the Olle Karlqvist family album). The pho-
tographs of Lowen and Teich were provided courtesy of Jeff Thiebauth and Boston
University, respectively.
We are indebted to a number of individuals who assisted us in our attempts to
secure various photographs. These include Tedros Tsegaye, who helped us obtain
a photograph of Conny Palm; Steven Rockey, Mathematics Librarian at the Cornell
University Mathematics Library, who tracked down a photograph of Alfr´
ed Haar in

viii PREFACE
a collection of Haar’s works (Szo´´kefalvi-nagy, 1959); and Jan van der Spiegel and
Nader Engheta at the University of Pennsylvania, who valiantly attempted to secure
a photograph of Gleason Willis Kenrick from the University archives.
Finally, we extend our special thanks to those individuals who kindly provided
photographs of themselves: Sir David R. Cox of Nuffield College at the University of
Oxford, John W. Van Ness of the University of Texas at Dallas, Benoit B. Mandelbrot
of Yale University and the IBM Corporation, and David W. Allan of Fountain Green,
Utah.
Acknowledgments
We greatly appreciate the efforts of the seven reviewers who carefully read an early
version of the manuscript and provided invaluable feedback: Jan Beran, Patrick
Flandrin, Conor Heneghan, Eric Jakeman, Bradley Jost, Michael Shlesinger, and
Xueshi Yang. We acknowledge valuable discussions with Bahaa Saleh and Benoit
Mandelbrotrelating tothe presentationof the material. Bo Ryu and David Starobinski
made many helpful suggestions pertinent to the material contained in Chapter 13.
Arnold Mandell brought to our attention a number of publications related to fractals
in medicine. Carl Anderson, Darryl Veitch, Murad Taqqu, Larry Abbott, Luca Dal
Negro, and David Bickel alerted us to many useful references in a broad variety of
fields. Iver Brevik and Arne Myskja supplied information regarding Tore Engset’s
contributions to teletraffic theory. Wai Yan (Eliza) Wong provided extensive logistical
supportand craftedmagnificent diagramsand figures. AnnaSwan kindlyassisted with
translation from the Swedish. Many at Wiley, including Stephen Quigley, Susanne
Steitz, and Heather Bergman have been most helpful, patient, and encouraging. We
especiallyappreciate the attentivenessand thoroughnessthat Melissa Yanuzzibrought
to the production process. Most of all, our families provided the love and support that
nurtured the process of creating this book, from inception to publication. Our thanks
go to them.
We gratefully acknowledge financial support provided by the National Science
Foundation; the Center for Telecommunications Research (CTR), an Engineering
Research Center at Columbia University supported by the National Science Founda-
tion; the U.S. Joint Services Electronics Program through the Columbia Radiation
Laboratory; the U.S. Office of Naval Research; the Whitaker Foundation; the Inter-
disciplinary Science Program at the David & Lucile Packard Foundation; the Office
of National Drug Control Policy; the National Institute on Drug Abuse; the Center
for Subsurface Sensing and Imaging Systems (CenSSIS), an Engineering Research
Center at Boston University supported by the National Science Foundation; and the
Boston University Photonics Center.
STEVEN BRADLEY LOWEN
MALVIN CARL TEICH
Boston, Massachusetts

Contents
Preface v
List of Figures xv
List of Tables xix
Authors xxi
1 Introduction 1
1.1 Fractals 2
1.2 Point Processes 4
1.3 Fractal-Based Point Processes 4
Problems 6
2 Scaling, Fractals, and Chaos 9
2.1 Dimension 11
2.2 Scaling Functions 13
2.3 Fractals 13
2.4 Examples of Fractals 16
2.5 Examples of Nonfractals 23
2.6 Deterministic Chaos 25
2.7 Origins of Fractal Behavior 32
2.8 Ubiquity of Fractal Behavior 39
Problems 46
ix

xCONTENTS
3Point Processes: Definition and Measures 49
3.1 Point Processes 50
3.2 Representations 51
3.3 Interval-Based Measures 54
3.4 Count-Based Measures 63
3.5 Other Measures 70
Problems 79
4 Point Processes: Examples 81
4.1 Homogeneous Poisson Point Process 82
4.2 Renewal Point Processes 85
4.3 Doubly Stochastic Poisson Point Processes 87
4.4 Integrate-and-Reset Point Processes 91
4.5 Cascaded Point Processes 93
4.6 Branching Point Processes 95
4.7 L´evy-Dust Counterexample 95
Problems 96
5 Fractal and Fractal-Rate Point Processes 101
5.1 Measures of Fractal Behavior in Point Processes 103
5.2 Ranges of Power-Law Exponents 107
5.3 Relationships among Measures 114
5.4 Examples of Fractal Behavior in Point Processes 115
5.5 Fractal-Based Point Processes 120
Problems 126
6 Processes Based on Fractional Brownian Motion 135
6.1 Fractional Brownian Motion 136
6.2 Fractional Gaussian Noise 141
6.3 Nomenclature for Fractional Processes 143
6.4 Fractal Chi-Squared Noise 145
6.5 Fractal Lognormal Noise 147
6.6 Point Process from Ordinary Brownian Motion 149
Problems 150
7 Fractal Renewal Processes 153
7.1 Power-Law Distributed Interevent Intervals 155
7.2 Statistics of the Fractal Renewal Process 157

CONTENTS xi
7.3 Nondegenerate Realization of a Zero-Rate Process 164
Problems 166
8 Processes Based on the Alternating Fractal Renewal Process 171
8.1 Alternating Renewal Process 174
8.2 Alternating Fractal Renewal Process 177
8.3 Binomial Noise 179
8.4 Point Processes from Fractal Binomial Noise 182
Problems 183
9 Fractal Shot Noise 185
9.1 Shot Noise 186
9.2 Amplitude Statistics 189
9.3 Autocorrelation 194
9.4 Spectrum 195
9.5 Filtered General Point Processes 197
Problems 198
10 Fractal-Shot-Noise-Driven Point Processes 201
10.1 Integrated Fractal Shot Noise 204
10.2 Counting Statistics 205
10.3 Time Statistics 212
10.4 Coincidence Rate 214
10.5 Spectrum 215
10.6 Related Point Processes 216
Problems 219
11 Operations 225
11.1 Time Dilation 228
11.2 Event Deletion 229
11.3 Displacement 241
11.4 Interval Transformation 247
11.5 Interval Shuffling 252
11.6 Superposition 256
Problems 261
12 Analysis and Estimation 269
12.1 Identification of Fractal-Based Point Processes 271

xii CONTENTS
12.2 Fractal Parameter Estimation 273
12.3 Performance of Various Measures 281
12.4 Comparison of Measures 309
Problems 310
13 Computer Network Traffic 313
13.1 Early Models of Telephone Network Traffic 315
13.2 Computer Communication Networks 320
13.3 Fractal Behavior 324
13.4 Modeling and Simulation 332
13.5 Models 334
13.6 Identifying the Point Process 337
Problems 351
Appendix A Derivations 355
A.1 Point Processes: Definition and Measures 356
A.2 Point Processes: Examples 358
A.3 Processes Based on Fractional Brownian Motion 360
A.4 Fractal Renewal Processes 362
A.5 Alternating Fractal Renewal Process 371
A.6 Fractal Shot Noise 376
A.7 Fractal-Shot-Noise-Driven Point Processes 382
A.8 Analysis and Estimation 394
Appendix B Problem Solutions 397
B.1 Introduction 398
B.2 Scaling, Fractals, and Chaos 401
B.3 Point Processes: Definition and Measures 404
B.4 Point Processes: Examples 412
B.5 Fractal and Fractal-Rate Point Processes 427
B.6 Processes Based on Fractional Brownian Motion 441
B.7 Fractal Renewal Processes 447
B.8 Alternating Fractal Renewal Process 454
B.9 Fractal Shot Noise 459
B.10 Fractal-Shot-Noise-Driven Point Processes 463
B.11 Operations 473
B.12 Analysis and Estimation 486
B.13 Computer Network Traffic 494

CONTENTS xiii
Appendix C List of Symbols 505
C.1 Roman Symbols 506
C.2 Greek Symbols 510
C.3 Mathematical Symbols 511
Bibliography 513
Author Index 567
Subject Index 577

List of Figures
1.1 Coastline of Iceland at different scales 3
1.2 Vehicular-traffic point process 5
2.1 Cantor-set construction 17
2.2 Realization of Brownian motion 20
2.3 Fern: a nonrandom natural fractal 21
2.4 Grand Canyon: a random natural fractal 22
2.5 Realization of a homogeneous Poisson process 23
2.6 Nonchaoticsystemwithnonfractal attractor: time course 27
2.7 Chaotic system with nonfractal attractor: time course 27
2.8 Chaotic system with fractal attractor 29
2.9 Chaotic system with fractal attractor: time course 30
2.10 Nonchaotic system with fractal attractor 31
2.11 Nonchaotic system with fractal attractor: time course 32
3.1 Point-process representations 52
3.2 Rescaled-range analysis: pseudocode 60
3.3 Rescaled-range analysis: illustration 61
3.4 Detrended fluctuation analysis: pseudocode 62
3.5 Detrended fluctuation analysis: illustration 63
3.6 Construction of normalized variances 67
4.1 Stochastic-rate point processes 89
xv

xvi LIST OF FIGURES
4.2 Cascaded point process 94
5.1 Representative rate spectra 118
5.2 Representative normalized Haar-wavelet variances 119
5.3 Normalized Daubechies-wavelet variances 121
5.4 Fractal and nonfractal point processes 122
5.5 Fractal-rate and nonfractal point processes 125
5.6 Estimated normalized-variance curves 127
5.7 Representative interval spectra 128
5.8 Representative interval wavelet variances 129
5.9 Representative interevent-interval histograms 130
5.10 Representative capacity dimensions 131
5.11 Generalized dimensions for an exocytic point process 132
6.1 Realizations of fractional Brownian motion 140
6.2 Realizations of fractional Gaussian noise 143
7.1 Power-law interevent-interval densities 156
7.2 Fractal-renewal-process spectra 158
7.3 Fractal-renewal-process coincidence rate 160
7.4 Fractal-renewal-process normalized variance 161
7.5 Fractal-renewal-process Haar-wavelet variance 162
7.6 Fractal-renewal-process counting distributions 163
7.7 Minimal covering of fractal renewal process 164
7.8 Interevent-interval density for an interneuron 167
7.9 Normalized Haar-wavelet variance for an interneuron 168
8.1 Realization of an alternating renewal process 173
8.2 Alternating fractal-renewal-process spectrum 178
8.3 Sum of alternating renewal processes: binomial noise 180
8.4 Convergence of binomial noise to Gaussian form 182
9.1 Linearly filtered Poisson process: shot noise 187
9.2 Power-law-decaying impulse response functions 188
9.3 Stable amplitude probability densities 193
9.4 Fractal-shot-noise spectra 196
10.1 Shot-noise-driven Poisson process 203
10.2 Integrated power-law impulse response function 205
10.3 Fractal-shot-noise-driven-Poisson count distributions 207
10.4 Fractal-shot-noise-driven-Poisson variances 210
10.5 Fractal-shot-noise-driven-Poisson wavelet variances 211
10.6 Fractal-shot-noise-driven-Poisson interval densities 213
10.7 Fractal-shot-noise-driven-Poisson coincidence rates 215

LIST OF FIGURES xvii
10.8 Fractal-shot-noise-driven-Poisson spectra 216
10.9 Hawkes point process 218
10.10 Generation of ˇ
Cerenkov radiation 221
11.1 Event deletion in point processes 230
11.2 Interval histograms for randomly deleted data 233
11.3 Spectra for randomly deleted data 234
11.4 Haar-wavelet variances for randomly deleted data 235
11.5 Event-time displaced point process 243
11.6 Interval histograms for event-time-displaced data 244
11.7 Spectra for event-time-displaced data 245
11.8 Haar-wavelet variances for event-time-displaced data 246
11.9 Interval-transformed point process 248
11.10 Spectra for exponentialized data 250
11.11 Haar-wavelet variances for exponentialized data 251
11.12 Randomly shuffled point process 252
11.13 Spectra for randomly shuffled data 253
11.14 Haar-wavelet variances for randomly shuffled data 254
11.15 Superposition of point processes 256
11.16 Interevent-interval density for an interneuron 265
11.17 Normalized Haar-wavelet variance for an interneuron 266
11.18 Generalized dimensions for an interneuron 267
12.1 FGP-driven-Poisson estimated Haar-wavelet variance 277
12.2 FGP-driven-Poisson estimated normalized variance 284
12.3 FGP-driven-Poisson estimated count autocovariance 286
12.4 FGP-driven-Poisson estimated rescaled range 288
12.5 FGP-driven-Poisson estimated detrended fluctuation 290
12.6 FGP-driven-Poissonestimated interval waveletvariance 292
12.7 FGP-driven-Poissonestimated interval-based spectrum 294
12.8 Fluctuationsof estimated varianceand wavelet variance 297
12.9 FGP-driven-Poisson estimated rate spectrum 305
13.1 M/M/1 overflow probability vs. service ratio 320
13.2 M/M/1 overflow probability vs. maximum queue length 321
13.3 Representation of major nodes of the Internet 322
13.4 Cascaded point process in computer network traffic 324
13.5 Computer-network-traffic estimated rate spectrum 326
13.6 Computer-network-traffic estimated wavelet variance 327
13.7 Computer-network-traffic statistics: BC-pOct89 340
13.8 Computer-network-traffic statistics: BC-pAug89 341

xviii LIST OF FIGURES
13.9 Computer-network-traffic statistics: Bartlett–Lewis 348
13.10 Computer-network-traffic statistics: Neyman–Scott 349
B.1 Length of Icelandic coastline at different scales 398
B.2 Approaching the perimeter of a circle 399
B.3 Capacity dimension for simulated point processes 430
B.4 Estimatedinterevent-intervaldensity for an interneuron 451
B.5 Estimated Haar-wavelet variance for an interneuron 452
B.6 Generating a fractal-shot-noise spectrum 461
B.7 Fractal spectrum comprising two contributions 474
B.8 Block-shuffled fractal-rate-process spectrum 480
B.9 Event-time-displaced fractal-rate-process spectrum 481
B.10 Dead-time-deleted Poisson counting distributions 483
B.11 Bias in normalized-variance estimates 489
B.12 Fractal-renewal and Poisson spectral estimates 491
B.13 Fractal-renewalandPoissonwavelet-varianceestimates 492
B.14 Fractal-renewal and Poisson rescaled-range estimates 493
B.15 M/M/1 queue-length histogram 495
B.16 FGPDP/M/1 queue-length histogram (ρµ= 0.9)496
B.17 FGPDP/M/1 queue-length histogram (ρµ= 0.5)497
B.18 SHUFFLED-FGPDP/M/1 queue-length histogram 498
B.19 RFSNDP/M/1 queue-length histogram (ρµ= 0.9)499
B.20 MODULATED-FGPDP/M/1 queue-length histogram 500
B.21 Monofractal least-squares fitto wide-range bifractal 502
B.22 Monofractal least-squares fitto narrow-range bifractal 502

List of Tables
1.1 Length of Icelandic coastline at different scales 6
1.2 Polygon approximation for perimeter of circle 7
2.1 Representative objects: measurements and dimensions 11
9.1 Classes of fractal shot noise 189
12.1 Fractal-exponent estimatesfromHaar-wavelet variance 278
12.2 Fractal-exponent estimates from normalized variance 285
12.3 Fractal-exponent estimates from count autocovariance 287
12.4 Fractal-exponent estimates from rescaled range 289
12.5 Fractal-exponent estimatesfromdetrended fluctuations 291
12.6 Fractal-exponent estimates from NIWV 293
12.7 Fractal-exponent estimates from interval spectrum 295
12.8 Optimized fractal-exponent estimates from NHWV 299
12.9 Oversampled fractal-exponent estimates from NHWV 303
12.10 Optimizedfractal-exponentestimatesfromrate spectrum 307
13.1 Computer network traffic simulation parameters 347
13.2 Ethernet-trafficintervalstatistics:data and simulations 350
B.1 Fractal behavior following shuffling and interval
transformation 474
xix

Authors
Steven Bradley Lowen received the B.S. degree in elec-
trical engineering from Yale University in 1984, Magna
cum Laude and with distinction in the major. He was
elected to Tau Beta Pi that same year. Following
two years with the Hewlett–Packard Company he en-
tered Columbia University, from which he received the
M.S. and Ph.D. degrees in 1988 and 1992, respectively,
both in electrical engineering. Lowen was awarded the
ColumbiaUniversity ArmstrongMemorial Prize in1988
and in 1990 he was the recipient of a Joint Services Elec-
tronics Program Fellowship in the Columbia Radiation
Laboratory.
He began his research career by examining fractal patterns in the sequences of
action potentials traveling along auditory nerve fibers. Recognizing that efforts to
understand these fractal processes were hampered by the lack of a solid theoretical
framework, he set out to develop the relevant mathematical models. This effort led to
the development of alternating fractal renewalprocesses and fractal shot noise, as well
as point-process versions thereof. In connection with this effort he also investigated
fractal renewal point processes and several other fractal-based processes. This body
of work served as the foundation for his Ph.D. thesis, entitled Fractal Point Processes
(Lowen, 1992), as well as the basis of a number of journal articles and the core of
several chapters in this book.
xxi

xxii AUTHORS
After receiving the Ph.D. degree, Dr. Lowen continued his research at Columbia
as an Associate Research Scientist. He then joined Boston University as a Senior
Research Associate in the Department of Electrical and Computer Engineering in
1996. He was elected to Sigma Xi in 1994.
With a collection of models for fractal-based point processes in hand, Lowen fo-
cused on establishing appropriate methods for their analysis and synthesis. This work
quantified the performance of fractal estimators for point processes and highlighted
the practical realities of generating realizations for these processes. He also stud-
ied the interaction between dead time (refractoriness) and fractal behavior in point
processes.
Concurrently, working with various collaborators, he returned to examining appli-
cations for point processes with fractal characteristics by adapting the mathematical
frameworkhe developedto a numberof biomedical point processes. He demonstrated
that suitably modified fractal-based point processes serve to properly characterize
action-potential sequences on auditory nerve fibers. He then turned his attention to
signaling in the visual system by identifying fractal models that could describe the
neural firing patterns of individual cells in this system, as well as collections of such
cells, and detailing how the fractal patterns affect information transmission in this
network. Using a similar approach, he also examined human heartbeat patterns and
investigated how different measures of these fractal data sets could serve as markers
of the cardiovascular health of the subject. Finally, he explored neurotransmitter se-
cretion at the neuromuscular junction, and developed a suitable model showing that
it, too, exhibits fractal characteristics.
Dr. Lowen also applied his fractal models to physical phenomena. These included
charge transport in amorphous semiconductors and noise in infrared CCD cameras;
he developed multidimensional versions of his fractal-based point processes for the
latter. He also devoted substantial efforts to the modeling, synthesis, and analysis of
computer network traffic.
In 1999 Dr. Lowen joined McLean Hospital and the Harvard Medical School,
where he is currently Assistant Professor of Psychiatry. He has brought his fractal
expertise to bear on attention-deficit and hyperactivity disorder, and the analysis of
data collected with functional magnetic resonance imaging (fMRI). He is currently
investigating fractal and other aspects of these applications, as well as carrying out
research on drug abuse.
Dr. Lowen has authored or co-authored some 30 refereed journal articles as well as
a collection of book chapters and proceedings papers. He holds a number of patents,
and serves as a reviewer for several technical journals and funding agencies. Over
the course of his career, he has supervised three graduate students.

AUTHORS xxiii
Malvin Carl Teich received the S.B. degree in physics
from MIT in 1961, the M.S. degree in electrical engi-
neering from Stanford University in 1962, and the Ph.D.
degree from Cornell University in 1966. His bachelor’s
thesis comprised a determination of the total neutron
cross-section of palladium metal while his doctoral dis-
sertation reported the first observation of the two-photon
photoelectric effect in metallic sodium. His first profes-
sional affiliation was with MIT’s Lincoln Laboratory in
Lexington, Massachusetts, where he demonstrated that
heterodyne detection could be achieved in the middle-
infrared region of the electromagnetic spectrum.
Teich joined the faculty at Columbia University in 1967, where he served as a
member of the Electrical Engineering Department (as Chairman from 1978 to 1980),
the Applied Physics Department, the Columbia Radiation Laboratory, and the Fowler
Memorial Laboratory for Auditory Biophysics. Extending his work on heterodyning,
he recognized that the interaction could be understood in terms of the absorption of
individual polychromatic photons, and demonstrated the possibility of implementing
the process in a multiphoton configuration. He developed the concept of nonlinear
heterodyne detection — useful for canceling phase or frequency noise in an optical
system.
During his tenure at Columbia, he also carried out extensive work in point pro-
cesses, with particular application to photon statistics, the generation of squeezed
light, and noise in fiber-optic amplifiers and avalanche photodiodes. Among his
achievements is a description of luminescence light in terms of a photon cluster
point process. This perspective led him to suggest that detector dead time could
be used advantageously to reduce the variability of this process and thereby lumi-
nescence noise. This approach was incorporated in the design of the star-scanner
guidance system for the Galileo spacecraft, which was subjected to high radio- and
beta-luminescence background noise as a result of bombardment by copious Jovian
gamma- and beta-ray emissions. In the domain of quantum optics he developed the
concept of pump-fluctuation control in which the variability of a pump point process
comprising a beam of electrons is reduced by making use of self-excitation in the
form of Coulomb repulsion. Using a space-charge-limited version of the Franck–
Hertz experiment in mercury vapor he demonstrated the validity of this concept by
generating the first source of unconditionally sub-Poisson light. His work on fiber-
optic amplifiers led to an understanding of the properties of the photon point process
that emerges from the laser amplifier and thereby of the performance characteristics
of these devices.
Teich’s interest in point processes in the neurosciences was fostered by a chance
encounter in 1974 with William J. McGill, then Professor of Psychology and Presi-
dent of Columbia University. This, in turn, led to a long-standing collaboration with
Shyam M. Khanna, Director of the Fowler Memorial Laboratory for Auditory Bio-
physics and Professor in the Department of Otolaryngology at the Columbia College
of Physicians & Surgeons. Together, Teich and Khanna carried out animal experi-

xxiv AUTHORS
ments over many years in which spike trains in the peripheral auditory system were
recorded. Analysis of these data led to the discovery that, without exception, action-
potential sequences in the auditory system exhibited fractal features. Teich and his
students, including Lowen, developed suitable point-process models to accommodate
these data and to offer a fresh mathematical perspective of sensory neural coding. In
a collaboration with researchers at the Karolinska Institute in Stockholm, they also
conducted heterodyne velocity measurements of the vibratory motion of individual
sensory cells in the cochlea, discovering that these cells can vibrate spontaneously,
even in the absence of a stimulus.
In 1995 Teich was appointed Professor Emeritus of Engineering Science and Ap-
plied Physics at Columbia. He joined Boston University, where he is currently teach-
ing and pursuing his research interests as a faculty member with joint appointments
in the Departments of Electrical and Computer Engineering, Physics, Cognitive and
Neural Systems, and Biomedical Engineering. He is Co-Director of the Quantum
Imaging Laboratory and a Member of the Photonics Center, the Hearing Research
Center, the Program in Neuroscience, and the Center for Adaptive Systems. He also
serves as a consultant to government and private industry.
His current efforts in the domain of quantum optics are directed toward developing
imaging systems that make use of entangled photon pairs generated in the nonlinear
optical process of parametric down-conversion. His work in fractals and wavelets is
directed toward understanding biological phenomena such as the statistical properties
of neurotransmitter exocytosis at the synapse, action-potential patterns in auditory-
and visual-system neurons, and heart-rate-variability analysis of patients who suffer
from cardiovascular-system dysfunction.
Teich is a Fellow of the Acoustical Society of America, the American Association
for the Advancement of Science, the American Physical Society, the Institute of
Electrical and Electronics Engineers, and the Optical Society of America. He is
a member of Sigma Xi and Tau Beta Pi. In 1969 he received the IEEE Browder
J. Thompson Memorial Prize for his paper “Infrared Heterodyne Detection.” He
was awarded a Guggenheim Fellowship in 1973. In 1992 he was honored with the
Memorial Gold Medal of Palack´
y University in the Czech Republic, and in 1997 he
received the IEEE Morris E. Leeds Award.
He has authored or coauthored some 300 journal articles and holds a number of
patents. He is the coauthor, with Bahaa Saleh, of Fundamentals of Photonics (Wiley,
1991).
Among his professional activities, he served as a member of the Editorial Advisory
Panel for the journal Optics Letters from 1977 to 1979, as a Member of the Editorial
Board of the Journal of Visual Communication and Image Representation from 1989
to 1992, and as Deputy Editor of Quantum Optics from 1988 to 1994. He is currently
a Member of the Editorial Board of the journal Jemn´a Mechanika a Optika and a
Distinguished Lecturer of the IEEE Engineering in Medicine and Biology Society.

1
Introduction
Albrecht Penck (1858–1945), a
German geographer and geologist
known particularly for his studies of
glaciation in the Alps, recognized
thatthe lengthof a coastlinedepends
on the scale at which it is measured.
Lewis Fry Richardson (1881–
1953), a British mathematician and
Quaker pacifist, studied turbulence,
weather prediction, the statistics of
wars, and the relationship between
length and measurement scales.
1

2INTRODUCTION
1.1 Fractals 2
1.2 Point Processes 4
1.3 Fractal-Based Point Processes 4
Problems 6
1.1 FRACTALS
What is the length of a coastline? Albrecht Penck, a Professor of Geography at
the University of Vienna, observed more than a hundred years ago that the apparent
length of a coastline grew larger as the size of the map increased (Penck, 1894). He
concluded that this apparently simple question has a complex answer — the outcome
of the measurement depends on the scale at which the coastline is measured.
Using a detailed map of a given coastline, and carefully tracing all of its bays and
peninsulas, a sufficiently patient geographer could follow the features and arrive at a
number for its length. A more hurried geographer, using a map of lower resolution
that follows the coastline less closely, would obtain a smaller result since many of the
features seen by the first geographer would be absent. In general, higher measurement
precision yields a greater number of discernible details, and consequently results in
a coastline of greater length. The question “What is the length of a coastline?” has
no single answer.
This phenomenon is illustrated in Fig. 1.1 for a section of the Icelandic coastline
between the towns of Seyðisfj¨
orður and H¨
ofn. The three maps, illustrated in different
shades of gray, are identical; only the scale used to measure the length of the coastline
differs. The measurement indicated by the white curve on the dark-gray map traces
features with a scale of 0.694 km. Following all of the features of the map at this scale
requires 769 segments, each of length 0.694 km, which yields an overall coastline
length of 534 km. This measurement closely hugs the coastline. The medium-gray
map, whose boundary (black curve) is measured with a 6.94-km scale, requires just
over 45 segments, leading to a coastline length of 314 km. This coarser measurement
follows the coastline more approximately; the result therefore appears more jagged
and yields a shorter length. Finally, a measurement made at a scale of 69.4 km,
represented on the light-gray map, yields the shortest length of the three: 133 km.
Were the scale to increase further, a minimum distance of 125 km would obtain
for scales in excess of 125 km, since that is the point-to-point distance between the
two towns. At the opposite extreme, if the measurement scale were to reach below
0.694 km, the length of the coastline would grow beyond 534 km, as ever smaller
bays and peninsulas, rocks, and grains of sand were taken into account.
Although coastlines do not have well-defined lengths, as recognized by Penck
(1894), and subsequently by Steinhaus (1954) and Perkal (1958a,b), an empirical
mathematical relation connecting the measured coastline length and the measurement
scalewas discovered by Lewis Fry Richardson. In his Appendix to Statistics of Deadly
Quarrels, which appeared in print some years after his death, Richardson (1961)

FRACTALS 3
MEASUREMENT
SCALE (km)
MEASURED
LENGTH (km)
0.694
6.94
69.4
534
314
133
ATL ANTIC
OCEAN
ICELAND
SEYÐISFJÖRÐUR
HÖFN
Fig.1.1 The coastlineof Iceland betweenSeyðisfj¨
orðurand H¨
ofn,measured atthree different
scales. The finest-scale measurement is shown as the white curve on the dark-gray (top) map.
The inset table indicates the measured coastline length for the three different scales. The finer
the scale of the measurement, the greater the detail captured, and the larger the outcome for
the length of the coastline.
showed that this relation takes the form of a power-law function,
d∝sc,(1.1)
where disthe lengthof the coastline, sis the measurement scale, and cis a (negative)
power-law exponent.1A circle, in contrast, does not fit this form, suggesting that real
coastlines do not resemble simple geometrical shapes, and that Richardson’s relation
is not spurious.
1In Statistics of Deadly Quarrels, Richardson (1960) had previously demonstrated that the magnitude of
a war related to its frequency of occurrence by means of a power-law function. For each tenfold increase
in size, he found roughly a threefold decrease in frequency. He also determined that the occurrences of
wars closely follow a Poisson process (see Sec. 4.1), albeit with a quasi-periodic modulation of the rate.
He further concluded that states bordering a large number of contiguous states tended to become involved
in wars more often — hence, his attention to the lengths of frontiers and coastlines. A biographical sketch
of Richardson is provided by Mandelbrot (1982, Chapter 40).

4INTRODUCTION
Not long after Richardson’s work was published, BenoitMandelbrot (1967a, 1975)
revisited the issueof coastline length, and beganto lay the groundworkfor what would
later be called fractal analysis.2The dependence of a measurement outcome on the
scale chosen to make that measurement is the hallmark of a fractal object, and
coastlines are indeed fractal. The power-law relationship provided in Eq. (1.1) offers
a useful description of coastlines and other fractal objects, as we will see in Chapter 2.
1.2 POINT PROCESSES
After a long day measuring coastlines, our geographer drives home for the night.
Unfortunately, many other people have chosen this same hour to drive, and our geog-
rapher encounters traffic. Since this is a recurring problem, the local government has
decided to charter a study of the traffic flow patterns along the road from the coast.
What is the best method for describing the traffic?
Certain details of the vehicles, such as their color or the number of occupants, are
irrelevant to the traffic flow. To first order, a listing of the times at which a vehicle
crosses a given point on the road provides the most salient information. Such a record,
in the form of a set of marks on a line, is called a point process.3The mathematical
theory of point processes forms a surprisingly rich field of study despite its seeming
simplicity.4
Figure 1.2 illustrates the process of reducing a moving set of vehicles on a roadway
to a point process. The figure comprises snapshots of the same stretch of single-lane
roadway, at different times as successive vehicles pass a fixed measurement location
(indicated by vertical dashed line). Vehicles yet to reach the measurement location
are shown as white. As each vehicle passes this location, it turns light gray, and the
point-process record at the right accrues a corresponding mark at that time, indicating
the passing event. Many of the vehicles (labeled by letters) appear in several of the
snapshots. Dark gray indicates a vehicle that passed the dashed line elsewhere in the
figure whereas black indicates a vehicle that passed this location yet earlier.
1.3 FRACTAL-BASED POINT PROCESSES
This book concerns fractal-based point processes — processes with fractal proper-
ties comprised of discrete events, either identical or taken to be identical.
2A photograph of Mandelbrot stands at the beginning of Chapter 7. A recent interview by Olson (2004)
offers some personal reminiscences about his life and career.
3A point process is sometimes called a time series, although this latter designation usually refers to a
discrete-time process.
4Point processes relevant to vehicular traffic flow have been studied, for example, by Chandler, Herman
& Montroll (1958); Komenani & Sasaki (1958); Bartlett (1963, 1972); Newell & Sparks (1972); Bovy
(1998); and Kerner (1998, 1999).

FRACTAL-BASED POINT PROCESSES 5
POINT
PROCESS
A
B
C
D
E
TIME
t
D C B A
F E D C B A
F E D C B A
G F E D C B
H G F E D C
Fig. 1.2 Generation of a point process from vehicular traffic. Each horizontal line depicts
vehicles traveling along the same stretch of single-lane road, but at different times. As each
successive vehiclepasses a measurement location(indicated by the vertical dashed line), it turns
light gray in color and a corresponding mark appears in the final point-process representation
(short horizontal arrow at far right), denoting the occurrence of an event at that time. In each
depiction of the roadway, vehicles that have not yet passed the dashed line are shown as white,
whereas those that already passed it elsewhere in the figure appear as dark gray. Vehicles
shown in black passed the dashed line at a yet earlier time.
A more detailed introduction to fractals, and their connection to chaos, is provided
in Chapter 2. We define point processes, and consider measures useful for charac-
terizing them, in Chapter 3.Chapter 4 sets forth a number of important examples
of point processes. With an understanding of fractals and point processes in hand,
we address their integration in Chapter 5. Mathematical formulations for several im-
portant fractal-based point-process families follow, in Chapters 6–10. An exposition
of how various operations affect these processes appears in Chapter 11.Chapter 12
considers techniques for the analysis and estimation of fractal-based point processes.
Finally, Chapter 13 is devoted to computer network traffic, an important application
that serves as an illustration of the various approaches and models set forth in earlier
chapters.

6INTRODUCTION
Problems
1.1 Length of Icelandic coastline at different measurement scales Table 1.1
provides results for the length of a portion of the east coast of Iceland, measured
between the towns of Seyðisfj¨
orður and H¨
ofn, at different measurement scales.
Measurement Scale sNumber of Segments Coastline Length d
(km) (km)
0.694 769 534
1.39 306 425
2.78 140 389
6.94 45.2 314
13.9 20.3 282
27.8 6.33 176
50.0 2.84 142
69.4 1.92 133
Table 1.1 Length of the Icelandic coastline between the towns of Seyðisfj¨
orður and H¨
ofn,
determined using eight different measurement scales. The finer the scale, the greater the length.
These measurements were made from a map with a resolution of 0.694 km, as determined by
the edge length of the minimum pixel size. The point-to-point distance between the two towns
is 125 km.
1.1.1. Plot the coastline length dvs. the measurement scale son doubly loga-
rithmic coordinates.
1.1.2. Use the plot generated in Prob. 1.1.1, together with Eq. (1.1), to determine
the power-law exponent cthat characterizes the eastern Icelandic coastline.
1.1.3. Using this same form of analysis, Richardson (1961, p. 169) showed that
Eq. (1.1) indeed characterized several coastlines. He reported the following results:
(1) c≈ −0.02 for the South African coastline between Swakopmund and Cape Santa
Lucia; (2) c≈ −0.13 for the Australian coastline; and (3) c≈ −0.25 for the west
coast of Great Britain. Compare the scaling exponent cyou obtain for the east coast
of Iceland with those determined by Richardson. Consult an atlas to estimate the
relative roughness of the four coastlines and relate this to the power-law exponents c.
1.2 Circles and fractals Suppose that we calculate entries for a table similar to
the ones that appear in Fig. 1.1 and Table 1.1, but for a circle of unit circumference.
To measure the circumference with nequal line segments, inscribe a regular polygon
of nsides into the circle, and calculate the perimeter of the polygon. This procedure
yields a perimeter that increases as the polygon side length decreases, as shown in
Table 1.2.

PROBLEMS 7
Polygon Side Length Number of Sides Polygon Perimeter
0.033 30 0.998
0.098 10 0.984
0.187 5 0.935
0.276 3 0.827
Table 1.2 Polygon approximation for the perimeter of a circle.
1.2.1. Calculatean exact expressionfor the side lengthand total estimated perime-
ter as a function of the number of sides, and verify that the perimeter monotonically
increases with the number of sides.
1.2.2. Is a circle a fractal? Why?

2
Scaling, Fractals, and
Chaos
Georg Cantor (1845–1918), a
celebrated German mathematician,
founded set theory and recognized
the distinction between countably
infinite and uncountably infinite
sets, such as the sets of rational and
real numbers, respectively.
The French mathematician Henri
Poincar´
e (1854–1912) established
that certain deterministic nonlinear
dynamical systems exhibit an acute
sensitivity to initial conditions; this
characteristic is now recognized as
a hallmark of deterministic chaos.
9

10 SCALING, FRACTALS, AND CHAOS
2.1 Dimension 11
2.1.1 Capacity dimension 12
2.2 Scaling Functions 13
2.3 Fractals 13
2.3.1 Fractals, scaling, and long-range dependence 14
2.3.2 Monofractals and multifractals 15
2.4 Examples of Fractals 16
2.4.1 Cantor set 17
2.4.2 Brownian motion 19
2.4.3 Fern 21
2.4.4 Grand Canyon river network 22
2.5 Examples of Nonfractals 23
2.5.1 Euclidean shapes 23
2.5.2 Homogeneous Poisson process 23
2.5.3 Orbits in a two-body system 24
2.5.4 Radioactive decay 24
2.6 Deterministic Chaos 25
2.6.1 Nonchaotic system with nonfractal attractor 26
2.6.2 Chaotic system with nonfractal attractor 28
2.6.3 Chaotic system with fractal attractor 28
2.6.4 Nonchaotic system with fractal attractor 30
2.6.5 Chaos in context 31
2.7 Origins of Fractal Behavior 32
2.7.1 Fractals and power-law behavior 32
2.7.2 Physical laws 33
2.7.3 Diffusion 34
2.7.4 Convergence to stable distributions 35
2.7.5 Lognormal distribution 36
2.7.6 Self-organized criticality 37
2.7.7 Highly optimized tolerance 37
2.7.8 Scale-free networks 37
2.7.9 Superposition of relaxation processes 38
2.8 Ubiquity of Fractal Behavior 39
2.8.1 Fractals in mathematics and in the physical sciences 39
2.8.2 Fractals in the neurosciences 41
2.8.3 Fractals in medicine and human behavior 43
2.8.4 Recognizing the presence of fractal behavior 44
2.8.5 Salutary features of fractal behavior 45
Problems 46

DIMENSION 11
2.1 DIMENSION
The word “dimension,” at least among the technically inclined, generally conjures
up an image of a familiar elemental shape such as a line or rectangle. These objects
have dimensions that correspond to the measurements used to quantify them: meters
and square meters, respectively. Table 2.1 presents four representative “Euclidean”
objects along with their dimensions (Mandelbrot, 1982).
Object Measurement Dimension
Point meters00
Line meters11
Square meters22
Cube meters33
Table 2.1 Representative objects: measurements and dimensions.
The union of two objects that have a particular dimension is characterized by that
same dimension. Thus, any finite number of points retains a dimension of zero, and
three squares connected end to end as a whole maintain a dimension of two. What
happens to the dimension as we deform the objects, converting a line into a curve,
say, or a square into an ellipse? Common sense suggests that the dimension of an
object is robust in the face of such manipulations, and topological theory bears this
out. A curve in three-dimensional space, such as a helix, maintains a dimension of
unity since uncoiling the helix yields a line of that dimension.
The foregoing discussion illustrates a general property: the dimension of an object
cannot exceed the dimension of the space in which it resides (the Euclidian dimen-
sion). An infinite collection of points immediately adjacent to each other yields a
curve, and one can generate a circle from the union of all possible points equidistant
from a given point. In both cases, the component objects have a dimension smaller
than that of the space. However, one can form a square from a collection of smaller
squares, and all squares have a dimension of two. These examples lead to another
property: the dimension of each member of a group of objects (the topological
dimension) cannot exceed the dimension of the object formed by their union.
Taken together, the two properties reveal that a collection of objects of a particu-
lar dimension, embedded in an object with another dimension, will have an overall
dimension that lies between the two. For example, a collection of points (dimension
zero) lying within a square (dimension two) can yield a line (dimension one). Yet, a
different collection of points could yield a square of smaller size (dimension two), or
a single point (dimension zero). In all cases, however, the dimension of the resulting
object lies between zero and two inclusive, in accord with the properties set forth
above.

12 SCALING, FRACTALS, AND CHAOS
2.1.1 Capacity dimension
Although the concept of dimension as discussed above has intuitive appeal, it is im-
portant to develop a more rigorous approach for quantifying dimension. We illustrate
one measure, initially introduced by Pontrjagin & Schnirelmann (1932), called the
capacity dimension or box-counting dimension (this technique is discussed in more
detail in Sec. 3.5.4). Imagine an ellipse (including its interior) in a plane; we know
this object has a dimension of two. Suppose we draw a grid over the ellipse and the
surrounding region, yielding a collection of square boxes, and then count the number
of squares that overlap at least one point in the ellipse. For squares of a given edge
size ², we obtain a number M(²). (The precise value of this number depends on
the alignment of the grid with respect to the ellipse, but this fact does not affect the
following argument.)
Now repeat the process with squares half the size of the original ones. The number
of squares required to cover the ellipse will increase roughly by a factor of four, since
four new squares cover one of the original ones. Thus, M(²/2) ≈4M(²). In the
limit, as the size of the squares decreases towards zero, we find
M(²)→C²−2,(2.1)
where the constant Crepresents the area of the ellipse, and the alignment of the grid
does not affect this result. To extract the exponent from Eq. (2.1), we first take the
logarithm, which yields a relation linear in the exponent. Dividing by the logarithm of
the inverse box size and taking the limit for small boxes yields the desired exponent:
lim
²→0
ln£M(²)¤
ln(1/²)=2,(2.2)
which coincides with the dimension of an ellipse. This suggests a general method for
obtaining the capacity dimension that will report the correct exponent even when it
may not be readily apparent in the functional form of M(²).
Now suppose that we repeat the process with a curve in a plane. In this case, the
number of squares required increases linearly with 1/², whereupon Eq. (2.2) yields
a value of unity. Similarly, a finite collection of npoints requires no more than n
squares, no matter how small ²becomes, resulting in a dimension of zero.
In general, the box-counting technique of determining the dimension proceeds by
covering the set in question with “boxes,” namely cubes, squares, line segments, or
other forms, depending on the space within which the shape lies. The relationship
between the number of boxes that contain part of the set and the size of those boxes,
as the size decreases to zero, determines the capacity dimension D0of the set:
M(²)→C²−D0.(2.3)
Thus far, the outcome agrees with our intuition about dimension. When applied to
fractals, however, we will see that this approach leads to noninteger values, although
it is always bounded from above by the Euclidian dimension, and from below by the
topological dimension.

SCALING FUNCTIONS 13
2.2 SCALING FUNCTIONS
Fractals turn out to have close connections to the scaling behavior observed in some
functions. A function is said to “scale” when shrinking or stretching both axes (by
possibly different amounts, neither equal to unity) yields a new graph that coincides
with the original. Scaling leads mathematically to power-law1dependencies in the
scaled quantities, as we now proceed to show.
Consider a function fthat depends continuously on the scale sover which we
take measurements. Suppose that changing the scale by any factor aeffectively
changes the function by some other factor g(a), which depends on the factor abut is
independent of the original scale s:
f(as) = g(a)f(s).(2.4)
The only nontrivial solution of this scaling equation for real functions and arguments,
and for arbitrary aand s, is (see Prob. 2.5)
f(s) = b g(s),(2.5)
with
g(s) = sc(2.6)
for some constants band c(Lowen & Teich, 1995; Rudin, 1976). Equations (1.1) and
(2.3) provide examples of this relationship.
Restricting ato a fixed value in Eq. (2.4) yields a larger set of possible solutions
(Shlesinger & West, 1991):
g(s;a) = sccos[2πln(s)/ln(a)].(2.7)
2.3 FRACTALS
The concept of a fractal involves three closely related characteristics, each of which
could serve as a definition in its own right. Indeed, a variety of definitions for fractals
exist (Mandelbrot, 1982). Furthermore, fractals can be deterministic orrandom. They
can also be static, such as the Icelandic coastline, or arise from a dynamical process
such as Brownian motion.
First, fractals possess a form of self-scaling: parts of the whole can be made to fit
to the whole in some nontrivial way by shifting and stretching. If stretching equally
in all directions yields such a fit, then an object is said to be self-similar. If the fit
requires anisotropic stretching, then the object is said to be self-affine (Mandelbrot,
1Power-law functions have many aliases, including “algebraic,” “hyperbolic,” and “allometric.” When
applied to distributions, the term “heavy-tailed” often (but not always) refers to the same functional form.

14 SCALING, FRACTALS, AND CHAOS
1982). For deterministic fractals, the fit is exact. Random fractals, in contrast, fit
statistically; transformed parts resemble the whole and have similar probabilistic
characteristics, although they do not precisely coincide with it. The coast of Iceland,
for example, contains similar features over a range of sizes. As illustrated in Fig. 1.1,
what would appear to be a simple bay on a large-scale (coarse-grained) map turns
into a meandering connection of inlets and other invaginations when displayed more
finely. Examining a length of coastline on a map (without intimate knowledge of
the particular section under study) does not provide information about the scale of
the map despite knowledge of the size of the entire object, in this case Iceland. In
contrast, examining of a section of nonfractal object, such as a circle, readily yields
the scale in terms of the size of the object.
Second, the statistics that are used to describe fractals scale with the measurement
size employed. For example, the length of the east coast of Iceland follows the form
of Eqs. (2.5) and (2.6) with an empirical fractal exponent c≈ −0.30, as shown in
Prob. 1.1. Statistics with power-law forms are thus closely related to fractals. Indeed,
we often highlight this connection by presenting various measures using logarithmic
axes for both the ordinate and abscissa; power-law functions become straight lines on
such doubly logarithmic graphs and their slopes provide the power-law exponents.
This characteristic of fractals proves quite useful.
Third, the fractal exponent that corresponds to a particular statistic, one of the
generalized dimensions (see Sec. 3.5.4), assumes a noninteger value. As their size
decreases, the number of boxes required to cover the Icelandic coastline increases in
such a way that the capacity dimension D0≈1.30 (see Secs. 2.1 and 3.5.4). This
scaling exponent assumes a noninteger value lying between that of a line (D0= 1)
and that of a plane (D0= 2).
2.3.1 Fractals, scaling, and long-range dependence
Fractal behavior, such as an object containing smaller copies within itself, can extend
down to arbitrarily small sizes in an abstract mathematical construct. However, real-
world fractals generally exhibit minimum sizes beyond which fractal behavior is not
obeyed. For example, decreasing the length scale used to measure the length of a
coastline will eventually lead to a breakdown in scaling behavior. The geological
forces at work over kilometer scales differ from those operating over much smaller
length scales, leading to different appearances over these smaller lengths. Certainly
at a scale corresponding to individual atoms, the emergent features are expected to
bear little resemblance to those at macroscopic length scales.
There are also limits at large scales. Fractal behavior can have a maximum scale,
one that often corresponds to the size of the fractal object itself. The minimum and
maximum scales that bound fractal behavior are known as the lower cutoff (or inner
cutoff) and the upper cutoff (or outer cutoff), respectively.
Moreover, anydata set collected from a real-life physical or biological experiment
will perforce have lower and upper cutoffs, corresponding to the resolution limits of
the measurement apparatus and the extent of the entire data set, respectively. These
lower and upper measurement cutoffs impose limits on observable fractal behavior

FRACTALS 15
that sometimes prove more restrictive than those of the fractal object under study
itself. In the case of the Icelandic coastline portrayed in Fig. 1.1, for example, the
resolution of the map from which the measurement is constructed (0.694 km), which
is determined by the edge length of the minimum pixel size (see Prob. 1.1), imposes
a lower cutoff. An upper cutoff is imposed by the size of the island itself.
Generating a point process from a rate (see Chapter 4) necessarily involves some
loss of information, and can also set an effective minimum scale. In both the dou-
bly stochastic and integrate-and-reset point processes considered in Chapter 4, for
example, fractal features present in the rate process over time scales shorter than the
average time between events will be greatly attenuated in the resultant point process.
In a more rigorous mathematical context, fine distinctions are sometimes drawn
between fractals and scaling (Flandrin, 1997; Flandrin & Abry, 1999). The term
”scaling” is used when both lower and upper cutoffs exist, “fractal” denotes objects
for which no small-size cutoff exists, and “long-range dependence” corresponds to
the lack of a large-size cutoff.2Since essentially all of the applications we consider
derive from limited measurements, our discussion might be more properly framed in
terms of scaling rather than fractal behavior. Following common usage, however, we
generally do not make this distinction. The L´
evy dust, considered in Sec. 4.7, and the
zero crossings of ordinary and fractional Brownian motion, considered in Sec. 6.1,
are the sole exceptions. These two collections of points, which, properly speaking,
are not point processes, are fractal in the strict sense of the term.
2.3.2 Monofractals and multifractals
Thescaling behaviordiscussed thusfar involves asingle stretchingor shiftingrule, and
a single exponent for each statistic. For some objects, the rule and exponents depend
on the position within the object, or on the size of the component. Each such object
can thus contain a range of fractal behaviors, and is therefore called a multifractal
(Mandelbrot, 1999; Sornette, 2004). In this context, a simpler fractal object described
in our earlierdiscussions is calleda monofractal. Although examplesof multifractals
can be found, in practice relatively few point-process data sets contain sufficient
information to accurately characterize their multifractal spectrum.
Perhaps the best method for attempting such a characterization leads to a mul-
tifractal spectrum by simulating a number of surrogate data sets with different pa-
rameters, and selecting the best fitting parameters as estimates of the multifractal
behavior (Roberts & Cronin, 1996). This method yields good accuracy with as few
as N= 100 points. However, its inherent parametric approach limits its usefulness
in general, since the algorithm requires a priori knowledge of the form of the mul-
2More precisely, a process is said to have long-range dependence when its autocorrelation has an infinite
integral (for continuous-time processes) or an infinite sum (for discrete-time processes) (Cox, 1984).
Theoretically, a process could have long-range dependence without exhibiting power-law behavior, but
this is uncommon.

16 SCALING, FRACTALS, AND CHAOS
tifractal spectrum. Indeed, it estimates only two parameters from the data set rather
than an arbitrary form for the multifractal spectrum.
Finally, such methods typically require that the point processes themselves, rather
than merely the rates of these processes, exhibit (multi)fractal behavior. Such fractal
point processes (see Sec. 5.5.1) form an important subclass of the class of fractal-
based point processes (see Sec. 5.5) that we explore, but they do not describe a large
number of data sets.
As a consequence of these limitations, we concentrate principally on monofractals
in this book. Computer network traffic is a notable exception: the availability of
extensive, long data records allow a valid multifractal analysis to be carried out (see
Sec. 13.3.8).
2.4 EXAMPLES OF FRACTALS
Fractals abound in many fields: mathematics (Mandelbrot, 1982; Stoyan & Stoyan,
1994; Peitgen, J¨
urgens & Saupe, 1997; Barnsley, 2000; Mandelbrot, 2001; Falconer,
2003; West, Bologna, Grigolini & MacLachlan, 2003; Doukhan, 2003); physics
(Mandelbrot, 1982; Feder, 1988; Schroeder, 1990; Sornette, 2004); geology (Tur-
cotte, 1997); imaging science (Peitgen & Saupe, 1988; Turner, Blackledge & An-
drews, 1998; Flake, 2000); electronic devices and systems (Buckingham, 1983; van
der Ziel, 1986, 1988; Weissman, 1988; Kogan, 1996); complex electronic and pho-
tonic media (Berry, 1979; Merlin, Bajema, Clarke, Juang & Bhattacharya, 1985;
Kohmoto, Sutherland & Tang, 1987); materials growth (Kaye, 1989; Vicsek, 1992);
signal processing (Flandrin & Abry, 1999); engineering (L´
evy V´
ehel, Lutton &
Tricot, 1997); vehicular-traffic behavior (Musha & Higuchi, 1976; Bovy, 1998);
computer networks (Mandelbrot, 1965a; Leland, Taqqu, Willinger & Wilson, 1994;
Albert, Jeong & Barab´
asi, 1999; Park & Willinger, 2000); biology and physiol-
ogy (Musha, 1981; Turcott & Teich, 1993; Bassingthwaighte, Liebovitch & West,
1994; West & Deering, 1994, 1995; Collins, De Luca, Burrows & Lipsitz, 1995;
Turcott & Teich, 1996; Liebovitch, 1998; Vicsek, 2001; Teich, Lowen, Jost, Vibe-
Rheymer & Heneghan, 2001; Shimizu, Thurner & Ehrenberger, 2002); behavior
and psychiatry (Paulus & Geyer, 1992; West & Deering, 1995; Gottschalk, Bauer
& Whybrow, 1995; Teicher, Ito, Glod & Barber, 1996; Anderson, Lowen, Renshaw,
Maas & Teicher, 1999; Anderson, 2001); neuroscience (Verveen, 1960; Evarts, 1964;
Musha, Takeuchi & Inoue, 1983; L¨
auger, 1988; Millhauser, Salpeter & Oswald, 1988;
Teich, 1989; Lowen & Teich, 1996a; Teich, Turcott & Siegel, 1996; Teich, Heneghan,
Lowen, Ozaki & Kaplan, 1997; Thurner, Lowen, Feurstein, Heneghan, Feichtinger
& Teich, 1997; Lowen, Cash, Poo & Teich, 1997b); fractals also play important roles
in other fields.
We proceed to consider four examples of fractals, one each of the possible com-
binations of
•artificial and natural
•deterministic and random.

EXAMPLES OF FRACTALS 17
2.4.1 Cantor set
The Cantor set, discovered by Georg Cantor (1883), provides an example of an
artificial, deterministic, one-dimensional fractal structure that extends to arbitrarily
small scales. One particular mathematical construction of this set has as its starting
point the closed unit interval
C0≡[0,1].(2.8)
From C0, we form C1, the next step in the formation of the triadic Cantor set, by
removing the middle third of this interval:
C1≡£0
3,1
3¤S£2
3,3
3¤,(2.9)
where ∪represents the set union operation. We then obtain C2from C1by removing
the middle thirds of both segments, so that
C2≡£0
9,1
9¤S£2
9,3
9¤S£6
9,7
9¤S£8
9,9
9¤.(2.10)
The set Cndenotes the nth stage in this process. This procedure is continued indefi-
nitely, leading to the Cantor set itself, C, which is defined as the limit
C ≡ lim
n→∞ Cn.(2.11)
Fig. 2.1 The first six stages in the construction of a triadic Cantor set. The process begins
with the unit interval; removing the middle third of each segment at a given stage yields the
following stage. Continuing this process indefinitely yields the Cantor set itself as a limit.
The first six stages in the construction of a triadic Cantor set are displayed in
Fig. 2.1. The Cantor set Cconsists of two exact copies of itself, in the intervals £0,1
3¤
and £2
3,1¤,respectively, each of which is one-third the size of the whole. It also
contains four copies of itself, each one-ninth the size of the whole. In fact it has 2n
copies, each 3−nthe size of the original set, for all nonnegative integers n. For the
triadic Cantor set, increasing the length scale ²by a factor of 3 decreases the number
of copies N(²)by a factor of 2, so that N(²)∼²−D0with D0≡log(2)/log(3) .
=

18 SCALING, FRACTALS, AND CHAOS
0.630930. There is, in fact, an entire family of generalized dimensions Dq(see
Sec. 3.5.4) but for monofractals such as the Cantor set, these all coincide so that
Dq=Dfor all q.
A set with infinitely many copies of itself, each of vanishing size, can exhibit
counterintuitive properties. Such seeming paradoxes often occur in the study of
abstract fractals, and we now proceed to examine one in the light of the Cantor set:
this set has a total length of zero, but just as many points as the unit interval employed
as the first stage in its construction. To show this, we begin with the total length
of the Cantor set. The initial stage of the Cantor set consists of the unit interval,
and therefore has a Lebesgue measure or length L(C0)of unity. The second stage
has 2 segments of length 1
3,for a total length L(C1) = 2
3;the nth stage comprises
2nsegments each of length 1/3n, yielding L(Cn)=(2
3)n.Continuing this process
indefinitely leads to the total length of the Cantor set itself as a limit:
L(C) = lim
n→∞ L(Cn) = lim
n→∞ ¡2
3¢n=0.(2.12)
So the Cantor set has zero total length.
Turning now to the number of points in the Cantor set, consider a ternary expansion
of the points in the original interval C0≡[0,1]. Each point xin this interval may be
represented by a corresponding sequence of digits 0.a1a2a3a4. . ., where
x=X
k
ak¡1
3¢k,(2.13)
with each akeither 0, 1 or 2. Points of C0contained in the open interval ¡1
3,2
3¢will
not appear in C1, and all have a 1 in the first position after the decimal point of their
ternary expansions. (The point 1
3,the upper limit of the first segment, also has a 1
in the first position, but remains in C1and in Cas well. We will return to the issue
of endpoints shortly.) Points with a 1 in the second position after the decimal point
correspond to the middle third of both segments of C1, and will not appear in C2or in
subsequent stages. Thus, Cncontains only those points without a 1in any of the first
npositions of the corresponding expansions.
In the limit, then, the Cantor set Ccontains only those points that do not display a 1
in any position of the corresponding expansion; the ternary expansion of any point in
Cconsists solely of the symbols 0 and 2. For example, the point corresponding to the
expansion 0.020202 . . . (base 3) =1
4belongs to C, whereas 0.111111 . . . (base 3) =
1
2does not. Points in the original interval C0may also be expanded in binary format,
with each digit chosen from the set {0,1}. Therefore, there exists a one-to-one
mapping between the points in the Cantor set Cand those in the original unit interval
C0; simply replace the “2” symbols in the ternary expansion for the former with
“1” symbols in the binary expansion of the latter. In particular, the Cantor set has
uncountably many points. The endpoints, mentioned earlier, form only a countable
subset of the Cantor set, and therefore do not change its cardinality.
Variants of the Cantor set described above, in which each stage in the construction
removes a fraction of the points removed in constructing the ordinary Cantor set (see,
for example, Rana, 1997), are known as fat Cantor sets. Consider removing only the

EXAMPLES OF FRACTALS 19
middle c/3nof the segments employed in the construction of C, for example, where
0< c < 1. Here we remove a total of 2n−1segments of width c/3nin constructing
the nth stage of the fat Cantor set CF. The total width remaining becomes
L(CF
n)=1−
n
X
m=1
2m−1c/3m
= lim
n→∞ L(CF
n)
= lim
n→∞ Ã1−
n
X
m=1
2m−1c/3m!
= 1 −(c/3) /¡1−2
3¢
=1−c . (2.14)
For c= 1 we recover the result that L(C) = 0. The set CFtherefore has a Lebesgue
measure of 1−c, but with an uncountably infinite number of points missing when
compared with the unit interval C0. In particular, in this set an infinite number of
intervals of infinitesimal size exist near any point that belongs to C. For the Cantor
set itself, each point in Chas an infinite number of neighbors in Cthat are arbitrarily
close to it, but Clacks intervals of any length.
2.4.2 Brownian motion
We use Brownian motion as an example of an artificial, random fractal. Brown-
ian motion has a long and storied history in the annals of several scientific dis-
ciplines: biology (Brown, 1828), financial mathematics (Bachelier, 1900), physics
(Einstein, 1905; Perrin, 1909), and mathematics (Wiener, 1923; Kolmogorov, 1931;
L´
evy, 1948). The first observation of this phenomenon appears to have been made in
1785 by Jan Ingenhousz, a Dutch physician, in the course of examining the behavior
of powdered charcoal on the surface of alcohol (see Klafter, Shlesinger & Zumofen,
1996, p. 33), but the term Brownian motion arose following the Scottish botanist
Robert Brown’s (1828) description of the movement of pollen grains in water.
In accordance with general usage, however, we denote as Brownian motion a par-
ticular continuous-time random process known as a Wiener–L´
evy process (Wiener,
1923; L´
evy, 1948) [alternate appellations are Wiener process (Wiener, 1923) and
Bachelier process (Bachelier, 1900, 1912)]. Thus, Brownian motion, like the Cantor
set, is considered to be an abstract construction, although it does closely approximate
much experimental data. Unlike the Cantor set, however, Brownian motion involves
randomness in its definition. Different realizations of the Brownian-motion process
appear different, although all are governed by the same statistical properties.
One definition of Brownian motion B(t), t ≥0, involves the following three
properties. First, Brownian motion is a Gaussian process; this signifies that a vector
{B(t1), B(t2),..., B (tk)}for any positive integer kand any set of times {t1, t2,..., tk}
has a joint Gaussian (normal) distribution. Second, the mean is zero: E[B(t)] = 0

20 SCALING, FRACTALS, AND CHAOS
Fig. 2.2 A realization of Brownian motion. Time increases towards the right, with the origin
at the left.
for all t, where E[·]represents expectation or mean. Third, the autocorrelation3of
the process at two times sand tequals the smaller of the two times
E[B(s)B(t)] = min(s, t)(2.15)
for all sand t, where min(x, y)returns the smaller of xand y. Figure 2.2 displays a
realization of Brownian motion.
We can derive a number of other characteristics from these three properties. In
particular, Brownian motion contains statistical copies of itself. To see this, define a
new function B∗(t)≡B(at), a version of Brownian motion with a rescaled time axis.
The random process B∗(t)also belongs to the Gaussian family of random processes,
with a mean of zero and an autocorrelation
E[B∗(s)B∗(t)] = E[B(as)B(at)] = amin(s, t).(2.16)
Now consider rescaling the amplitude of B∗(t): define B†(t) = a−1/2B∗(t) =
a−1/2B(at). Like B(t)and B∗(t),B†(t)is a zero-mean Gaussian process. The
autocorrelation for B†(t)is therefore written as
E[B†(s)B†(t)] = E[a−1/2B∗(s)a−1/2B∗(t)]
=a−1E[B(as)B(at)]
=a−1amin(s, t)
= min(s, t),(2.17)
which is identical to that of the original process B(t).
3We define autocorrelation as the expectationof the product of a process at two different times or delays,
while autocovariance denotes the result with the mean value removed. For zero-mean processes, the two
coincide.

EXAMPLES OF FRACTALS 21
Since B†(t)and B(t)are Gaussian processes with the same mean and autocorre-
lation, the two processes are statistically identical (Feller, 1971). Thus, changing the
time axis by a scale aand the amplitude axis by a scale aH, with H=1
2,yields the
same result, and B(t)contains statistical copies of itself at any scale. In Chapter 6 we
consider a generalization of Brownian motion, called fractional Brownian motion, in
which the parameter Hcan assume any value between zero and unity.
2.4.3 Fern
We move now from abstract mathematical fractal objects, such as the Cantor set
and Brownian motion, to natural fractal objects. These are ubiquitous in the real
world. A simple fern provides a particularly clear example. Figure 2.3 displays the
main frond of a fern (oriented vertically), which contains many sub-fronds (oriented
horizontally), each a miniature copy of the whole.
Fig. 2.3 A fern, an example of a natural fractal with little randomness. The main frond com-
prises many sub-fronds, each a miniature copy of the whole. This fern, Athyrium filix-femina
(Lady Fern), was collected from the backyard of the first author’s residence in Massachusetts.
It is well described as a deterministic fractal.
This scaling continues; eachsub-frond contains sub-sub-fronds(oriented vertically
again), and at the bottom of the figure there is evidence for a fourth level of detail.

22 SCALING, FRACTALS, AND CHAOS
Like all objects in the physical world, ferns have a minimum scale for their fractal
behavior, here at the fourth level. The copies, while not perfect replicas of the whole,
do not differ much from it; the fern is well described as a deterministic fractal.
2.4.4 Grand Canyon river network
Fig. 2.4 An overhead view of the Grand Canyon, a random natural fractal gouged out by the
Colorado river. The main canyon contains many sub-canyons, each resembling the whole in
a statistical manner. This photograph was taken from space by astronauts during U.S. Space
Shuttle Flight STS61A. Image obtained from: Earth Sciences and Image Analysis, NASA–
Johnson Space Center; 15 November 2004; “Astronaut Photography of Earth–Display Record.”
http://eol.jsc.nasa.gov/scripts/sseop/photo.pl?mission= STS61A&roll=201&frame=75
Whereas a fern provides an example of a deterministic natural fractal, most natural
fractals exhibit randomness. Consider the Grand Canyon (Arizona), shown in Fig. 2.4.
The main canyon, running from the top left, through the center, and exiting at the
lower left, contains a number of sub-canyons along its length. While each sub-canyon
appears different from the Grand Canyon itself and from the other sub-canyons, all
resemble each other. The sub-canyons resemble the whole in a statistical manner.
Again, the scaling continues, with the sub-canyons containing still smaller sub-sub-
canyons of similar appearance within them, and so forth, down to the resolution

EXAMPLES OF NONFRACTALS 23
limit of the photograph. Here the lower limit of fractal behavior derives from the
measurement, rather than from the fractal object itself. The Grand Canyon thus
provides an example of a random natural fractal.
2.5 EXAMPLES OF NONFRACTALS
Lest the reader gain the mistaken impression that all objects are fractals, we provide
four counterexamples, again employing one each of the possible combinations of
artificial and natural, and deterministic and random.
2.5.1 Euclidean shapes
Classical Euclidean shapes, such as circles, lines, and simple polyhedra have a single,
well-defined scale, and therefore do not exhibit similar behavior over different scales.
These artificial, deterministic objects do not reveal further detail upon magnification,
nor do they possess copies of themselves. Such shapes are, therefore, not fractal.
2.5.2 Homogeneous Poisson process
Remaining in the abstract realm but turning now to random objects, we consider the
one-dimensional homogeneous Poisson process (Parzen, 1962; Cox, 1962; Haight,
1967; Cox & Isham, 1980), perhaps the simplest of all point processes (see Sec. 4.1).
Like Brownian motion, different realizations of this process have a different appear-
ance, although each is governed by the same statistical properties. A single constant
positive quantity, the rate, denotes the number of events (points) expected to occur in
a unit interval, and this quantity completely characterizes the homogeneous Poisson
process. The absence of memory completes the definition of this process; given the
rate, knowledge of the entire history and future of a given realization of a homo-
geneous Poisson process yields no additional information about the behavior of the
process at the present.
-TIME t
6 6 66 6 6 66 6 6
Fig. 2.5 Schematic representation of a one-dimensional homogeneous Poisson process. The
time axis runs horizontally to the right, and the vertical arrows depict individual events (points)
as they occur in time.
A schematic representation of a realization of a one-dimensional homogeneous
Poisson process appears in Fig. 2.5. The vertical arrows depict individual events
(points) as they occur, while the horizontal axis represents time. Although the inter-
vals between the events vary they are associated with a fixed time scale via the rate
parameter, in contrast to a fractal object. In particular, decreasing the time scale used

24 SCALING, FRACTALS, AND CHAOS
to display the process yields a more sparse version of the original that appears quite
different from it. Unlike a fractal process, it does not appear to be a random copy of
the original. Further, the probability density function for the intervals follows an ex-
ponential form [see Eq. (4.3)], rather than the power-law form of Eqs. (2.5) and (2.6).
Like the Euclidean shapes considered above, the homogeneous Poisson process is not
fractal.
2.5.3 Orbits in a two-body system
The path followedby one of the bodiesin a two-bodyorbiting system, such as the earth
and the moon,4provides an example of a natural, deterministic, nonfractal object.
Newtonian physics predicts that the two bodies will orbit about their mutual center
of mass along trajectories described by perfect ellipses, and will do so indefinitely
(Newton, 1687; Feynman, Leighton & Sands, 1963, vol. I, pp. 7-1–7-8). These orbits
have a single scale; they do not exhibit similar behavior over different scales nor do
they contain copies of themselves. The paths resemble the abstract Euclidean shapes
exemplified in Sec. 2.5.1.
In contrast, the paths traced by planets in systems containing three or more bod-
ies do exhibit fractal characteristics. This is particularly evident when the bodies
have similar masses and are separated by similar distances. Such systems exhibit
deterministic chaos (see Sec. 2.6), and chaotic systems often have fractal movement
patterns.
2.5.4 Radioactive decay
Finally, for an example of a natural, random, nonfractal process, we turn to radioactive
decay (Feynman et al., 1963, vol. I, pp. 5-3–5-5). A single radioactive atom will
decay at some random time in the future, and, while the exact time of decay remains
unknowablea posteriori, the probability of decay by any specified time is well known.
Imagine, now, a collection of identical radioactive atoms, each undergoing decay at
a random time. The registrations of these decay events form a random point process.
It is associated with a single time constant or scale: the average decay time of the
atoms.
The emissions resemble the homogeneous Poisson process presented in Sec. 2.5.2
provided that the observation times are sufficiently smaller than the average decay
time (Rutherford & Geiger, 1910). Like the Poisson process, modifying the time
scale over which the radioactive decay process is observed results in a qualitatively
different process. Moreover, the probability density function for the times between
decays does not follow the power-law form of Eqs. (2.5) and (2.6). Radioactive decay
is not a fractal process.
4For simplicity of exposition we ignore perturbations induced by other celestial bodies and tides, as well
as minor relativistic effects.

DETERMINISTIC CHAOS 25
2.6 DETERMINISTIC CHAOS
Chaos and fractals are not synonymous, although the two concepts are often conflated.
Chaos is, of course, an important topic in its own right (Poincar´
e, 1908; Devaney,
1986; Glass & Mackey, 1988; Moon, 1992; Ott, Sauer & Yorke, 1994; Strogatz, 1994;
Schuster, 1995; Alligood, Sauer & Yorke, 1996; Peitgen et al., 1997; Thompson &
Stewart, 2002; Ott, 2002). Even though chaos does not play a central role in the
treatment of fractals, we compare and contrast these two phenomena to clarify their
relationship.
Chaos describes the behavior of a deterministic nonlinear dynamical system in the
presence of the following three features. First, small changes in the initial state of
the system must lead to quite different results at some later time. For two identical
systems beginning in slightly different states, the difference between them increases
exponentially over time. Poincar´
e (1908) was the first to allude to this “sensitive
dependence on initial conditions”.5Second, and related to the first, the prediction of
system dynamics becomes increasingly more difficult as the time of prediction moves
further into the future. And third, an infinite number of unstable periodic orbits exists.
With arbitrarily smallcontinual adjustments, the dynamics ofthe system can be forced
to follow any number of periodic paths, although in the absence of such adjustments
the dynamics quickly depart from all such orbits. The diversity of behavior offered
by a chaotic system has profound consequences. Conrad (1986) has categorized
five functional roles that chaos might play in biological systems: search, defense,
maintenance, cross-level effects, and dissipation of disturbance.
Given a dynamical system, it is customary to plot the state variables in phase
space, collapsing the time information in the process. The resulting graph provides
a window on the dynamics of the system. For dissipative systems, after an initial
transient period system activity converges to a restricted region of phase space called
the attractor of the system.
Some systems have fractal attractors, also known as strange attractors. The
dynamics such systems display a rich pattern in phase space; enlarging a section of
such an attractor continues to reveal new details without limit. Many (but not all)
systems exhibiting chaos have strange attractors. Similarly, many (but not all) strange
attractors derive from systems that are chaotic. However, neither feature necessarily
implies the other. Unfortunately, the literature is rife with misconceptions pertaining
to this issue.
We proceed to demonstrate the fundamental distinction between the two concepts
by presenting examples of all four possibilities: chaotic and nonchaotic systems with
both fractal and nonfractal attractors. To facilitate comparison between the various
systems we confine ourselves to the simple class of iterated-function systems, which
5“Avery small cause that escapes our notice has a considerable effect that we cannot fail to see, and we then
say that the effect arises from chance ...but it may happen that small differences in the initial conditions
produce very large differences in the final phenomena. A small error in the former then produces a very
large error in the latter and prediction becomes impossible ...”—Poincar´
e (1908)

26 SCALING, FRACTALS, AND CHAOS
follow the form
xn+1 =f(xn)(2.18)
or xn+1 =f(xn, yn) a)
yn+1 =g(xn, yn) b) (2.19)
for the one- and two-dimensional versions, respectively.
2.6.1 Nonchaotic system with nonfractal attractor
We begin with the logistic map, a particular example of Eq. (2.18) that takes the form
of a quadratic recurrence relation,
xn+1 =f(xn) = c xn(1 −xn).(2.20)
This function maps the unit interval [0,1] to itself, and exhibits behavior that varies
with the parameter c. It is a discrete version of the logistic equation of fame in ecology
(Verhulst, 1845, 1847).
We begin by examining the stability of Eq. (2.20). A fixed point satisfies the
equation xn+1 =xn=x∗, which yields
x∗=f(x∗)
=cx∗(1 −x∗)
0 = x∗£x∗−(1 −1/c)¤.(2.21)
Eliminating the degenerate value x∗= 0 provides x∗= 1 −1/c for the remaining
fixed point. What happens to values near the fixed point determines its stability; to
assess this, we use a test value xn=x∗+²n, where ²nis a value much smaller than
unity. We then have
xn+1 =f(xn)
x∗+²n+1 =f(x∗+²n)
=c(x∗+²n)£1−(x∗+²n)¤
=c x∗(1 −x∗) + c ²n(1 −2x∗−²n)
=x∗+c ²n(1 −2x∗−²n)
²n+1/²n=c(1 −2x∗−²n)
= 2 −c(1 + ²n).(2.22)
For the nonchaotic case, we choose c= 2. The fixed point then becomes x∗= 1 −
1/c =1
2,whereupon Eq. (2.22) yields ²n+1/²n=−2²nindicating a rapid (quadratic,
in fact) relaxation towards the fixed point. Since the fixed point is stable, the attractor
of the system consists of that single point only. Thus, a plot of all possible values
xn, after transient effects have subsided, yields a single point, xn=1
2.This zero-
dimensional object has no fractal qualities whatsoever, and forms a nonfractal (non-
strange) attractor. Furthermore, since all values of xnconverge rapidly to the fixed

DETERMINISTIC CHAOS 27
point x∗, any differences among starting values must decrease, rather than increase,
over time, thereby precludinga sensitivedependence on initial conditions. The system
of Eq. (2.20) with c= 2 therefore does not exhibit chaos. Figure 2.6 displays this
convergence by showing the sequence {xn}that results from two different starting
values, x0= 0.1and 0.4, and illustrates the lack of chaos in this system.
ITERATION NUMBER
n
ITERATE VALUE
x
n
1086420
1.0
0.8
0.6
0.4
0.2
0.0
Fig. 2.6 Time course of a logistic system with parameter c= 2, for two different starting
values: x0= 0.1and 0.4. Although the two initial points differ widely, they both converge to
the same value, the fixed point x∗=1
2.This system thus does not display sensitive dependence
to initial conditions, and does not exhibit chaos.
ITERATION NUMBER
n
ITERATE VALUE
x
n
50403020100
1.0
0.8
0.6
0.4
0.2
0.0
Fig. 2.7 Time course of a logistic system with parameter c= 4, for two different starting
values: x0= 0.1and 0.1 + 10−9. Although the two initial points differ only slightly, the
iterates diverge and are completely unrelated after 30 iterations. This system does indeed
display sensitive dependence to initial conditions, and exhibits chaos.

28 SCALING, FRACTALS, AND CHAOS
2.6.2 Chaotic system with nonfractal attractor
For the purposes of this example, we again consider the logistic map of Eq. (2.20), but
now with c= 4. The nonzero fixed point becomes x∗= 1 −1/c =3
4.The stability
analysis of Eq. (2.22) now yields ²n+1/²n=−2−4²n≈ −2so that deviations
about the fixed point double in magnitude with each iteration. This fixed point thus
does not comprise the attractor. In fact, for this value of cno limit cycles exist of
any finite period and the entire interval 0< xn<1forms the attractor (Schroeder,
1990, pp. 291–294). Except for a set of measure zero, iterates of any initial value x0
will eventually come arbitrarily close to any specified value in the unit interval. This
attractor forms a simple line segment and again has no fractal properties, establishing
that for the logistic map with c= 4 the attractor is nonfractal.
Despite not having a fractal attractor, the system nevertheless displays chaos
(Schroeder, 1990, pp. 291–294). The lack of fixed points or limit cycles suggests
this, but a graphical demonstration illustrates it well. Figure 2.7 presents the se-
quence of iterations {xn}resulting from two starting values: 0.1and a value just a
bit larger, 0.1 + 10−9. Although indistinguishable at first, the difference between
the two paths grows over time, and by iteration n= 30 the two sequences exhibit
no relation to each other. This sensitivity to initial conditions illustrates the chaotic
nature of the logistic system for the parameter value c= 4.
2.6.3 Chaotic system with fractal attractor
We next turn to the H´
enon attractor (H´
enon, 1976). This two-dimensional iterated-
function system follows the form of Eq. (2.19) with
xn+1 = 1.0 + ax2
n+byna)
yn+1 =xn,b) (2.23)
with a=−1.4and b= 0.3. We first establish the fractal nature of the attractor
by simulation. Starting with (x0, y0) = (1.08003,0.305372), we iterate Eq. (2.23)
1000 times, discarding these first results to eliminate any transient behavior, and
then iterate a further 3 000 times and retain these values. Figure 2.8a) illustrates the
attractor, which forms a boomerang shape bounded by −1.3< x, y < 1.3. The
initial pair (x0, y0) = (1.08003,0.305372) belongs to the attractor, as verified by
further iterations, justifying its choice as a starting value. Enlarging a small section
of Fig. 2.8a) (the box shown at the upper right) yields a banded structure [panel b)];
further enlargements yield substantially similar forms, as shown in panels c) and
d). This self-similarity provides evidence of fractal characteristics, and in fact this
attractor is indeed a fractal object (Peitgen et al., 1997).
We now proceed to consider the chaotic nature of the H´
enon system. As before, we
employ two different starting values, (x0, y0) = (1.08003,0.305372), as in Fig. 2.8,
and (x0+²x, y0)with ²x= 10−7. Figure 2.9 shows the xvalues of the iterates
diverging so that by n= 43 the two sequences have essentially no connection to
each other, despite being almost identical at n= 0; the results resemble those for the
logistic system with c= 4, displayed in Fig. 2.7.

DETERMINISTIC CHAOS 29
This establishes the sensitivity to initial conditions of the system, and thereby pro-
vides evidence of chaos. More detailed and rigorous analysis supports this conclusion
(Peitgen et al., 1997).
ITERATE x
a)
b)
)
d)
ITERATE y
Fig. 2.8 a) Three thousand x-ypairs in the attractor of the H´
enon system as shown in
Eq. (2.23), with a=−1.4and b= 0.3. An initial 1 000 iterations were discarded to eliminate
transient effects. b) An enlargement of the small area within the box in panel a): 3 000 values
of the attractor constrained to lie in the region 0.6≤x≤0.7and 0.5≤y≤0.7. A parallel
banded structure emerges. c) A further enlargement, of the area within the box in panel b):
3000 values within 0.64 ≤x≤0.65 and 0.61 ≤y≤0.63. An enlargement of the upper
band in panel b) yields a result similar to panel b). d) A final enlargement of the area within the
box in panel c): 3 000 values within 0.644 ≤x≤0.645 and 0.622 ≤y≤0.625. The upper
band in panel c) resolves into the same pattern as seen in the whole of panels b) and c). Hence,
the attractor has similar structures over many spatial scales, suggesting that it forms a fractal
object. This system is also chaotic, as illustrated by the time course displayed in Fig. 2.9.

30 SCALING, FRACTALS, AND CHAOS
ITERATION NUMBER
n
ITERATE VALUE
x
n
6050403020100
1.0
0.5
0.0
-0.5
-1.0
Fig. 2.9 Time course of the H´
enon system with parameters a=−1.4and b= 0.3, for two
different starting values: (x0, y0) = (1.08003,0.305372), and (x0+²x, y0)with ²x= 10−7.
Again, although the two initial points differ only slightly, the iterates diverge and appear
completely unrelated by the 43 iteration. Like the logistic system with c= 4 (see Fig. 2.7),
this system displays sensitive dependence to initial conditions, and exhibits chaos. The attractor
for this system is fractal, however, as illustrated in Fig. 2.8.
2.6.4 Nonchaotic system with fractal attractor
Finally we consider a nonchaotic system which nevertheless has an attractor with
fractal characteristics. We again employ the logistic map, Eq. (2.20), with the pa-
rameter c.
= 3.56995168804. As previously, we begin by simulating the system to
illustrate the fractal nature of the resulting attractor. Using an arbitrary starting value
x0= 0.31412577217182861803, we iterate Eq. (2.20) 3 000 times after discarding
the first 1000 iterates.
The attractor is illustrated in Fig. 2.10a) — it forms a set of disconnected regions
in the unit interval 0<x<1. Progressive enlargements of regions of the x-axis of
Fig. 2.10a) (delineated by the horizontal lines portrayed in the top three panels) yield
new detail. Although different in form from that of the H´
enon attractor, the evident
self-similarity suggests that the attractor is fractal.
Moving now to confirm the presence or absence of chaos in this system, we
again employ two different starting values: x0= 0.87951016911829671 and x0=
0.89087022021791951, chosen from the values shown in Fig. 2.10a), after 1000 it-
erations to eliminate transient effects. As shown in Fig. 2.11, unlike the results for
the logistic system with c= 4 and for the H´
enon system, different starting points do
not diverge. The system of Eq. (2.20) with c.
= 3.56995168804 therefore does not
exhibit sensitive dependence on initial conditions and is not chaotic. Other mathemat-
ical models of nonchaotic systems with fractal attractors have been set forth (Grebogi,
Ott, Pelikan & Yorke, 1984), as has a physical experiment exhibiting such behavior
(Ditto, Spano, Savage, Rauseo, Heagy & Ott, 1990).

DETERMINISTIC CHAOS 31
a)
b)
)
d)
ITERATE x
Fig. 2.10 a) Three thousand values in the attractor of the logistic system [Eq. (2.20)] with
parameterc.
= 3.56995168804. An initial 1 000 iterations were discardedto eliminate transient
effects. Each iterate is represented by a vertical line for clarity. The xaxis ranges from zero
to unity in this panel. Horizontal lines above and below the left-most cluster delineate the
interval enlarged in the subsequent panel. b) An enlargement of the original interval delineated
by the horizontal lines in panel a): 3 000 values of the attractor constrained to lie in the region
0.338 ≤x≤0.386. Increased detail emerges with a structure similar to that of the whole
attractor in a). c) and d) Further enlargements of the regions delineated by horizontal lines in
the preceding panels: 3 000 values of the attractor in the intervals 0.3424 ≤x≤0.3437 and
0.342544 ≤x≤0.342581, respectively. Fresh new structures that resemble those in panels
a) and b) continue to appear, suggesting that the attractor is fractal. This system is not chaotic,
however, as revealed by the time course displayed in Fig. 2.11.
2.6.5 Chaos in context
Considering the results presented to this point, we see that systems can exhibit chaotic
behavior or fractal (strange) attractors, or both, or neither. All four possibilities exist.
From a fundamental perspective, the term chaos describes certain nonlinear deter-
ministic dynamical systems whereas the term fractal describes certain objects. Thus,
chaos does not imply fractal nor does fractal imply chaos.

32 SCALING, FRACTALS, AND CHAOS
ITERATION NUMBER
n
ITERATE VALUE
x
n
50403020100
1.0
0.8
0.6
0.4
0.2
0.0
Fig. 2.11 Time course of the logistic system of Eq. (2.20) with parameter c.
=
3.56995168804, for two different starting values. The iterated values maintain a difference
roughly equal to that of the starting values, ≈0.01, neither converging nor diverging. Hence,
this system does not display sensitive dependence to initial conditions, and does not exhibit
chaos. The attractor for this system is fractal, however, as illustrated in Fig. 2.10.
Moreover, the presence of significant noise or random behavior in a system gen-
erally precludes a meaningful assertion that the system is chaotic. Noise experiences
the amplifying behavior of the system’s sensitivity to initial conditions, so that even
identical starting values experience rapidly diverging paths. Under such conditions,
the concept of chaos loses its usefulness. Instead, the random nature of the system,
imparted by the noise, becomes a key defining quality of its dynamics.
Since the topic of this treatise is random fractals, we do not consider chaos further.
2.7 ORIGINS OF FRACTAL BEHAVIOR
2.7.1 Fractals and power-law behavior
Why are fractal characteristics found in so many systems, both natural and synthetic?
A good part of the reason turns out to be the close connection between fractals and
scaling and hence between fractals and power-law behavior (see Secs. 2.2 and 2.3).
Indeed, close examination reveals that the fractal behavior associated with many
of the models considered throughout this book derives explicitly from the power-
law relationships embodied in these models. When no such direct link exists, it
turns out that other intrinsic properties of these models ultimately lead to power-law
relationships. Power-law relationships can sometimes be traced to the presence of a
cascade process in the underlying phenomenon.

ORIGINS OF FRACTAL BEHAVIOR 33
The essential notion of a fractal has historical antecedents in theory and exper-
iment alike (see Mandelbrot, 1982, Chapter 41). Consider Leibniz’s (1646–1716)
conception of fractional integro-differentiation and his definition of the straight line;
Kant’s (1724–1804) ruminations about the lack of homogeneity in the distribution of
matter; Laplace’s (1749–1827) suggestion that the scaling nature of Newton’s Law
of gravitation offered an axiom more natural than that of Euclid; and Weierstraß’s
(1815–1897) construction of a continuous, but nowhere differentiable, function.
In the empirical domain, we recall Weber’s (1835) finding that the relaxation of a
stretched silk thread follows a decaying power-law function of time, and Kohlrausch’s
(1854) observation that the decay of charge in a Leyden jar follows this very same
form. We appreciate that Omori (1895) long ago recognized that the rate of after-
shocks following an earthquake decays as an inverse function of time.
Indeed, power-law behavior is ubiquitous (see, for example, Malamud, 2004). It
occurs in many guises, including deterministic laws, first-order statistics, second-
order statistics, distributions, and nonlinear transformations. It is observed in the
dynamicalresponses of systems and intheir frequencyspectra. Pareto (1896)long ago
discovered that scale-invariant, power-law distributions characterize the income of
individuals in many societies.6Behavior in accord with the Paretodistribution,7and
its discrete counterpart, the zeta distribution, emerges in a broad array of contexts.
Examples include:
•The number of species in different genera (Willis, 1922).
•The number of publications by different authors (Lotka, 1926).
•The agricultural yields of different sized plots (Fairfield-Smith, 1938).
•The energies of earthquake occurrences (Gutenberg & Richter, 1944).
•The mass densities of yarns of different lengths (Cox, 1948).
•The frequencies of word usage in natural languages (Zipf, 1949).
•The sizes of computer files (Park, Kim & Crovella, 1996).
The question posed at the beginning of this section — “Why are fractal characteristics
found in so many systems?” — can thus be recast as: “Why is power-law behavior
found in so many systems?”
2.7.2 Physical laws
Several key laws of classical physics take the form of deterministic power-law func-
tions of the distance r,
F∝rc
,(2.24)
6A photograph of Pareto stands at the beginning of Chapter 7.
7A useful generalization of the Pareto distribution has been provided by Mandelbrot (1960, 1982), as will
be elaborated subsequently.

34 SCALING, FRACTALS, AND CHAOS
where Fis the force (or field) and ris the distance (see Feynman, 1965). Perhaps the
mostprominent exampleof this scaling relationis Newton’s (1687) Lawof gravitation,
which provides that the gravitational field Fassociated with an object at a distance r
follows an inverse-square law, F∝r−2, so that c=−2.
The Coulomb field associated with a charged particle also behaves in accordance
with Eq. (2.24), again with c=−2. Other charge configurations similarly lead to
power laws, but with different exponents; examples are an infinite line of charge
(c=−1); a charge dipole (c=−3); a charge quadrupole (c=−4); and the van der
Waals force between a pair of dipoles (c=−7). In the study of the mechanics of
materials, Hooke’s Law provides that the restoring force for an elastic medium also
obeys Eq. (2.24), where ris the deformation and c= +1 (see Gere, 2001). Also,
the Langmuir–Childs Law for space-charge-limited current flow in electronic devices
dictates that i∝V3/2, where iis the current and Vis the voltage (see Terman, 1947,
Sec. 5–5).
Some physical processes are conveniently described in terms of power-law func-
tions of time, as evidenced by the following examples: (1) the distance dtraveled by
an object falling under the force of gravity is characterized by d∝t2; (2) Kepler’s
Third Law of celestial mechanics specifies that the major axis bof an elliptical plan-
etary orbit is related to the orbital period Tvia b∝T2/3; and (3) the time course of
the mean photon flux density emitted by a charged particle via ˇ
Cerenkov radiation
varies as h(t)∝t−5(see Prob. 10.6).
In quantum mechanics, the allowed energy levels Ejof many systems are propor-
tional to some power of the quantum number j(see Saleh & Teich, 1991, Chapter 12),
Ej∝jc.(2.25)
Examples are the hydrogen atom (c=−2); the harmonic oscillator with a linear
restoring force (c= +1); the anharmonic oscillator with a cubic restoring force
(c= +4
3); and the infinite quantum well (c= +2). The rigid rotor behaves as
Ej∝j(j+ 1). The spatial scaling of the Lagrangian for these systems allows us to
deduce these exponents directly from the form of the potential energy function (see
Schroeder, 1990, pp. 66–67).
For simple physical systems, the exponents care typically integers or rational
numbers, although fractional exponents are not uncommon in semiconductor physics
(see Saleh & Teich, 1991, Chapter 15). In the biological sciences, fractionalexponents
are more the rule than the exception, as will become apparent subsequently.
2.7.3 Diffusion
In the domain of stochastic processes, diffusion offers a straightforward route to
achieving power-law dynamics (Whittle, 1962; Marinari, Parisi, Ruelle & Widney,
1983). In one-dimensional diffusion, an object moves randomly along an axis, with
no preferred direction, and with motion at each instant that is independent of motion at
all other times. The path of such an object coincides with Brownian motion, discussed
in Sec. 2.4.2. Equation (2.15) shows that the variance of the position grows linearly

ORIGINS OF FRACTAL BEHAVIOR 35
with time; given the zero-mean Gaussian nature of the process, this leads immediately
to a probability density for the particle position x:
px(x) = (4π∆t)−1/2expµ−x2
4∆ t¶,(2.26)
with ∆adiffusion constant. The peak height of the density decays with time as
t−1/2, an inverse power law. Given a concentration of small objects u0(particles,
for example) clustered tightly about a starting value x0, a simple modification of
Eq. (2.26) yields a particle concentration envelope u(x, t)given by
u(x, t) = u0(4π∆t)−1/2expµ−(x−x0)2
4∆ t¶.(2.27)
For diffusion in a multidimensional Euclidean space of (integer) dimension
DE, the motion along each of the component axes forms an independent realization
of Brownian motion. The corresponding concentration profile then becomes (see, for
example, Pinsky, 1984)
u(x, t) = u0(4π∆t)−DE/2expµ−|x−x0|2
4∆ t¶,(2.28)
where xand x0represent the general and initial position vectors, respectively. The
concentration decays as t−DE/2, providing exponents 1
2,1,and 3
2for DE=1,2,and
3, respectively. For diffusion on objects that are physical examples of fractals, the
fractal dimension of the object replaces the Euclidean dimension DEin Eq. (2.28),
thereby offering a larger set of allowable exponents. Problems 10.8 and 10.9 address
the ramifications of such diffusion processes.
Diffusion-limited aggregation (DLA) describes the aggregation and growth of
structures when diffusion dominates transport (Witten & Sander, 1981). This model
characterizes a broad variety of phenomena including electrodeposition, dielectric
breakdown, snowflake formation, mineral-vein formation in geologic structures, and
the growth of biological structures such as coral (see, for example, Vicsek, 1992;
Halsey, 2000).
Subdiffusion is an important form of anomalous diffusion in which the mean-
square displacement varies as tα(0< α < 1) rather than as t(see Bouchaud &
Georges, 1990). This process can be understood in a simple way by making use of
fractional Gaussian noise (see Sec. 6.2) in a generalized Langevin equation (Kou &
Xie, 2004).
2.7.4 Convergence to stable distributions
An important and far-reaching rationale for the emergence of power-law distributions
has its origins in the limit theorem developed by Paul L´
evy (1937, 1940) (see also
Gnedenko & Kolmogorov, 1968; Feller, 1971; Mandelbrot, 1982; Christoph & Wolf,
1992; Samorodnitsky & Taqqu, 1994; Bertoin, 1998; Sato, 1999; Sornette, 2004).

36 SCALING, FRACTALS, AND CHAOS
Sums of identical and independent continuous random variables are characterized
by stable distributions, which generally have power-law tails. The sole exception
is the Gaussian distribution (Gauss, 1809), which emerges (via the ordinary central
limit theorem) when the constituent random variables are endowed with finite second
moments.8
Discrete analogs of the family of continuous stable distributions have recently
been examined (Hopcraft, Jakeman & Tanner, 1999; Matthews, Hopcraft & Jakeman,
2003; Hopcraft, Jakeman & Matthews, 2002, 2004). These probability distributions
typically follow the form
p(n)∼1/nc,1< c < 2,(2.29)
for large n. They have zero mode and infinite mean in the absence of an upper cutoff.
However, for counting distributions with an upper cutoff, and therefore a finite mean,
sums converge to the Poisson distribution, which assumes the role played by the
Gaussian for the continuous stable distributions.
2.7.5 Lognormal distribution
A closely related rationale for the presence of power-law behavior stems from the fea-
tures of the lognormal distribution (Kolmogorov, 1941; Aitchison & Brown, 1957;
Gumbel, 1958). This distribution is often used as a model for characterizing systems
comprising products of random variables, via an argument that proceeds asfollows. A
product of random variables with finite second moments, under logarithmic transfor-
mation, becomes a sum. Application of the ordinary central limit theorem renders the
sum Gaussian (normal). The original product, then, obeys the lognormal distribution
since its logarithm has a normal distribution.
The lognormal distribution has a long tail and sums of independent lognormally
distributed random variables retain their lognormal form (Mitchell, 1968; Barakat,
1976); although these sums ultimately do converge to Gaussian form, the convergence
is exceedingly slow. Moreover, the tail of the lognormal distribution is closely mim-
icked by a power-law distribution over a wide range (Montroll & Shlesinger, 1982;
Shlesinger, 1987; West & Shlesinger, 1989, 1990); these authors further argue that
many data thought to obey an inverse power-law distribution instead obey the log-
normal law over a broad range and then ultimately transition to power-law behavior
at very large values of the random variable.
In the domain of discrete processes, the Poisson transform of the lognormal dis-
tribution has found widespread use in modeling the photon fluctuations of laser light
transmitted through random media such as the turbulent atmosphere (Diament &
Teich, 1970a; Teich & Rosenberg, 1971). The justification for using the lognormal
model here is the same as that provided above: in traveling from source to receiver,
the laser light encounters a large number of independent atmospheric layers with
random transmittances.
8Photographs of Gauss and L´
evy can be found at the beginning of Chapter 8. A biographical sketch of
L´
evy is provided by Mandelbrot (1982, Chapter 40).

ORIGINS OF FRACTAL BEHAVIOR 37
2.7.6 Self-organized criticality
Power-law behavior arises in other ways as well. Some systems spontaneously evolve
toward a critical state and thereby generate power-law distributions. A sandpile
provides the canonical example of this process, called self-organized criticality
(Bak, Tang & Wiesenfeld, 1987; Bak, 1996), and abbreviated soc; the addition of
grains of sand to the top of a sandpile results in the formation of a cone at exactly the
critical angle of repose. Some added grains merely stop where they land, but many
trigger avalanches with power-law varying sizes, maintaining the critical state.
Expansion-modification systems provide another example of such spontaneous
evolution. In this case, two processes operate simultaneously, one creating long-range
correlation, and the other destroying it; the resulting construct exhibits correlations
over all scales, and therefore fractal structure (Li, 1991). As an example, each element
ina binarysequence iseither invertedwithprobability p, orduplicated withprobability
1−p. Similar behavior may occur in a variety of artificial and natural systems, as
they evolve towards complex, critical states and produce power-law behavior. In a
related model, simple white noise perturbs the movement of activatedneural clusters
and competing dissipative and restorative forces ultimately generate 1/f-type noise
(Usher, Stemmler & Olami, 1995).
Asanother particularexample, acollection ofinterconnected processesthat evolves
according to the logistic equation generates power-law-distributed amplitudes over a
broad range of system parameters (Solomon & Richmond, 2002).
2.7.7 Highly optimized tolerance
Highly optimized tolerance (Carlson & Doyle, 1999; Doyle & Carlson, 2000; Carl-
son & Doyle, 2002) suggests another possible origin for power-law behavior. Accord-
ing to this theory, power-law behavior emerges naturally as a result of the evolution
of a complex system toward optimal performance and robustness. Natural selection
is said to drive the evolution for collections of living organisms, while engineering
design provides the optimizing impetus for artificial systems. This evolutionary pro-
cess leads to the emergence of specialized states (which would be rare in a random
system without design input) concomitantly with power-law behavior.
Power-law characteristics, and hence fractal behavior, can therefore emerge natu-
rally from system evolution via a number of different constructs (Gisiger, 2001).
2.7.8 Scale-free networks
Yet another way that power-law behavior comes into play is via scale-free networks
(Albert & Barab´
asi, 2002; Dorogovtsev & Mendes, 2003; Pastor-Satorras & Ves-
pignani, 2004). For such networks, no node is typical. Some have an enormous
number of connections whereas most are only weakly connected to others. Since
well-connected nodes, called hubs, can have hundreds, thousands, or millions of
links, there is no scale associated with the network. Connectivity in links per node
is described by a probability law, known as the degree distribution, that typically

38 SCALING, FRACTALS, AND CHAOS
follows a power-law form (Krapivsky, Rodgers & Redner, 2001):
p(n)∼1/nc,2< c < 3.5.(2.30)
There are many ways in which scale-free networks can come into being (see, for
example, Krapivsky, Redner & Leyvraz, 2000). The underlying features that lead
to the formation of such networks are continual development and the preferential
attachment to highly linked nodes. As new nodes are formed, the network continues
to evolve; each new node tends to connect to the more highly connected existing
nodes since these are most easily identified.
Examples of scale-free networks in the biological domain stretch from cellular
metabolic networks, in which biochemical reactions link a collection of molecules, to
the brain, in which axons and dendrites link a collection of neurons (Egu´
ıluz, Chialvo,
Cecchi, Baliki & Apkarian, 2005). Such networks are plentiful in the technological
arena: important examples are air transportation systems, the Internet, and the World
Wide Web (see Sec. 13.2.1). Scale-free networks are also pervasive in the social
domain: examples include scientific collaborations connected by joint publications;
scientific papers linked by citations; people connected by professional associations
or friendships; epidemics of contagious disease linked by family members; and busi-
nesses linked by joint ventures. They have the salutary feature of being robust against
accidental failures because random breakdowns selectively affect the most plentiful
nodes, which are the least connected. Such networks are, however, highly vulnerable
to coordinated attacks directed at the hubs, which are the most intricately connected
of the nodes (Albert & Barab´
asi, 2002).
Despite the evident diversity of these scale-free networks, their common architec-
ture brings them under the same mathematical umbrella: the power-law distribution
embodied in Eq. (2.30). The range of asymptotic power-law exponents is rather
narrow and differs from that for discrete stable distributions [compare Eqs. (2.29)
and (2.30)]. The convergence properties of sums of identical, independently dis-
tributed discrete zeta random variables that are suitable for characterizing scale-free
networks have recently been established. The limiting form turns out to be the Pois-
son distribution but the convergence can be exceptionally slow (Hopcraft et al., 2004),
much as with the convergence of sums of lognormal random variables to Gaussian
form (see Sec. 2.7.5). Problems involving discrete scale-invariant behavior should
be formulated in terms of discrete models since continuum models and mean-field
approximations can lead to erroneous results (Hopcraft et al., 2004).
Not all networks are scale-free, of course. Prominent exceptions include the lo-
cations of atoms in a crystal lattice, the U.S. highway system, the power grid in
the Western United States, and the neural network of the organism Caenorhabditis
elegans.
2.7.9 Superposition of relaxation processes
Finally, we note that the observation of first- and second-order statistics with power-
law behavior is often ascribed to a superposition of relaxation processes exhibiting
a spread of time constants. Maxwell’s student Hopkinson (1876) appears to have

UBIQUITY OF FRACTAL BEHAVIOR 39
originated this explanation, suggesting that the power-law decay of the charge in a
Leyden jar might be understood on the basis of various relaxation times for the differ-
ent silicate components of the glass through which the discharge occurred. However,
this argument was later abandoned as unworkable because of the large number of
exponentials required. von Schweidler (1907) resurrected this approach by consider-
ing a large number of relaxation processes with a wide spread of time constants. He
noted that the properties of the gamma function were such that a power-law function
could be represented in terms of a weighted collection of exponential functions with
different relaxation times.
In the context of semiconductor physics, van der Ziel (1950) used a correlation-
function version of this approach to explain the inverse-frequency form of the spec-
trum; Halford (1968) subsequently offered a generalization of this model. This con-
struct finds wide acceptance in the semiconductor-physics community by virtue of its
connection to trapping mechanisms, which offer an exceptionally wide range of time
constants (McWhorter, 1957, see also Prob. 7.10). Buckingham (1983, Chapter 6)
addressed the role of the weighting functions.
Many other materials and systems, physical and biological alike, display simi-
lar power-law behavior, as shown in Chapter 5. However, the relaxation-process
approach is rarely appropriate for characterizing these processes because of the enor-
mous range of time constants required to yield 1/f behavior over a reasonable range
of frequencies. A ratio of time constants of 106, for example, yields 1/f behavior
only over four decades of frequency whereas a ratio of 1012 offers 10 decades (Buck-
ingham, 1983, Chapter 6; see also Prob. 9.1 and Fig. B.6). Few systems aside from
semiconductors offer the requisite range of time constants.
Another way of mitigating the presence of power-law behavior is to assume that
an exponential cutoff ultimately prevails. In practice this often turns out not to be
the case, however. Indeed, von Schweidler (1907) himself carried out extensive
experiments seeking such a cutoff in the decay of charge in Leyden jars, but found
none.
2.8 UBIQUITY OF FRACTAL BEHAVIOR
2.8.1 Fractals in mathematics and in the physical sciences
The most comprehensive treatments of fractals have principally been in mathemat-
ics and the physical sciences. Extensive treatments have appeared, for example, in
the following books: Mandelbrot (1982); Feder (1988); Peitgen & Saupe (1988);
Schroeder (1990); Peitgen et al. (1997); L´
evy V´
ehel et al. (1997); Turcotte (1997);
Turner et al. (1998); Flandrin & Abry (1999); Flake (2000); Barnsley (2000); Park &
Willinger (2000); Mandelbrot (2001); Falconer (2003); West et al. (2003). The appli-
cation of fractals in fields such as economics, finance, and hydrology is widespread
(see, for example, Mandelbrot, 1982, 1997; Mandelbrot & Hudson, 2004; Henry &
Zaffaroni, 2003; Montanari, 2003).

40 SCALING, FRACTALS, AND CHAOS
Fractal analysis in the physical sciences proves highly important, as indicated by
the following examples:
•We are all keenly aware of the fractal geometry of nature, thanks to the seminal
work of Benoit Mandelbrot (1982).
•The noise in many electronic components and systems exhibits fractal behavior
at low frequencies (Sec. 5.4.1).
•Semiconductor layered structures comprising stacks of materials of different
bandgaps have beenfabricated in the formof Cantor sets (Cantor, 1883), aswell
as Fibonacci (1202), Thue–Morse (Thue, 1906, 1912; Morse, 1921b,a), and
Rudin–Shapiro (Rudin, 1959; Shapiro, 1951) sequences. Such nonperiodic,
deterministic structures can exhibit fractal electronic, thermal, and magnetic
properties (see, for example, Merlin et al., 1985; Kohmoto et al., 1987; Kol´
aˇ
r,
Ali & Nori, 1991; Dulea, Johannson & Riklund, 1992).
•Photonic materials and devices consisting of layers of materials with different
refractive indices havealso been constructed in the form of Cantor sets, as well
as Fibonacci and Thue–Morse sequences (Jaggard & Sun, 1990; Kol´
aˇ
r et al.,
1991; Liu, 1997; Jaggard, 1997; Zhukovsky, Gaponenko & Lavrinenko, 2001).
For example, Hattori, Schneider & Lisboa (2000) suggested constructing a
fiber Bragg grating that takes the form of a Cantor set. Such nonperiodic and
deterministic photonic media can exhibit optical properties with unusual fea-
tures, including: (1) optical reflection and transmission withself-similar spectra
(Gellermann, Kohmoto, Sutherland & Taylor, 1994; Dal Negro, Oton, Gaburro,
Pavesi, Johnson, Lagendijk, Righini, Colocci & Wiersma, 2003; Ghulinyan,
Oton, Dal Negro, Pavesi, Sapienza, Colocci & Wiersma, 2005; Dal Negro,
Stolfi, Yi, Michel, Duan, Kimerling, LeBlanc & Haavisto, 2004); (2) complex
light dispersion (Hattori, Tsurumachi, Kawato & Nakatsuka, 1994); (3) band-
edge group-velocity reduction (Dal Negro et al., 2003; Ghulinyan et al., 2005);
(4) pseudo-bandgaps and omnidirectional reflection (Dal Negro et al., 2004);
and (5) light emission with uncommon spectral characteristics (Dal Negro, Yi,
Nguyen, Yi, Michel & Kimerling, 2005).
•Light scattered or refracted by passage through a random fractal phase screen
exhibits fractal wave properties (Berry, 1979; Jakeman, 1982); Berry (1979)
coined the term diffractal to describe the resulting wave.
•Errors in telephone networks often occur as fractal clusters (Prob. 7.7).
•The photon statistics of ˇ
Cerenkov radiation exhibit fractal characteristics under
certain conditions (Prob. 10.6).
•Analysis of the fractal statistics of earthquake patterns can assist in the predic-
tion of future earthquake occurrences (Prob. 10.7).
•Computer communication networks evolve into scale-free forms and the traffic
resident on these networks exhibit fractal characteristics (Chapter 13).

UBIQUITY OF FRACTAL BEHAVIOR 41
2.8.2 Fractals in the neurosciences
Therehave beenfewer comprehensivetreatments of fractals inthe biological sciences;
we explicitly note those of Bassingthwaighte et al. (1994), West & Deering (1995),
and Liebovitch (1998). Fractals play an important role in biological sciences such
as ecology (see, for example, Halley & Inchausti, 2004), which has often been a
breeding ground for novel mathematical approaches.
In this and the following section, respectively, we examine a number of examples
of fractal behavior in the neurosciences and in medicine and human behavior.
Power-law behavior is common in the neurosciences. Featured at levels from
the molecular to the organism, it is manifested in many systems. Neural systems
evidently benefit from the flexibility of being able to match the time scale of a current
stimulus while incorporating the memories of past stimuli. We present a number of
examples, emphasizing those that fall in the class of fractal-based point processes:
•Ion channels reside in biological cell membranes, permitting ions to diffuse in
or out of a cell (Sakmann & Neher, 1995). Power-law behavior characterizes
various features of ion-channel behavior (Liebovitch, Fischbarg & Koniarek,
1987; Liebovitch, Fischbarg, Koniarek, Todorova & Wang, 1987; Liebovitch
& T´
oth, 1990; Liebovitch, Scheurle, Rusek & Zochowski, 2001; L¨
auger, 1988;
Millhauser et al., 1988). Many ion channels exhibit independent power-law-
distributed closed times between open times of negligible durations, and are
well described by a fractal renewal point process (Lowen & Teich, 1993c).
When the open times have significant duration, the alternating fractal renewal
process serves as a suitable model instead (Lowen & Teich, 1993c, 1995;
Thurner et al., 1997). Moreover, the time constant attendant to the recovery of
certain ion channels depends on the duration of prior activity in a power-law
fashion (Toib, Lyakhov & Marom, 1998).
•Fractal behavior exists in excitable-tissue recordings for various biological sys-
tems in vivo, from the microscopic to the macroscopic (Bassingthwaighte et al.,
1994; West & Deering, 1994). Membrane voltages vary randomly in time, of-
ten exhibiting Gaussian fluctuations with power-law spectra (Verveen, 1960;
Verveen & Derksen, 1968; Stern, Kincaid & Wilson, 1997; Lowen, Cash, Poo
& Teich, 1997a). Superpositions of alternating fractal renewal processes, rep-
resenting collections of ion-channel openings and closings, provide a plausible
model for this process (Lowen & Teich, 1993d, 1995).
•Communication in the nervous system is generally mediated by the exocyto-
sis of multiple vesicular packets (quanta) of neurotransmitter molecules at the
synapse between cells, either spontaneously (Fatt & Katz, 1952) or in response
to an action potential at the presynaptic cell (Katz, 1966). Neurotransmitter
packets induce miniature end-plate currents (MEPCs) at the postsynaptic mem-
brane, and their rate of flow exhibits fractal behavior such as power-law spectra,
that can be described by a fractal-based point process (Lowen et al., 1997a,b).

42 SCALING, FRACTALS, AND CHAOS
•Power-lawbehaviorcharacterizes the second-orderstatistics of action-potential
sequences in isolated neuronal preparations and isolated axons; the spectrum
often follows a form close to 1/f over a broad range of frequencies (Musha,
Kosugi, Matsumoto & Suzuki, 1981; Musha et al., 1983). Moreover, the spike
rate in response to a step-function input in many sensory neurons follows a
power-law decay during the course of adaptation (Chapman & Smith, 1963),
frequently varying as t−1/4(Biederman-Thorson & Thorson, 1971; Thorson
& Biederman-Thorson, 1974).
•Auditory nerve-fiber action potentials from essentially all in vivo preparations
display neural-spike clusters (Teich & Turcott, 1988) and fractal-rate behavior
over time scales greater than about 1 sec, under both spontaneous and driven
conditions (Teich, 1989; Teich, Johnson, Kumar & Turcott, 1990; Teich, 1992;
Teich & Lowen, 1994; Lowen & Teich, 1992a, 1996a; Powers & Salvi, 1992;
Kelly, Johnson, Delgutte & Cariani, 1996). This behavior could arise from
superpositionsof fractalion-channel transitions(Teich,Lowen& Turcott,1991;
Lowen & Teich, 1993b, 1995) or via fractal-rate vesicular exocytosis (Lowen
et al., 1997a,b).
•As in the auditory system, spontaneous and driven visual-system action poten-
tials also exhibit fractal-rate characteristics. This behavior appears in all retinal
ganglion cells and lateral-geniculate-nucleus cells in the thalamus (Teichet al.,
1997; Lowen, Ozaki, Kaplan, Saleh & Teich, 2001), as well as in cells of the
striate cortex (Teich et al., 1996). Moreover, insect visual-system interneurons
generate spike trains with fractal-rate characteristics under both spontaneous
and driven conditions (Turcott, Barker & Teich, 1995). Motion-sensitive neu-
rons in the fly visual system adapt over a wide range of time scales that are
established by the stimulus rather than by the neuron (Fairhall, Lewen, Bialek
& de Ruyter van Steveninck, 2001a,b).
•Fractal features appear in action-potential sequences associated with many
central-nervous-system neurons operating under a broad variety of conditions,
including those in the cortex, thalamus, hippocampus, amygdala, pyramidal
tract, medulla, and mesencephalic reticular formation (see, for example, Evarts,
1964; Yamamoto & Nakahama, 1983; Yamamoto, Nakahama, Shima, Ko-
dama & Mushiake, 1986;Kodama, Mushiake, Shima, Nakahama &Yamamoto,
1989;Gr¨
uneis,Nakao, Yamamoto,Musha& Nakahama,1989; Gr¨
uneis,Nakao,
Mizutani, Yamamoto, Meesmann & Musha, 1993; Lewis, Gebber, Larsen &
Barman, 2001; Orer, Das, Barman& Gebber,2003; Fadel, Orer, Barman, Vong-
patanasin, Victor & Gebber, 2004; Bhattacharya, Edwards, Mamelak & Schu-
man, 2005).
•Networks of rat cortical neurons contained in slice cultures exhibit brief neu-
ronal avalanches whose spatiotemporal patterns are stable and repeatable for
many hours; these power-law distributed structures may serve as a substrate
for memory (Beggs & Plenz, 2003, 2004).

UBIQUITY OF FRACTAL BEHAVIOR 43
•Power-law behavior has a strong presence in the domain of sensory perception.
Although the transduction of a stimulus at the first synapse in a neural system
often follows a logarithmic form, stimulus estimation and detection are usually
characterized by power-law functions of the stimulus intensity with sub-unity
exponents (Stevens, 1957, 1971; Barlow, 1957; McGill & Goldberg, 1968;
Moskowitz, Scharf & Stevens, 1974; McGill & Teich, 1995).
•The natural course of forgetting in humans is well described by a decaying
power-lawfunctionof time (Wickelgren,1977; Wixted& Ebbesen, 1991, 1997;
Wixted, 2004).
There appear to be many origins of fractal activity in the nervous system; power-
law fluctuations at the level of the protein may play an underlying role.
2.8.3 Fractals in medicine and human behavior
The quantitative analysis of the fractal characteristics of biomedical signals can yield
information that assists with the diagnosis of disease and with the determination
of its severity. This information, in turn, can have vital implications regarding the
appropriate treatment regimen, and can influence the outcome of treatment. We
provide a number of examples:
•Fractal analysis of the fluctuations in human standing (Musha, 1981; Shimizu
et al., 2002) reveals age-related changes not evident using conventional, non-
fractal methods (Collins et al., 1995). A different constellation of changes ap-
pears in Parkinson’s disease (Mitchell, Collins, De Luca, Burrows & Lipsitz,
1995). After correcting for age, the fractal dynamics of human gait (walking)
reveal the severity of Huntington’s disease in patients, and appear to correlate
with the degree of impairment (Hausdorff, Mitchell, Firtion, Peng, Cudkowicz,
Wei & Goldberger, 1997).
•Fluctuations in mood show evidence of fractal behavior in their spectra, which
display quantitative differences between bipolar-disorder patients and normal
controls (Gottschalk et al., 1995). That these fluctuations follow a fractal form
may lead to better methods for predicting and controlling mood disorders (see
Sec. 2.8.5).
•Evidence of fractal behavior in the spectrum of the human heartbeat has been
known for more than two decades (Kobayashi & Musha, 1982). Fractal meth-
ods do differentiate between normal and diseased patients with some degree
of success (Turcott & Teich, 1993; Peng, Mietus, Hausdorff, Havlin, Stanley
& Goldberger, 1993; Peng, Havlin, Stanley & Goldberger, 1995; Turcott &
Teich, 1996). However, nonfractal measures (based on a fixed time scale of
about twenty seconds) are superior for indicating the presence of cardiovascu-
lar dysfunction (Thurner, Feurstein & Teich, 1998; Thurner, Feurstein, Lowen

44 SCALING, FRACTALS, AND CHAOS
& Teich, 1998; Ashkenazy, Lewkowicz, Levitan, Moelgaard, Bloch Thomsen
& Saermark, 1998; Teich et al., 2001).
•Fractal measures of activity have successfully quantified changes in the move-
ment patterns of laboratory rats induced by drugs of abuse (Paulus & Geyer,
1992), and these same fractal measures help improve the diagnosis of attention
deficit hyperactivity disorder (ADHD) in children (Teicher et al., 1996).
•Normal prenatal development may, in fact, require that fractal activity patterns
be established in the brain. Developmental disorders such as autism could
possibly result from a failure in the generation of these patterns (Anderson,
2001).
•Developmental insults, such as early abuse, quantitatively alter fractal parame-
ters measured in experimental animals (Anderson, 2001). Evidence also exists
that the fractal patterns of brain activity change with emotional state (Anderson
et al., 1999), with implications for psychiatric diagnoses.
2.8.4 Recognizing the presence of fractal behavior
Fractal activity directly influences how systems operate. It is therefore important to
recognize its presence, and to understand its features, so that system performance can
be properly evaluated and controlled.
For computer network traffic (see Chapter 13) and vehicular traffic, for example,
estimates of the fractal parameters provide measures of performance and useful de-
sign guidelines. Detailed analysis of fractal activity has proven to be indispensable.
Dealing with fractal behavior in a system is not a trivial enterprise, however. Even
seemingly simple tasks, such as calculating the mean and variance of the rate for
a fractal process, offer unique challenges. The low-frequency nature of the noise
indicates that nearby values are highly correlated so that obtaining reliable estimates
often requires a prohibitive number of samples (see Chapter 12).
In some cases, fractal behavior serves as a source of unavoidable noise that dimin-
ishes system performance. An example is 1/f -type noise in electronic components
and circuits (see Sec. 5.4.1). The presence of fractal noise places restrictions on the
information throughput of such systems, the calculation of which requires fractal
analysis.
Finally, it is important to recognize the possible presence of fractal noise to avoid
drawing erroneous conclusions. A case in point is the landmark study conducted by
Fatt & Katz in 1952, in which the authors carried out an investigation of the statistical
behavior of sequences of miniature endplate currents (MEPCs) at the neuromuscular
junction. In the course of describing the methods used to analyze their data, they
carefully noted that each segment of data selected for analysis was sufficiently short to
exclude, as they putit, the “occasional occurrences of shorthigh-rate bursts” of events,
and to avoid “progressive changes of the mean.” In fact, fractal-rate fluctuations

UBIQUITY OF FRACTAL BEHAVIOR 45
do exist in exocytic behavior and MEPCs (Lowen et al., 1997a,b), and the MEPCs
observed by Fatt & Katz (1952) almost certainly exhibited such behavior (see Lowen
et al., 1997b, for an analysis). Unaware of the presence or importance of these
fluctuations, however, they removed most traces of them by selecting relatively short
segments of data for analysis and, moreover, chose precisely those segments that
exhibited minimal fluctuations. The observation of fractal-rate behavior requires
long data sets, and the presence of both bursts and apparent trends lie at its very core.
2.8.5 Salutary features of fractal behavior
Fractal behavior is ubiquitous and its study reveals much about our surroundings. We
discover, for example, that natural scenes and natural sounds exhibit fractal properties
in space and time, and sensory systems have adapted to this property (Musha, 1981;
Teich, 1989; Dan, Atick & Reid, 1996; Taylor, 2002; Simoncelli & Olshausen, 2001;
Yu, Romero & Lee, 2005).
Given the ubiquity of fractal activity, what biological advantages might accrue
from its presence?
Fractal behavior offers tolerance to noise and errors. The deleterious effects of
noise diminish in importance because the concentration of power at lower frequencies
assures increased predictability. New scales introduced by errors are less disruptive
in fractal processes since they already exist in the initial distributions (West, 1990).
Scale-free networks are robust against accidental failures, as pointed out in Sec. 2.7.8.
Moreover, the presence of fractal noise can serve to optimize the throughput of a
system, with examples in both neural signaling and vehicular traffic (Ruszczynski,
Kish & Bezrukov, 2001).
Developmentally, internallygenerated fractalsignals [suchas neuralsignals arising
during rapid-eye-movement (REM) sleep] provide a prenatal stimulus that mimics
natural signals and assists the brain in developing normally. An animal can thus
emerge at birth with its visual system attuned to the world it enters (Anderson, 2001).
Search patterns executed by animals and by humans, which often have fractal prop-
erties (Cole, 1995; Viswanathan, Afanasyev, Buldyrev, Murphy, Prince & Stanley,
1996; Aks, Zelinsky & Sprott, 2002), appear optimal given the likely distributions of
targets.
Finally, the salutary features of fractal behavior in medicine have been documented
in a number of cases. When used for relieving pain via transcutaneous electrical
nerve stimulation, 1/f noise outperforms white noise (Musha, 1981). This is also
true for sensitizing baroreflex function in the human brain (Soma, Nozaki, Kwak &
Yamamoto, 2003). The flexibility of response offered by fractal behavior may also
serve as a harbinger of health (see West & Deering, 1995).

46 SCALING, FRACTALS, AND CHAOS
Problems
2.1 Fractal and nonfractal objects Comment on the fractal properties, or the lack
thereof, in each of the following:
1. the aorta, all the arteries it branches into, the arterioles, and the capillaries in a
rabbit;
2. a tree trunk and all its branches and twigs, as visualized in the winter when it
is devoid of leaves;
3. a lock of hair;
4. a brick;
5. a sand dune in the Namibian desert, without vegetation;
6. a cumulus cloud;
7. the Himalayan mountains;
8. the path of a curve ball thrown by a major-league baseball pitcher;
9. a randomized version of C3— we generate the third iteration towards the
Cantor set, which contains eight segments, but add an independent random
value uniformly distributed over [−0.01,+0.01] to the beginning and ending
value of each segment.
2.2 Logistic to tent map Consider the logistic map, Eq. (2.20), with c= 4, as
studied in Sec. 2.6.2.
2.2.1. Show that the substitution
y≡π−1arccos(1 −2x)(2.31)
converts Eq. (2.20) into a tent map (Schroeder, 1990, p. 291):
yn+1 =½2yn0≤yn≤1
2
2−2yn1
2<yn≤1.(2.32)
2.2.2. Find the ratio |²n+1/²n|.
2.3 Cantor variant Imagine a variant of the Cantor set described in Sec. 2.4.1,
denoted C0. At each stage in the construction of the variant set we remove the middle
half of each remaining interval. Thus, the intervals £0
4,1
4¤S£3
4,4
4¤comprise the
result C0
1after the first step in its construction.
2.3.1. What total length (Lebesgue measure) remains in the limiting set C0?
2.3.2. How many points remain in C0compared with the original unit interval?
2.3.3. What value of D0does C0have?

PROBLEMS 47
2.4 Cantor-set membership Consider the point x= 0.002002 . . .3, where the
subscript 3indicates a ternary expansion.
2.4.1. To what fraction does xcorrespond?
2.4.2. Does xbelong to the endpoints of C?
2.4.3. Does xbelong to the interior of C(in other words, in Cbut not an endpoint
of it)?
2.4.4. Does Ccontain irrational numbers?
2.5 Scaling solution Show that Eqs. (2.5) and (2.6) form the only solution to
Eq. (2.4) for arbitrary aand x.

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Author Index
Abbe (1878), 64, 513
Abeles et al. (1983), 227, 513
Abry et al. (2002), 314, 513
Abry & Flandrin (1996), 58, 68, 513
Abry et al. (2000), 58, 274, 331, 332,
513
Abry et al. (2003), 58, 114, 274, 513
Abry & Sellan (1996), 139, 514
Aiello et al. (2001), 321, 514
Aitchison & Brown (1957), 36, 147, 514
Aizawa (1984), 173, 176, 514
Aizawa & Kohyama (1984), 173, 514
Akay (1997), 58, 514
Aks et al. (2002), 45, 514
Albert & Nelson (1953), 236, 514
Albert & Barab´
asi (2002), 37, 38, 321,
514
Albert et al. (1999), 16, 321, 514
Aldroubi & Unser (1996), 58, 514
Alexander & Orbach (1982), 471, 514
Allan (1966), 68, 276, 514
Alligood et al. (1996), 25, 514
Anderson (2001), 16, 44, 45, 514
Anderson et al. (1999), 16, 44, 515
Arecchi & Califano (1987), 173, 515
Arecchi & Lisi (1982), 173, 515
Argoul et al. (1989), 75, 515
Arlitt & Jin (1998), 330, 515
Arneodo et al. (1988), 58, 515
Arrault & Arneodo (1997), 75, 515
Ashkenazy et al. (2001), 275, 515
Ashkenazy et al. (1998), 44, 275, 515
Asmussen (2003), 316, 328, 515
Ayache & L´
evy V´
ehel (1999), 123, 515
Azhar & Gopala (1992), 224, 472, 515
Baccelli & Br´
emaud (2003), 51, 515
Bachelier (1900), 19, 515
Bachelier (1912), 19, 516
Bacry et al. (1993), 75, 516
Bak (1996), 37, 516
Bak et al. (1987), 37, 516
Barakat (1976), 36, 516
Bardet et al. (2000), 280, 516
Bardet et al. (2003), 114, 139, 280, 516
Barlow (1957), 43, 516
Barndorff-Nielsenetal. (1978), 156, 516
Barnes & Allan (1966), 68, 144, 516
Barnsley (2000), 16, 39, 516
Bartlett (1955), 50, 82, 87, 516
Bartlett (1963), 4, 72, 88, 94, 516
Bartlett (1964), 72, 94, 95, 202, 517
Bartlett (1972), 4, 517
Barton & Poor (1988), 137, 143, 517
567

568 AUTHOR INDEX
Bassingthwaighte et al. (1994), 16, 41,
517
Bassingthwaighte & Raymond (1994),
60, 517
Bateman (1910), 64, 517
Beggs & Plenz (2003), 42, 517
Beggs & Plenz (2004), 42, 517
Bell (1960), 116, 517
Bell (1980), 116, 517
Benassi et al. (1997), 123, 517
Beran (1992), 309, 517
Beran (1994), 60, 309, 517
Beran et al. (1995), 325, 517
Berger & Mandelbrot (1963), 154, 167,
517
Berry (1979), 16, 40, 517
Berry et al. (1980), 151, 517
Bertoin (1998), 35, 174, 518
Bharucha-Reid (1997), 236, 518
Bhattacharya et al. (2005), 42, 518
Bickel (1999), 122, 123, 518
Bickel (2000), 168, 452, 518
Bickel & West (1998a), 168, 452, 518
Bickel & West (1998b), 168, 452, 518
Biederman-Thorson & Thorson (1971),
42, 518
Bienaym´
e (1845), 95, 518
Blair & Erlanger (1932), 92, 518
Blair & Erlanger (1933), 92, 518
Bouchaud & Georges (1990), 35, 518
Bovy (1998), 4, 16, 518
Boxma (1996), 334, 518
Bracewell (1986), 51, 519
Br´
emaud & Massouli´
e (2001), 217, 519
Brichet et al. (1996), 328, 519
Brillinger (1981), 51, 519
Brillinger (1986), 78, 519
Brockmeyer et al. (1948), 316, 519, 524,
525
Brown (1828), 19, 519
Buckingham (1983), 16, 39, 107, 116,
172, 173, 196, 519
Buldyrev et al. (1995), 290, 519
Burgess (1959), 231, 519
Burlatsky et al. (1989), 471, 519
C¸ inlar (1972), 84, 256, 258, 519
Caccia et al. (1997), 60, 519
Campbell (1909a), 186, 520
Campbell (1909b), 186, 520
Campbell (1939), 50, 520
Cantor et al. (1975), 237, 520
Cantor & Teich (1975), 237, 263, 520
Cantor (1883), 17, 40, 520
Carlson & Doyle (1999), 37, 520
Carlson & Doyle (2002), 37, 520
Carson (1931), 195, 520
Castaing (1996), 123, 520
ˇ
Cerenkov (1934), 220, 520
ˇ
Cerenkov (1937), 220, 520
ˇ
Cerenkov (1938), 220, 468, 520
Chandler et al. (1958), 4, 520
Chandrasekhar (1943), 193, 520
Chapman & Smith (1963), 42, 520
Chistyakov (1964), 57, 521
Christoph & Wolf (1992), 35, 521
Cohen (1969), 316, 328, 521
Cohen (1973), 328, 521
Cohn (1993), 516, 521
Cole (1995), 45, 521
Collins et al. (1995), 16, 43, 521
Conrad (1986), 25, 521
Cooper (1972), 316, 521
Cox (1948), 33, 521
Cox (1955), 88, 521
Cox (1962), 23, 51, 82, 85, 231, 264,
521
Cox (1963), 241, 521
Cox (1984), 15, 336, 521
Cox & Isham (1980), vi, 23, 51, 54, 66,
71, 77, 82, 83, 85, 88, 95, 97,
231, 258, 259, 264, 407, 521
Cox & Lewis (1966), 51, 70, 82, 88, 273,
521
Cox & Smith (1953), 84, 521
Cox & Smith (1954), 84, 522
Crovella & Bestavros (1997), 325, 329,
335, 336, 522
Dal Negro et al. (2003), 40, 522
Dal Negro et al. (2004), 40, 522
Dal Negro et al. (2005), 40, 522
Daley (1974), 54, 522
Daley & Vere-Jones (1988), 51, 85, 94,
522
Dan et al. (1996), 45, 522
Daubechies (1988), 120, 522
Daubechies (1992), 58, 522
Davenport & Root (1987), 186, 189, 522
Davidsen & Schuster (2002), 149, 522

AUTHOR INDEX 569
Davies & Harte (1987), 139, 142, 522
Dayan & Abbott (2001), 77, 522
DeBoer et al. (1984), 57, 64, 282, 523
DeLotto et al. (1964), 237, 263, 523
Dettmann et al. (1994), 440, 523
Devaney (1986), 25, 523
Diament & Teich (1970a), 36, 523
Diament & Teich (1970b), 501, 523
Ding & Yang (1995), 138, 523
Ditto et al. (1990), 30, 523
Doob (1948), 85, 523
Doob (1953), 189, 523
Dorogovtsev & Mendes (2003), 37, 321,
523
Doukhan (2003), 16, 523
Doyle & Carlson (2000), 37, 523
Duffy et al. (1994), 317, 523
Dulea et al. (1992), 40, 523
Ebel et al. (2002), 321, 524
Eccles (1957), 91, 524
Efron (1982), 255, 524
Efron & Tibshirani (1993), 255, 524
Egu´
ıluz et al. (2005), 38, 524
Einstein (1905), 19, 524
Ellis (1844), 97, 524
Embrechts et al. (1997), 57, 524
Engset (1915), 315, 316, 524
Engset (1918), 316, 524
Erlang (1909), 315, 524
Erlang (1917), 316, 318, 319, 524
Erlang (1920), 316, 525
Erramilli et al. (1996), 328, 525
Evans et al. (2001), 227, 525
Evarts (1964), 16, 42, 525
Fadel et al. (2004), 42, 525
Fairfield-Smith (1938), 33, 525
Fairhall et al. (2001a), 42, 525
Fairhall et al. (2001b), 42, 525
Falconer (2003), 16, 39, 525
Faloutsos et al. (1999), 321, 525
Fano (1947), 66, 525
Fatt & Katz (1952), 41, 44, 45, 151, 525
Feder (1988), 16, 39, 525
Feldmann et al. (1998), 323, 331, 335,
526
Feller (1941), 85, 526
Feller (1948), 50, 237, 526
Feller (1951), 60, 526
Feller (1968), vi, 87, 93, 174, 179, 526
Feller (1971), vi, 21, 35, 51, 56, 82, 86,
138, 146, 157, 164, 165, 192,
237, 526
Feynman (1965), 34, 526
Feynman et al. (1963), 24, 526
Fibonacci (1202), 40, 526
Field et al. (2004a), 334, 526
Field et al. (2004b), 334, 526
Fisher (1972), 82, 526
Flake (2000), 16, 39, 526
Flandrin (1989), 138, 141, 526
Flandrin (1992), 139, 143, 527
Flandrin (1997), 15, 527
Flandrin & Abry (1999), 15, 16, 39, 527
Fleckenstein (1969), vii, 527
Fourier (1822), 102, 527
Franken (1963), 84, 527
Franken (1964), 84, 527
Franken et al. (1981), 84, 527
Fr´
echet (1940), 50, 527
Furry (1937), 95, 527
Gardner (1978), 116, 527
Garrett & Willinger (1994), 330, 527
Gauss (1809), 36, 173, 527
Gellermann et al. (1994), 40, 527
Gere (2001), 34, 528
Gerstein & Mandelbrot (1964), 485, 528
Ghulinyan et al. (2005), 40, 528
Gilbert (1961), 154, 167, 528
Gilbert & Pollak (1960), 186, 187, 189,
190, 378, 528
Gilden (2001), 116, 528
Gillespie (1994), 168, 452, 528
Gisiger (2001), 37, 528
Glass & Mackey (1988), 25, 249, 528
Gnedenko & Kolmogorov (1968), 35,
528
Good (1961), 192, 528
Gottschalk et al. (1995), 16, 43, 528
Gradshteyn & Ryzhik (1994), 104, 166,
420, 448, 528
Grandell (1976), 88, 90, 197, 528
Grassberger (1985), 199, 528
Grassberger&Procaccia (1983), 75, 528
Grebogi et al. (1984), 30, 529
Greenwood & Yule (1920), 64, 147, 529
Greiner et al. (1999), 57, 529
Greis & Greenside (1991), 126, 529
Grigelionis (1963), 84, 529

570 AUTHOR INDEX
Gross & Harris (1998), 317, 529
Grossglauser & Bolot (1996), 327, 529
Gr¨
uneis (1984), 218, 335, 529
Gr¨
uneis (1987), 218, 529
Gr¨
uneis (2001), 218, 335, 529
Gr¨
uneis & Baiter (1986), 218, 335, 529
Gr¨
uneis & Musha (1986), 218, 219, 529
Gr¨
uneis et al. (1993), 42, 529
Gr¨
uneis et al. (1989), 42, 529
Gumbel (1958), 36, 57, 147, 529
Gurland (1957), 95, 530
Gutenberg & Richter (1944), 33, 530
Haar (1910), 67, 74, 102, 104, 530
Haight (1967), 23, 82, 530
Halford (1968), 39, 530
Halley & Inchausti (2004), 41, 530
Halsey (2000), 35, 530
Harris (1971), 242, 530
Harris (1989), 95, 530
Hattori et al. (2000), 40, 530
Hattori et al. (1994), 40, 530
Hausdorff et al. (1997), 43, 530
Hawkes (1971), 217, 530
Heath et al. (1998), 334, 335, 530
Heneghan et al. (1996), 113, 296, 530
Heneghan et al. (1999), 59, 531
Heneghan & McDarby (2000), 62, 531
H´
enon (1976), 28, 531
Henry & Zaffaroni (2003), 39, 531
Heyde & Seneta (2001), vii, 531
Hille (2001), 151, 531
Hohn et al. (2003), 335, 336, 346, 531
Holden (1976), 91–93, 149, 151, 531
Holtsmark (1919), 193, 531
Holtsmark (1924), 193, 531
Hon & Lee (1965), 275, 531
Hooge (1995), 169, 531
Hooge (1997), 169, 531
Hopcraft et al. (2002), 36, 531
Hopcraft et al. (2004), 36, 38, 532
Hopcraft et al. (1999), 36, 532
Hopkinson (1876), 38, 532
Hs¨
u & Hs¨
u (1991), 116, 532
Hu et al. (2001), 62, 532
Huberman & Adamic (1999), 321, 532
Humbert (1945), 192, 532
Hurst (1951), 59, 60, 116, 137, 287, 532
Hurst (1956), 59, 60, 137, 532
Hurst et al. (1965), 59, 60, 137, 532
Jaggard (1997), 40, 532
Jaggard & Sun (1990), 40, 532
Jakeman (1982), 40, 532
Jelenkovi´
c & Lazar (1999), 334, 532
Jelley (1958), 221, 222, 532
Jenkins (1961), 78, 532
Jensen (1992), 316, 533
Johnson (1925), 115, 533
Jost (1947), 237, 533
Kabanov (1978), 77, 533
Kagan & Knopoff (1987), 222, 533
Kallenberg (1975), 232, 533
Kang & Redner (1984), 472, 533
Kastner (1985), 169, 533
Katz (1966), 41, 151, 533
Kaulakys (1999), 149, 533
Kaye (1989), 16, 533
Kelly et al. (1996), 42, 533
Kendall (1949), 95, 533
Kendall (1953), 317, 533
Kendall (1975), 95, 518, 533
Kendall & Stuart (1966), 249, 533
Kenrick (1929), 172, 534
Kerner (1998), 4, 534
Kerner (1999), 4, 534
Khinchin (1934), 172, 534
Khinchin (1955), 84, 534
Kiang et al. (1965), 249, 534
Kingman (1993), 51, 534
Klafter et al. (1996), 19, 534
Kleinrock (1975), 316, 317, 534
Knoll (1989), 223, 534
Kobayashi & Musha (1982), 43, 116,
275, 534
Koch (1999), 91, 534
Kodama et al. (1989), 42, 534
Kogan (1996), 16, 116, 534
Kohlrausch (1854), 33, 534
Kohmoto et al. (1987), 16, 40, 534
Kol´
aˇ
r et al. (1991), 40, 535
Kolmogorov (1931), 19, 535
Kolmogorov (1940), 136, 137, 535
Kolmogorov (1941), 36, 535
Kolmogorov &Dmitriev (1947), 95, 535
Komenani & Sasaki (1958), 4, 535
Kou & Xie (2004), 35, 535
Krapivsky et al. (2000), 38, 535
Krapivsky et al. (2001), 38, 535
Krishnam et al. (2000), 179, 535

AUTHOR INDEX 571
Kumar & Johnson (1993), 123, 147, 535
Kurtz (1996), 335, 535
Kuznetsov & Stratonovich (1956), 70,
535
Kuznetsov et al. (1965), 70, 535
Lapenna et al. (1998), 155, 222, 536
Lapicque (1907), 91, 93, 536
Lapicque (1926), 91, 536
Latouche & Remiche (2002), 336, 536
L¨
auger (1988), 16, 41, 536
Lawrance (1972), 95, 202, 249, 536
Lax (1997), 186, 536
Leadbetter et al. (1983), 51, 536
Leland et al. (1993), 325, 536
Leland et al. (1994), 16, 325, 536
Leland & Wilson (1989), 117, 325, 340,
341, 345, 350, 536
Leland & Wilson (1991), 117, 325, 340,
341, 345, 350, 536
Levy & Taqqu (2000), 334, 335, 536
L´
evy (1937), 35, 192, 537
L´
evy (1940), 35, 192, 537
L´
evy (1948), 19, 537
L´
evy V´
ehel et al. (1997), 16, 39, 537
L´
evy V´
ehel & Riedi (1997), 331, 335,
537
Lewis et al. (2001), 42, 537
Lewis (1964), 94, 537
Lewis (1967), 94, 537
Lewis (1972), 51, 82, 88, 537
Li & Teich (1993), 147, 537
Li (1991), 37, 537
Libert (1976), 236, 263, 264, 537
Liebovitch (1998), 16, 41, 537
Liebovitch et al. (1987), 41, 537
Liebovitch et al. (2001), 41, 173, 538
Liebovitch & T´
oth (1990), 41, 538
Likhanov (2000), 497, 538
Likhanov et al. (1995), 337, 538
Little (1961), 318, 538
Liu (1997), 40, 538
Lotka (1926), 33, 538
Lotka (1939), 50, 85, 538
Lowen (1992), xxi, 86, 87, 155, 157,
169, 175–177, 538
Lowen (1996), 73, 239, 240, 538
Lowen (2000), 139, 142, 538
Lowen et al. (1997a), 41, 42, 45, 148,
149, 152, 446, 538
Lowen et al.(1997b), 16, 41, 42, 45, 117,
132, 147–149, 152, 360, 446,
538
Lowen et al. (1999), 178, 539
Lowen et al.(2001), 42, 77, 78, 117, 121,
486, 539
Lowen et al. (1998), 77, 539
Lowen & Teich (1989a), 186, 187, 191,
192, 539
Lowen & Teich (1989b), 186, 187, 196,
539
Lowen & Teich (1990), 186, 187, 189–
191,193–196, 216, 376–378,
471, 539
Lowen & Teich (1991), 90, 95, 186, 193,
204–209,212–214, 336, 377,
539
Lowen & Teich (1992a), 42, 117, 249,
539
Lowen & Teich (1992b), 169, 173, 539
Lowen & Teich (1993a), 73, 115, 308,
539
Lowen & Teich (1993b), 42, 145, 174,
539
Lowen & Teich (1993c), 41, 173, 539
Lowen & Teich (1993d), 41, 86, 155–
157, 164, 173, 178, 539
Lowen & Teich (1995), 13, 41, 42, 66,
115, 133, 173, 174, 273, 306,
440, 540
Lowen & Teich (1996a), 16, 42, 68, 540
Lowen & Teich (1996b), 145, 540
Lowen & Teich (1997), 145, 249, 540
Lubberger (1925), 50, 540
Lubberger (1927), 50, 540
Lukes (1961), 86, 197, 540
Lundahl et al. (1986), 139, 540
Maccone (1981), 144, 540
Machlup (1954), 172, 540
Malamud (2004), 33, 540
Malik et al. (1996), 274, 540
Mandel (1959), 146, 540
Mandelbrot (1960), 33, 154, 541
Mandelbrot (1964), 154, 541
Mandelbrot (1965a), 16, 154, 167, 541
Mandelbrot (1965b), 137, 541
Mandelbrot (1967a), 4, 541
Mandelbrot (1967b), 144, 541
Mandelbrot (1969), 335, 541

572 AUTHOR INDEX
Mandelbrot (1972), 154, 541
Mandelbrot (1974), 331, 541
Mandelbrot (1975), 4, 541
Mandelbrot (1982), 3, 11, 13, 16, 33,
35, 36, 39, 40, 59, 60, 75, 95,
116, 137, 140, 143, 150, 154,
541
Mandelbrot (1997), 39, 123, 154, 330,
541
Mandelbrot (1999), 15, 123, 541
Mandelbrot (2001), 16, 39, 60, 541
Mandelbrot & Hudson (2004), 39, 154,
541
Mandelbrot & Van Ness (1968), 136–
138, 141, 143, 144, 542
Mandelbrot & Wallis (1969a), 142, 542
Mandelbrot & Wallis (1969b), 60, 542
Mandelbrot & Wallis (1969c), 60, 542
Mannersalo & Norros (1997), 331, 542
Marinari et al. (1983), 34, 542
Masoliver et al. (2001), 189, 542
Matsuo et al. (1982), 95, 542
Matsuo et al. (1983), 95, 206, 542
Matsuo et al. (1984), 95, 542
Matthes (1963), 54, 542
Matthews et al. (2003), 36, 542
McGill (1967), 146, 147, 202, 542
McGill & Goldberg (1968), 43, 542
McGill & Teich (1995), 43, 543
McWhorter (1957), 39, 169, 173, 543
Merlin et al. (1985), 16, 40, 543
Mikosch et al. (2002), 336, 543
Millhauser et al. (1988), 16, 41, 543
Mitchell (1968), 36, 543
Mitchell et al. (1995), 43, 543
Molchan (2003), 136, 543
Montanari (2003), 39, 543
Montgomery (1952), 172, 543
Montroll & Shlesinger (1982), 36, 116,
543
Moon (1992), 25, 543
Moran (1967), 249, 543
Morant (1921), 237, 544
Moriarty (1963), 154, 544
Morse (1921a), 40, 544
Morse (1921b), 40, 544
Moskowitz et al. (1974), 43, 544
Moyal (1962), 50, 544
M¨
uller (1973), 236, 237, 263, 264, 544
M¨
uller (1974), 236, 263, 264, 544
M¨
uller (1981), 236, 237, 263, 544
Musha (1981), 16, 43, 45, 116, 544
Musha & Higuchi (1976), 16, 116, 544
Musha et al. (1985), 116, 544
Musha et al. (1981), 42, 544
Musha et al. (1983), 16, 42, 116, 544
Myskja (1998a), 316, 544
Myskja (1998b), 524, 545
Myskja & Espvik (2002), 524, 544, 545
Newell & Sparks (1972), 4, 545
Newton (1687), 24, 34, 545
Neyman (1939), 202, 426, 545
Neyman & Scott (1958), 93, 94, 202,
545
Neyman & Scott (1972), 93–95, 202,
545
Norros (1994), 328, 545
Norros (1995), 325, 331, 335, 545
Norsworthy et al. (1996), 91, 545
Olson (2004), 4, 545
Omori (1895), 33, 545
Oppenheim & Schafer (1975), 304, 306,
545
Orenstein et al. (1982), 169, 545
Orer et al. (2003), 42, 545
Oshanin et al. (1989), 471, 546
Ott (2002), 25, 402, 546
Ott et al. (1994), 25, 227, 546
Ovchinnikov & Zeldovich (1978), 472,
546
Palm (1937), 315, 546
Palm (1943), 50, 82, 84, 227, 231, 237,
256, 258, 316, 318, 546
Papangelou (1972), 228, 546
Papoulis (1991), 64, 186, 197, 375, 410,
546
Pareto (1896), 33, 154, 155, 546
Park & Gray (1992), 93, 546
Park (2000), 328, 546
Park et al. (1996), 33, 329, 335, 336, 546
Park et al. (2000), 329, 333, 546
Park & Willinger (2000), 16, 39, 314,
547
Parzen (1962), 23, 51, 82, 85, 97, 192,
197, 231, 236, 237, 264, 547
Pastor-Satorras&Vespignani(2004), 37,
321, 547
Paulus & Geyer (1992), 16, 44, 547

AUTHOR INDEX 573
Paxson & Floyd (1995), 325, 335, 336,
547
Pecher (1939), 92, 149, 547
Peitgen & Saupe (1988), 16, 39, 139,
275, 547
Peitgen et al. (1997), 16, 25, 28, 29, 39,
547
Peltier & L´
evy V´
ehel (1995), 123, 547
Penck (1894), 2, 547
Peng et al. (1995), 43, 61, 547
Peng et al. (1993), 43, 547
Peˇ
rina (1967), 147, 547
Perkal (1958a), 2, 547
Perkal (1958b), 2, 547
Perrin (1909), 19, 548
Petropulu et al. (2000), 192, 197, 548
Picinbono (1960), 186, 187, 189, 548
Pinsky (1984), 35, 223, 548
Pipiras & Taqqu (2003), 144, 548
Poincar´
e (1908), 25, 174, 548
Poisson (1837), 63, 83, 548
Pollard (1946), 192, 548
Pontrjagin & Schnirelmann (1932), 12,
75, 548
Powers & Salvi (1992), 42, 548
Press et al. (1992), 356, 548
Prucnal & Saleh (1981), 249, 548
Prucnal & Teich (1979), 110, 548
Prucnal & Teich (1980), 249, 548
Prucnal & Teich (1982), 202, 549
Prucnal & Teich (1983), 237, 263, 549
Quenouille (1949), 95, 549
Quine & Seneta (1987), 83, 549
Rammal & Toulouse (1983), 471, 549
Rana (1997), 18, 549
Rangarajan & Ding (2000), 126, 549
Raymond & Bassingthwaighte (1999),
144, 549
Reid (1982), vii, 549
Reiss (1993), 51, 549
R´
enyi (1955), 74, 549
R´
enyi (1956), 232, 549
R´
enyi (1970), 74, 549
Ricciardi & Esposito (1966), 263, 549
Rice (1944), 147, 172, 176, 186, 190–
192, 194, 195, 549
Rice (1945), 147, 172, 176, 186, 190–
192, 194, 195, 550
Rice (1983), 175, 550
Richardson (1960), 3, 550
Richardson (1961), 2, 6, 550
Riedi (2003), 123, 550
Riedi & L´
evy V´
ehel (1997), 331, 550
Riedi & Willinger (2000), 331, 550
Rieke et al. (1997), 77, 550
Roberts & Cronin (1996), 15, 550
Roughan et al. (1998), 328, 550
Rudin (1959), 40, 550
Rudin (1976), 13, 550
Ruszczynski et al. (2001), 45, 550
Rutherford & Geiger (1910), 24, 64, 551
Ryu & Elwalid (1996), 327, 551
Ryu & Lowen (1995), 204, 336, 551
Ryu & Lowen (1996), 259, 260, 334,
551
Ryu & Lowen (1997), 204, 334, 336,
551
Ryu & Lowen (1998), 115, 204, 334,
336, 551
Ryu & Lowen (2000), 189, 551
Ryu & Lowen (2002), 336, 551
Sakmann & Neher (1995), 41, 551
Saleh (1978), 51, 82, 88, 89, 146, 147,
551
Saleh et al. (1983), 88, 202, 551
Saleh et al. (1981), 202, 237, 551
Saleh & Teich (1982), 88, 90, 94, 95,
186, 189, 202, 205, 207–209,
486, 552
Saleh & Teich (1983), 93, 94, 202, 552
Saleh & Teich (1985a), 202, 552
Saleh & Teich (1985b), 227, 552
Saleh & Teich (1991), 34, 222, 552
Samorodnitsky&Taqqu(1994),35, 174,
552
Sapoval et al. (2004), 173, 552
Sato (1999), 35, 174, 552
Scharf et al. (1995), 68, 552
Schepers et al. (1992), 60, 552
Scher & Montroll (1975), 169, 552
Schick (1974), 173, 552
Schiff & Chang (1992), 227, 253, 552
Schmitt et al. (1998), 123, 173, 331, 552
Sch¨
onfeld (1955), 195, 553
Schottky (1918), 186, 553
Schreiber & Schmitz (1996), 253, 553
Schroeder (1990), 16, 28, 34, 39, 46,
116, 553

574 AUTHOR INDEX
Schuster (1995), 25, 553
Seidel (1876), 63, 553
Sellan (1995), 139, 553
Shapiro (1951), 40, 553
Shimizu et al. (2002), 16, 43, 553
Shlesinger (1987), 36, 116, 553
Shlesinger & West (1991), 13, 553
Sigman (1995), 54, 553
Sigman (1999), 57, 553
Sikula (1995), 173, 553
Simoncelli & Olshausen (2001), 45, 553
Smith (1958), 85, 554
Snyder & Miller (1991), 51, 82, 88, 197,
554
Solomon & Richmond (2002), 37, 554
Soma et al. (2003), 45, 554
Song et al. (2005), 321, 554
Sornette (2004), 15, 16, 35, 554
Srinivasan (1974), 51, 82, 554
Steinhaus (1954), 2, 554
Stepanescu (1974), 169, 173, 554
Stern et al. (1997), 41, 151, 554
Stevens (1957), 43, 554
Stevens (1971), 43, 554
Stoksik et al. (1994), 139, 554
Stoyan & Stoyan (1994), 16, 554
Strogatz (1994), 25, 554
Sz˝
okefalvi-nagy (1959), viii, 530, 554
Tak´
acs (1960), 85, 555
Takayasu et al. (1988), 199, 555
Taqqu (2003), 136, 555
Taqqu & Levy (1986), 177, 335, 555
Taqqu & Teverovsky (1998), 62, 289,
555
Taqqu et al. (1995), 274, 309, 555
Taqqu et al. (1997), 331, 555
Taubes (1998), 325, 555
Taylor (2002), 45, 555
Teich (1981), 202, 206, 555
Teich (1985), 249, 555
Teich (1989), 16, 42, 45, 555
Teich (1992), 42, 145, 249, 555
Teich & Cantor (1978), 227, 263, 555
Teich & Diament (1980), 237, 555
Teich & Diament (1989), 95, 556
Teichet al. (1997), 16, 42, 183, 204, 485,
556
Teich et al. (1996), 16, 42, 51, 58, 74,
113, 117, 296, 351, 484, 556,
558
Teich et al. (1990), 42, 145, 202, 204,
249, 427, 556–558
Teich & Khanna (1985), 227, 249, 556
Teich et al. (1993), 227, 232, 476, 556
Teich & Lowen (1994), 42, 249, 556
Teich & Lowen (2003), 145, 217, 556
Teichet al. (2001), 16, 44, 145, 227, 275,
556
Teich et al. (1991), 42, 556
Teich et al. (1978), 227, 237, 556
Teich & McGill (1976), 147, 237, 556
Teich et al. (1982a), 202, 557
Teich et al. (1982b), 202, 557
Teich & Rosenberg (1971), 36, 557
Teich & Saleh (1981a), 202, 557
Teich & Saleh (1981b), 202, 557
Teich & Saleh (1982), 227, 232, 233,
236, 557
Teich & Saleh (1987), 202, 206, 557
Teich & Saleh (1988), 88, 202, 557
Teich & Saleh (1998), 202, 557
Teich & Saleh (2000), 88, 202, 557
Teich et al. (1984), 202, 264, 483, 557
Teich & Turcott (1988), 42, 558
Teich& Vannucci (1978), 227, 237, 483,
501, 558
Teicher et al. (1996), 16, 44, 558
Telesca et al. (2004), 77, 558
Telesca et al. (1999), 155, 222, 558
Telesca et al. (2002a), 155, 558
Telesca et al. (2002b), 222, 558
Terman (1947), 34, 558
Tewfik & Kim (1992), 114, 139, 296,
297, 300, 558
Theiler (1990), 74, 75, 558
Theiler et al. (1992), 227, 253, 559
Theiler & Prichard (1996), 253, 559
Thi´
ebaut (1988), 155, 559
Thomas (1949), 206, 559
Thompson & Stewart (2002), 25, 559
Thorson & Biederman-Thorson (1974),
42, 559
Thue (1906), 40, 559
Thue (1912), 40, 559
Thurner et al. (1998), 43, 275, 559

AUTHOR INDEX 575
Thurner et al. (1997), 16, 41, 58, 66, 68,
71, 115, 146, 174, 217, 242,
273, 310, 394, 485, 559
Tiedje & Rose (1980), 169, 559
Timmer & K¨
onig (1995), 139, 559
Toib et al. (1998), 41, 560
Toussaint & Wilczek (1983), 472, 560
Tuan & Park (2000), 329, 560
Tuckwell (1988), 91, 93, 560
Tukey (1957), 249, 560
Turcott et al. (1995), 42, 167, 168, 265–
267, 451, 452, 560
Turcott et al. (1994), 112, 560
Turcott&Teich(1993),16, 43, 275, 487,
560
Turcott&Teich(1996),16, 43, 117, 227,
265, 275, 282, 487, 560
Turcotte (1997), 16, 39, 560
Turner et al. (1998), 16, 39, 560
Usher et al. (1995), 37, 560
van der Waerden (1975), 231, 560
van der Ziel (1950), 39, 560
van der Ziel (1979), 195, 561
van der Ziel (1986), 16, 561
van der Ziel (1988), 16, 116, 561
Vannucci & Teich (1978), 237, 238, 561
Vannucci & Teich (1981), 237, 238, 561
Veitch & Abry (1999), 274, 280, 561
Vere-Jones (1970), 94, 202, 204, 222,
223, 561
Verhulst (1845), 26, 561
Verhulst (1847), 26, 561
Verveen (1960), 16, 41, 92, 116, 173,
561
Verveen& Derksen (1968),41, 149, 151,
173, 561
Vicsek (1992), 16, 35, 561
Vicsek (2001), 16, 561
Ville (1948), 138, 561
Viswanathan et al. (1996), 45, 561
Voldman et al. (1983), 155, 562
von Bortkiewicz (1898), 83, 562
von Schweidler (1907), 39, 562
Voss (1989), 116, 562
Voss & Clarke (1978), 116, 562
Watson & Galton (1875), 95, 562
Weber (1835), 33, 562
Weiss (1973), 197, 562
Weiss et al. (1994), 192, 562
Weissman (1988), 16, 116, 562
Wescott (1976), 232, 562
West (1990), 45, 562
West & Bickel (1998), 168, 452, 562
West et al. (2003), 16, 39, 562
West & Deering (1994), 16, 41, 562
West& Deering (1995), 16, 41, 45, 116,
562
West & Shlesinger (1989), 36, 563
West & Shlesinger (1990), 36, 563
West et al. (1999), 116, 563
Whittle (1962), 34, 563
Wickelgren (1977), 43, 563
Wiener (1923), 19, 563
Wiener (1930), 172, 563
Willinger et al. (2004), 330, 563
Willinger et al. (2003), 314, 325, 329,
563
Willis (1922), 33, 563
Wise (1981), 485, 563
Witten & Sander (1981), 35, 199, 563
Wixted (2004), 43, 563
Wixted & Ebbesen (1991), 43, 563
Wixted & Ebbesen (1997), 43, 563
Wold (1948), 50, 564
Wold (1949), 50, 564
Wold (1965), 51, 564
Wolff (1982), 319, 564
Yamamoto & Nakahama (1983), 42, 564
Yamamoto et al. (1986), 42, 564
Yang et al. (2004), 222, 564
Yang & Petropulu (2001), 181, 184, 335,
564
Yu et al. (2005), 45, 335, 564
Yule (1924), 95, 564
Zhukovsky et al. (2001), 40, 440, 564
Zipf (1949), 33, 564
Zrelov (1968), 221, 222, 468, 564
Zucker (1993), 151, 565
Zuckerkandl & Pauling (1962), 168, 565
Zuckerkandl & Pauling (1965), 168, 565
Zumofen et al. (2004), 173, 565

Subject Index
absolute refractoriness, See event dele-
tion
action potentials, 50, 204
amygdala, 42
auditory nerve fiber, 42, 68, 117–
119,128–131, 145, 147, 233–
235,244–246, 249–251, 253,
254, 344, 345
central nervous system, 42
hippocampus, 42
integrate-and-reset model, Seeintegrate-
and-reset process(es)
lateral geniculate nucleus, 42, 77,
117–121,126–131, 183, 233–
235, 244–246, 250, 251, 253,
254
medulla, 42
reticular formation, 42
retinal ganglion cell, 42, 77, 117–
119,128–131, 183, 233–235,
244–246, 250, 251, 253, 254,
344
somatosensory cortex, 42
striate cortex, 42, 117–119, 128–
131,233–235, 244–246, 250,
251, 253, 254, 344, 345, 351
surrogate data analysis, 227
thalamus, 42
visual-systeminterneuron, 42,167–
168, 265–267, 344, 345
Allan, David W., viii, 68, 269, 276
Allan factor, 68
Allan variance, 68, 269
alternating fractal renewal process(es),
172–173, 177–182
autocorrelation, 183–184
autocovariance, 178, 183
chain of Markov processes, 178–
179
computernetworktraffic, 173, 334–
335
dwell times, 174
fractal binomial noise, 173, 181
fractalGaussian process, 173, 181–
182
fractal test signals, 173, 184
ion channels, 41, 173
nanoparticle fluorescence fluctua-
tions, 173
nerve-membranevoltagefluctuations,
173
rainfall, 173
577

578 SUBJECT INDEX
semiconductor noise, 173
spectrum, 177, 183
sums of, 173, 181
systemswith fractal boundaries,173
alternatingrenewalprocess(es), 172–182
alternatingfractal renewalprocess,
Seealternating fractal renewal
process(es)
autocorrelation, 175
Bernoulli random variables, 174
binomial noise, 173, 179–181
burst noise, 172
characteristic function, 175, 183
dwell times, 174
exponential dwell times, 176–177
extreme asymmetry, 176
Gaussian process, 181–182
moments, 174–175
on–off process, 172
random telegraph signal, 172
relation to renewal point process,
176
spectrum, 175–177, 183
sums of, 173, 179–181
amygdala, See action potentials
attention-deficit hyperactivity disorder,
44
auditory nerve fiber, See action poten-
tials
Barnes, James, 269, 276
Bartlett, Maurice, 94, 201, 202
Bartlett–Lewisprocess, See cascadedpro-
cess(es)
Berger, Jay, 153, 154
Bernoulli random deletion, Seeeventdele-
tion
Bernoulli, Jakob, 225, 226
binomial noise
asa sum ofalternating renewalpro-
cesses, 173, 179–181
autocorrelation, 179–181
binomial distribution, 179
convergenceto aGaussian process,
181–182
fractal, See fractal binomial noise
moments, 179
bivariate point process, See point pro-
cess(es)
block shuffling, See operations on point
processes
blocked counter, See event deletion
bootstrapmethod, Seeoperationson point
processes
box-counting dimension, See dimension
branchingprocess, Seecascadedprocess(es)
Brownian motion, 19–21
as a neuronal threshold, 149–150
Bachelier process, 19
definition, 19–20
diffusion process, See power-law
behavior
fractal-based point process from,
149–150
generation of, 150
history, 19
relationto fractional Brownianmo-
tion, 137
Wiener–L´
evy process, 19
zero crossings, 15
Burgess variance theorem, 231
burst noise, See alternating renewal pro-
cess(es)
Cantor, Georg, 9
Cantor set, 17–19
fat, 18
Hausdorff–Besicovitchdimension,
75
membership, 47
photonic multilayer-structure ver-
sion, 40
randomized version, 95
semiconductormultilayer-structure
version, 40
triadic, 17
variant, 46, 133–134
capacity dimension, See dimension
capacity-dimension scaling function, See
dimension
cascaded process(es), 93–95
applications of, 202, 323
Bartlett–Lewis, 94, 98–99, 324
branching, 95
cluster, 93
compound, 93
doubly stochastic Poisson process
version, 95, 333, 336, 346

SUBJECT INDEX 579
fractalBartlett–Lewis, 218–219, 335–
336, 345–351
fractalNeyman–Scott, 336–337, 345–
351
Neyman–Scott, 93, 202, 204, 324
Poisson branching, 95
Thomas, 95, 206
Yule–Furry, 95
central limit theorem, 36, 173, 174, 181,
188, 191, 216, 306
central nervous system, See action po-
tentials
ˇ
Cerenkov radiation, Seephotonstatistics
chaos, 25–32
fractal attractors, 25
fractals, connection to, 24–32
functional roles, 25
phase-randomizationsurrogate, 227
strange attractors, 25
characteristicfunction, Seeintervalstatis-
tics
clusterprocess, Seecascadedprocess(es)
coastline(s)
Australian, 6
British, 6
fractal, 2–4
H¨
ofn, 2, 6
Icelandic, 2, 6, 13–15
length of, 2–4, 6, 14
Seyðisfj¨
orður, 2, 6
South African, 6
compound process, See cascaded pro-
cess(es)
computer cache misses, 155
computer communication networks, See
computer network traffic
computer network traffic, 313–354
alternatingfractal renewalprocess,
173, 334–335
analysis and synthesis, 332
applications layer, 323
arrival process, 317
as a point process, 50
bit transmission, 323
blocking probability, 319
buffer occupancy, 316
buffer overflow probability, 319–
321, 325, 334, 351–352
buffer size, 316
CAIDA, 322
capacity-dimension scaling func-
tion, 343
cascaded-processmodels, 335–337,
345–351
characteristic features of, 342–343
computercommunication networks,
320–323, 328, 332
data sources, 117, 325
detrended fluctuations, 338, 340,
341, 348, 349
drop probability, 319
Ethernettraffic, 117–119, 315, 325,
337, 340–342, 345, 350, 351
event clustering, 345
exponentializeddata, 250, 251, 326,
327, 337, 343, 345
extendedalternatingfractal renewal
process, 335
feedback, 332
file transfers, 323
flow control, 333
fluid-flow models, 331
forwardKolmogorovequation,318,
351
fractalBartlett–Lewisprocess, 218,
335–336, 345–351
fractal exponents, 44, 126, 343–
344
fractal features, 16, 40, 324–332
fractal-Gaussian-process-drivenPois-
son process, 335, 352–353
fractalNeyman–Scottprocess, 204,
328, 333, 335–337, 345–353
fractal-rate point process, 342
fractal renewal process, 334
fractal-shot-noise-drivenPoissonpro-
cess, 204, 335–337
fractional Brownian motion, 325,
331
FTP, 323, 329, 335, 336
generalarrivalandservice processes,
317
generalized dimension, 343
geometricqueue-length distribution,
318, 319, 328, 351–352
heavy-tailedservicetimes, 328,333,
337
HTTP, 323, 335

580 SUBJECT INDEX
internetwork layer, 323
interval histogram, 130, 131, 233,
244, 338, 340–344, 348, 349,
351
interval sequence, 337, 340, 341,
348, 349
interval spectrum, 128, 339–341,
344, 348, 349
interval statistics, 350–351
intervalwaveletvariance,129, 338,
340, 341, 344, 348, 349
IP, 322–323
ISP, 322
link layer, 323
Little’s law, 318
local-area network, 325
Markovprocess, 317, 325, 327, 328
message-loss probability, 328, 332
model complexity, 332–333
modeling, 332–337, 345–351
modulatedfractal-Gaussian-process-
driven Poisson process, 353
monofractal approximation, 353–
354
multifractalfeatures, 16,329, 331–
332, 353–354
multiple data sets, 341–342
multiple servers, 317, 319, 351
multiple statistical measures, 337–
340
normalizedHaar-waveletvariance,
119,235, 325–327, 330, 339–
341, 344, 348, 349, 351, 353
normalized variance, 126, 127
packets, 50, 315, 323, 325
PASTA, 319
periodicities, 342
persistence, 329
physical layer, 323
point-processdescription, 330–331
point-processidentification, 271,337–
351
power-law file sizes, 33, 323, 329–
330, 335, 336
power-law queue-length distribu-
tion, 328, 352
predictability, 329
queue length, 316–318
queue waiting-time jitter, 328
queue-lengthdistribution, 317, 318,
325, 328, 334, 351, 353
queueingtheory,316–320, 327–329,
336–337
randomly deleted data, 234, 235,
344
randomly displaced data, 245, 246
rate-process description, 330–331,
334, 338, 340, 341, 348, 349
rate spectrum, 118, 234, 325–326,
339–341, 344, 348, 349, 351
rescaled range, 338, 340, 341, 348,
349
resemblanceto striate-cortex action
potentials, 345, 351
scale-freenetworks, 37–38, 40, 321–
322
scaling cutoffs, 330
second-orderstatistics, 325–328,340,
341, 348, 349
server utilization, 318
service process, 317
service ratio, 318, 328
shuffled data, 253, 254, 326, 327,
337, 342–344, 351, 352
simulations, 332–333, 345–353
SSH, 323, 329
static representation, 322
TCP, 323, 329, 334
teletraffic theory, 315
TELNET, 323, 335
transport layer, 323
UDP, 329
vertical layers, 323
video traffic, 325, 330
waiting number mean, 318
waiting time mean, 318
wide-area network, 325
World Cup access log, 330
WWW, 325
correlation dimension, See dimension
counting statistics, 63–70
α-particle counting, 64
accidents, 64, 147
Allan factor, 68
Allan variance, 68, 269
autocorrelation, 69, 71, 72, 106–
107, 121, 124, 132, 222, 232,
282, 296, 311

SUBJECT INDEX 581
autocovariance, normalized form,
69, 285–287
count sequence, 51, 63
counting distribution, 64, 65, 146–
147,163, 205–206, 218, 263–
264
counting-timeincrements, 297–298
counting-time oversampling, 302–
304
counting-timeweighting, 298–302
cross-spectrum, 78
dead-time-modified point process,
145, 227, 238, 240, 263–264
decimated point process, 263–264
dispersion ratio, 66
doubly stochastic Poisson process,
See doubly stochastic Pois-
son process(es)
factorial moments, 65, 83, 87, 161
Fano factor, 66
fractal renewal process, See fractal
renewal process(es)
fractal-shot-noise-drivenpointpro-
cess, See fractal-shot-noise-
driven point process(es)
generalized rates, 64
generalized version of normalized
Haar-wavelet variance, 69
homogeneous Poisson process, See
homogeneousPoisson process
index of dispersion, 66
integrate-and-reset process, Seeintegrate-
and-reset process(es)
kurtosis, 65
moments, 65–66
negativebinomial distribution,146
Neyman Type-A distribution, 202,
206, 209, 212
noncentral negative binomial dis-
tribution, 147
normalizedDaubechies-waveletvari-
ance, 120
normalizedgeneral-waveletvariance,
74, 113–114, 120, 296–297
normalizedHaar-waveletcovariance,
77–78
normalizedHaar-waveletvariance,
62, 66–69, 71, 73, 80, 97,
103–105,107, 111–112, 114,
115,117–119, 121, 122, 124–
126, 133, 167, 168, 232, 235,
236, 240, 246, 247, 249, 251,
254, 265, 266, 270, 275–282,
284–289,291, 293, 296–304,
306,307, 309–312, 325–327,
330, 339–344, 348, 349, 351,
353
normalizedHaar-waveletvariance,
relation to normalized vari-
ance, 62, 68–69, 112, 209–
211, 284, 296, 344
normalizedvariance, 66, 68–69, 71,
73, 79, 80, 96–99, 105, 107,
109–110, 112, 121, 124, 126,
127, 133, 134, 212, 219, 222,
231,234, 236, 282–284, 286–
288, 296, 311, 312, 344
normalizedwaveletcross-correlation
function, 77
periodic processes, 64
periodogram, 70
rate-based measures, 64
relationto interval statistics, 65, 344
relationshipamong measures, 114–
115
renewal process, See renewal pro-
cess(es)
sample rate, 64
shot-noise-driven Poisson process,
See doubly stochastic Pois-
son process(es)
skewness, 65
spectrum, 39, 43, 64, 70, 73, 80,
87, 88, 97, 103, 111, 113–
117, 121, 122, 124–126, 133,
134, 234, 236, 245, 247, 249,
250, 253, 254, 271, 275, 282,
304–312,325–326, 339–341,
344, 348, 349, 351
Thomas distribution, 206
variance-to-mean ratio, 66
Cox, David R., vi, viii, 81, 88
Cox process, See doublystochastic Pois-
son process(es)
cross-spectrum, See counting statistics
data-transmission errors, See fractal re-
newal process(es)

582 SUBJECT INDEX
dead-time deletion, See event deletion
decimation, See event deletion
deletion, See event deletion
detrended fluctuation analysis, See inter-
val statistics
developmental disorders, 44
developmental insults, 44
diffusion processes, See power-law be-
havior
dilation, See operations on point pro-
cesses
dimension
box-counting, 12, 14, 75, 164, 237
Cantor set, 18
Cantor-set variant, 46
capacity, 12, 14, 75, 96, 131, 164,
237
capacity-dimension scaling func-
tion, 131, 132, 343
correlation, 75, 96
Euclidian, 11, 23, 35, 75, 223
generalized, 74–76, 96, 130–132,
256, 332, 343
generalized-dimensionscaling func-
tion, 76, 131, 132, 266, 267
Hausdorff–Besicovitch, 75
information, 75
Kolmogorov entropy, 75
monofractal, 18, 75, 121
multifractal, 75
of a space, 11
of an object, 11
of diffusion processes, 34–35
of point processes, 75–76, 96, 111,
121, 126, 130–132, 256, 272,
332, 343
R´
enyi entropy, 74
topological, 11, 75
wavelet estimate of, 75
Dirac delta function, special property of,
92, 228
dispersion ratio, See counting statistics
displacement, See operations on point
processes
doubly stochastic Poisson process(es),
87–90
autocorrelation, 88
cascaded-processisomorph, 95,333,
336, 346
coincidence rate, 88
counting statistics, 88–89
dead-time-modified, 237–241
exponentialintervaldensity,89–90,
125, 249, 272, 281
factorial moments, 88
fractal-binomial-noise-driven,174,
182, 183, 272
fractal-Gaussian-process-driven,124–
125, 145, 183, 217, 229, 249,
270, 275–281, 310, 328, 335,
352–353
fractal-lognormal-noise-driven,250
fractal-rate-driven, 124, 262
fractal-shot-noise-driven,See fractal-
shot-noise-driven point pro-
cess(es)
integrated rate, 88
interval density, 89
interval statistics, 89–90
multistage shot-noise-driven, 95
random deletion of, 236
rate coefficient ofvariation, 89–90,
281
renewal version of, 90
shot-noise-driven, 90, 202–204
simulation of, 270, 310
spectrum, 88
superpositionof, Seesuperposition
drug abuse, 44
earthquakes, 33, 40, 77, 150, 155, 204,
222–223
emotional state, 44
equilibrium counter, See event deletion
Erlang,Agner Krarup,97, 313,315, 316,
319
Euclidian dimension, See dimension
event deletion, 226, 229–241
Bernoullirandom deletion, 226, 227,
230–236, 262–263, 344
blocked counter, 236, 264
Burgess variance theorem, 231
dead-time deletion, 226, 227, 230,
236–241, 262–264
decimation,97, 226, 227, 230–232,
262–264
decimation parameter, 231

SUBJECT INDEX 583
doubly stochastic Poisson process,
See doubly stochastic Pois-
son process(es)
effectsonfractal features, 229–231,
236
equilibrium counter, 236, 264
experimental interval histograms,
232–233
experimentalnormalizedHaar-wavelet-
variance curves, 235
experimental rate spectra, 234
fractal onset frequency, 232, 241
fractal onset time, 232, 241
fractal renewal process, See fractal
renewal process(es)
general results, 229–231
homogeneous Poisson process, See
homogeneousPoisson process
limitof a homogeneousPoisson pro-
cess, 232
periodic process, 232–236
renewal process, See renewal pro-
cess(es)
type-pdead time, 236
unblocked counter, 236, 263, 264
excitable-tissue recordings, 41
expansion-modification systems, 37
exponentialization,See operations onpoint
processes
extended dead time, See event deletion
Fano factor, 66
Fatt & Katz, 45
Feller, William, vi, 225, 237
fern, 22
Fibonacci sequences, See photonic ma-
terials, See semiconductors
fixed dead time, See event deletion
fluorescence fluctuations of nanoparti-
cles,See alternatingfractalre-
newal process(es)
Fourier, Jean-Baptiste, 101–102
fractal analysis, Seefractalparameter es-
timation
fractal-based point processes, See point
process(es)
fractal binomial noise
as a rate function, 174, 182–183,
272, 334
as a sum of alternating fractal re-
newal processes, 173, 181
convergenceto aGaussian process,
181–182
fractal-binomial-noise-drivengammapro-
cess, 183
fractal chi-squared noise, 145–147
as a rate function, 150–151
fractal exponential noise, 146
fractalnoncentral chi-squared noise,
147
fractal noncentral Rician-squared
noise, 147
negative binomial counting distri-
bution, 146
noncentralnegativebinomialcount-
ing distribution, 147
fractal exponent(s)
auditory nerve fiber, See action po-
tentials
computernetworktraffic, Seecom-
puter network traffic
cutoffs, 14–15, 36, 103, 105, 107–
108, 133, 274, 330
diffusion, See power-law behavior
estimation of, See fractal parame-
ter estimation
for fractal Bartlett–Lewis process,
219
for fractal point process, 121
for fractal-rate process, 124
for fractal shot noise, See fractal
shot noise
for multifractals, 15, 75, 331
for nonstationary nonfractal pro-
cesses, 110, 112, 133
fornormalized general-waveletvari-
ance, 113–114
for normalized Haar-wavelet vari-
ance, 111–114
from autocorrelation, 110–111
from count-based autocovariance,
287
from interval spectrum, 126–128,
295
fromnormalized Daubechies-wavelet
variance, 120
from normalized detrended fluctu-
ations, 291

584 SUBJECT INDEX
fromnormalized Haar-waveletvari-
ance,117–119, 235, 246, 251,
254, 278, 299, 303
from normalized interval wavelet
variance, 127–129, 293
fromnormalized rate spectrum,116–
118, 234, 245, 250, 253
fromnormalized rescaled range,289
fromnormalized variance, 109–110,
126, 127, 285
from rate spectrum, 307
human heartbeat, See heartbeat
Hurstexponent, 137, 143–144, 287,
289
lateral geniculate nucleus, See ac-
tion potentials
limited range of, 109–111
negative values of, 107–109, 133
observed values of, 109
range of values, 107–114
relations among, 105, 107, 114–
115, 133
relativestrengthoffluctuations, 103,
273
retinal ganglion cell, Seeactionpo-
tentials
same exponent from different frac-
tal renewal processes, 166
spectrum, 133
striate cortex, See action potentials
superposition, See superposition
time varying, 331
underexponentialization, 250, 251,
264
under general deletion, 229–231
under random deletion, 234, 235
under random displacement, 245,
246
under shuffling, 253, 254
values in biological systems, 34
vesicular exocytosis, See vesicular
exocytosis
visual-system interneuron, See ac-
tion potentials
fractal exponential noise, 146
fractal Gaussian process(es), 144–145
as a rate function, 145, 216–217
as a sum of alternating fractal re-
newal processes, 173, 181–
182
nomenclaturefor fractional processes,
143–145
fractal lognormal noise, 147–149
as a rate function, 148–149, 151–
152
rate statistics, 147–148
fractalnetworks, See scale-freenetworks
fractalnoncentral chi-squared noise,147
fractal noncentral Rician-squared noise,
147
fractal parameter estimation, 269–312
asymptote subtraction, 312
autocovariance, 285–287
bias from cutoffs, 274
bias/variance tradeoff, 311
choice of scaling range, 274
coincidence-rate limitations, 311
comparison of measures, 309–310
count-based measures, 282–287
counting-timeincrements, 297–299,
303
counting-time oversampling, 302–
304
counting-timeweighting, 298–302
detrended fluctuations, 289–291
discrete-time processes, 274
estimator bias, 274, 278–280, 284,
285, 287, 289, 291, 293, 295,
307
estimator root-mean-square error,
278, 285, 287, 289, 291, 293,
295, 299, 303, 307
estimator standard deviation, 278,
285, 287, 289, 291, 293, 295,
307
estimator variance, 273
fractal exponents, 107, 126, 270,
273–281, 285, 287, 289, 291,
293, 295, 299, 303, 307
heart rate variability, 274–275
interval-based measures, 287–296
interval spectrum, 294–296
intervalwaveletvariance,291–293
limitations of, 310
maximum-likelihoodapproach, 274
nonparametric approach, 273–274

SUBJECT INDEX 585
normalizedgeneral-waveletvariance,
296–297
normalizedHaar-waveletvariance,
276–281, 296–304, 344
normalizedvariance, 127, 282–285,
296, 311, 344
optimal measures, 271, 309
rate spectrum, 133, 304–309, 311
rescaled range, 287–289
robustness/error tradeoff, 311
simulations, 270, 275–278, 284–
295, 297, 299, 303, 305, 307,
310, 312
speed/accuracy tradeoff, 274
fractal point processes, See point pro-
cess(es)
fractal-rate point processes, Seepointpro-
cess(es)
fractalrenewalprocess(es), 87, 124, 131,
132, 154–166, 281
capacity dimension, 164
characteristic function, 155, 156,
166
coincidence rate, 159–160
comparisonwith homogeneous Pois-
son process, 122
computer cache misses, 155
computer network traffic, 334
counting distribution, 163
data-transmission errors, 40, 154,
166–167
earthquake occurrences, 155
effectofinterval-densityexponent,
157
factorial moments, 160–161
features of, 122
forward recurrence time, 262
fractal exponents, 158
fractal onset frequency, 166
generalized inverse Gaussian den-
sity, 156
generalized Pareto density, 165
interneuron counterexample, 167–
168, 265–267
interval density, 155–157
intervaldensitywith abrupt cutoffs,
155
interval density with smooth tran-
sitions, 156–157
interval moments, 155, 156
molecular evolution, 168–169
nondegeneraterealization, 164–166
normalizedHaar-waveletvariance,
162
normalized variance, 160–162
Pareto density, 154–155
point-process spectrum, 157–159,
166
random deletion of, 236, 263
same fractal exponent from differ-
ent interval densities, 166
simulation time, 166, 312
stable distribution, 157
superpositionof, Seesuperposition
trapping in semiconductors, 169,
224
Wald’s Lemma, 164
fractal shot noise, 186–197
amplitude statistics, 189–193
as a rate function, 90, 202–205
autocorrelation, 194–195
characteristic function, 189–190
cumulants, 190
degenerate, 188, 193
fractal exponents, 195–197
Gaussian limit, 145, 188
impulseresponse function, 187–188,
202, 205
integrated, 204–205
mass distributions, 198–199
multifractalimpulse response func-
tion, 331
parameter ranges, 188, 189
point processes from, See doubly
stochasticPoisson process(es)
power-law-duration variant, 188–
189, 198, 336, 352
spectrum, 188, 195–197
stable distribution, 188, 192, 193,
197
sums of, 198
fractal-shot-noise-driven integrate-and-
resetprocess, Seefractal-shot-
noise-drivenpointprocess(es)
fractal-shot-noise-drivenpointprocess(es),
202–217
applications of, 204

586 SUBJECT INDEX
applications of the Neyman Type-
A distribution, 202
applicationsof the shot-noise-driven
Poisson process, 202
ˇ
Cerenkov radiation, 220–222
coincidence rate, 214
computernetworktraffic, 328, 335–
337, 352–353
counting distribution, 205–206
counting statistics, 205–212
design of, 220
diffusion, 223
earthquakes, 222–223
factorial moments, 207–208
forward recurrence time, 212–213
fractal exponents, 209, 211, 214,
215
fractal-Gaussian-process-drivenlimit,
216–217
fractal-shot-noise-drivenintegrate-
and-reset process, 217
fractal-shot-noise-drivenPoissonpro-
cess, 90, 202–217
Hawkes point process, 217
impulse response function without
cutoffs, 220
intervaldensity,212–213,219, 272
multifractal version, 331
Neyman–Scott process, 202, 204
Neyman Type-A distribution, 202,
206
normalizedHaar-waveletvariance,
210–212
normalizedvariance, 208–209, 219
self-exciting point process, 217
semiconductorparticle detectors, 223–
224
spectrum, 215–216
fractal-shot-noise-drivenPoissonprocess,
See fractal-shot-noise-driven
point process(es)
fractals
and Kant, 33
and Kohlrausch, 33
and Laplace, 33
and Leibniz, 33
and Weber, 33
and Weierstraß, 33
artificial, 16–21
chaos, connection to, 24–32
coastlines, 2–4, 6
convergenceto stabledistributions,
35–36
deterministic, 13, 16–19, 21–22
diffusion processes, 34–35
dynamical processes, 13
examples of fractals, 16–23, 28–
30, 33, 115–120
examplesofnonfractals, 23–24,26–
28
expansion-modificationsystems, 37
highly optimized tolerance, 37
historical antecedents, 32–33
in art, 45
in ecology, 26, 41
in human behavior, 43–44
in mathematics, 39–40
in medicine, 43–44
in music, 116
in the biological sciences, 41–44
in the neurosciences, 41–43
in the physical sciences, 39–40
in the psychological sciences, 42,
43, 45, 116
in vehicular-traffic flow, 4, 44, 45,
50, 116
laws of physics, 33–34
lognormal distribution, 36, 147
long-range dependence, 14–15
natural, 16, 21–23
noninteger dimension, 14
objects, 4
onset frequencies, 114–115
onset times, 114–115
origins of fractal behavior, 32–39,
329–330
Pareto’s Law, 33
pink noise, 115–116
power-lawbehavior, connection to,
14, 32–39
putative exponential cutoff, 39
random, 13, 16, 19–23
rangeof time constants,38–39, 332
recognizing the presence of fractal
behavior, 44–45
salutary features of fractal behav-
ior, 41, 45
scale-free networks, 37–38, 45

SUBJECT INDEX 587
scaling, connection to, 13–15
self-organized criticality, 37
static, 13
ubiquity of fractal behavior, 39–44
fractals in human behavior
attention-deficit hyperactivity dis-
order, 44
developmental disorders, 44
developmental insults, 44
drug abuse, 44
mood fluctuations, 43
fractals in mathematics
convergenceto stabledistributions,
35–36
fractal geometry, 40
lognormal distribution, 36
fractals in medicine
blood flow, 116
congestive heart failure, 275
fluctuations in human standing, 43
heart rate variability, 43–44, 270,
274–275
pain relief, 45
sensitizationof baroreflex function,
45
fractals in the neurosciences
action potentials in auditory nerve
fibers, 42, 131, 145, 147, 249
actionpotentials in central-nervous-
system neurons, 42, 131
action potentials in isolated prepa-
rations, 41–42
action potentials in visual-system
neurons,42, 77, 131, 183, 217,
267
cognitive processes, 43, 116
electroencephalogramfluctuations,
116
excitable-tissuefluctuations, 41, 92,
116, 149, 151, 173
ion-channeltransitions, 41,151, 173
neuronalavalanches in slice prepa-
rations, 42
sensory detection and estimation,
42–43, 45
vesicular exocytosis, 41, 131, 132,
149, 151–152
fractals in the physical sciences
ˇ
Cerenkov radiation, 34, 40, 204,
220–222
computer network traffic, 40, 313–
354
data-transmission errors, 40, 154,
166–167
diffusionprocesses, 34–35, 204, 223
earthquakeoccurrences, 33, 40, 77,
150, 155, 204, 222–223
highly optimized tolerance, 37
laws of physics, 33–34
light scattering, 36, 40, 173
photonics, 40
self-organized criticality, 37, 222
semiconductors, 34, 39, 40, 116,
169, 172, 173, 223–224
fractional Brownian motion, 136–141
as a model for computer network
traffic, 325, 331
as a rate function, 140–141
autocorrelation, 137, 150
autocorrelation coefficient, 150
definition, 21, 137
generalized dimensions, 139–140
generationby fractional integration,
144
history, 136
Hurst exponent, 137
level crossings, 138
nomenclaturefor fractional processes,
143–145
ordinary Brownian motion, SeeBrow-
nian motion
properties, 138–139
realizations, 139–140
relation of Hurst and scaling expo-
nents, 143–144
relationto fractional Gaussiannoise,
141
relation to ordinary Brownian mo-
tion, 137
self-similarity, 138
stationary increments, 137, 150
synthesis, 139
Wigner–Ville spectrum, 138–139
zero crossings, 15
fractional Gaussian noise, 141–142
as a rate function, 142
definition, 141

588 SUBJECT INDEX
generalized dimensions, 142
generationby fractional integration,
144
in a Langevin equation, 35
nomenclaturefor fractional processes,
143–145
properties, 141–142
realizations, 142–143
relation of Hurst and scaling expo-
nents, 143–144
relationto fractional Brownianmo-
tion, 141
synthesis, 142
Wigner–Ville spectrum, 141–142
gammarenewalprocess, See renewalpro-
cess(es)
Gauss, Carl Friedrich, 36, 171, 173
generalized dimension, See dimension
generalized-dimension scaling function,
See dimension
generalizedinverseGaussiandensity,156
Grand Canyon river network, 22
Greenwood, Major, 49, 64, 147
Gutenberg–Richter Law, 33
Haar, Alfr´
ed, 68, 101, 102
Hausdorff–Besicovitch dimension, See
dimension
heart rate variability, 44, 270, 274–275
heartbeat, 43, 50, 64, 79, 116–119, 128–
131,145, 232–235, 244–246,
250,251, 253, 254, 264, 274–
275, 293, 345
heavy-tailed distributions, See interval
statistics
highly optimized tolerance, 37
hippocampus, See action potentials
Holtsmark distribution, 193
homogeneous Poisson process, 3, 23–
24, 71, 82–85, 96–97, 108,
122, 124, 125, 131, 132, 167,
198, 231, 236, 246, 249, 255,
258,260, 276, 278, 281, 316–
320, 323, 325, 328, 335, 352
dead-time-modified,237–238, 249,
262–264
decimated, 97, 263–264
factorial moments, 84
moments, 83
human standing, 43
Hurst, Harold Edwin, 59, 269, 287
Hurst exponent, See fractal exponent(s)
hypothesistesting, Seeoperationson point
processes
Icelandic coastline, 14
index of dispersion, See counting statis-
tics
information dimension, See dimension
integrate-and-reset process(es), 91–93
dead-time-modified, 237, 241
decimated, 231
fractal-binomial-noise-driven,174,
183
fractal-Gaussian-process-driven,145,
243
fractal-shot-noise-driven, 217
gamma-distributed rate, 98
identification of, 273
interval density, 92
interval moments, 92
interval statistics, 91–92
kernel for heartbeat model, 293,
310
leaky, 93
model for action potentials, 91
modulated rate, 98, 110, 112
normalized variance, 96
oversampled sigma-delta modula-
tor, 91
packet generation, 334
point-process spectrum, 91
randomly deleted, 236
time-varyingthreshold, 92–93, 149
interevent-intervaltransformation, See op-
erations on point processes
interneuron, See action potentials
interval statistics, 54–62
autocorrelation, 20, 57, 83, 282
characteristic function, 55–56
coefficient of variation, 55, 231,
233, 236, 345
cumulants, 55, 56
density, 55, 89–90, 121, 129–130,
227, 281
detrended fluctuation pseudocode,
62

SUBJECT INDEX 589
detrended fluctuation statistic, 61–
62, 79, 282
detrended fluctuation statistic, nor-
malized form, 62, 289–291
discriminating among fractal-rate
processes, 310–311
distribution, 281
doubly stochastic Poisson process,
See doubly stochastic Pois-
son process(es)
exponential density, 83, 89–90, 92,
97, 125
fractal renewal process, See fractal
renewal process(es)
fractal-shot-noise-drivenpointpro-
cess, See fractal-shot-noise-
driven point process(es)
heavy-tailed distributions, 13, 56,
57, 328, 333, 337
homogeneous Poisson process, See
homogeneousPoisson process
infinite moments, 56, 79, 165
integrate-and-reset process, Seeintegrate-
and-reset process(es)
intervalordering, 90, 227, 247–254,
256, 281, 345
kurtosis, 55, 79, 175, 179
limitations of, 122–124, 281–282,
344
moments, 55, 60
normalized wavelet variance, 59
Pareto distribution, 57, 138, 154,
155, 165
periodic processes, 64, 96
periodogram, 70
power-law distribution, See fractal
renewal process(es)
recurrence time, 56, 65, 79, 80, 96
relation to counting statistics, 65,
344
rescaled range pseudocode, 60
rescaled range statistic, 59–60, 79,
282
rescaledrange statistic, normalized
form, 60, 287–289
semi-invariants, 55
serial correlation coefficient, 57
shot-noise-driven Poisson process,
See doubly stochastic Pois-
son process(es)
skewness, 55, 79, 175, 179
spectrum, 42, 43, 58–59, 62, 64,
80, 83, 116, 126–128, 271,
275,282, 294–296, 339–341,
344, 348, 349, 351
subexponential distributions, 57
survivorfunction,55–57, 165,238,
259–262
wavelet transform, 58
wavelet variance, 58–59, 62, 127–
129, 275, 282, 291–293, 344
Weibull distribution, 57, 328
interval transformation, See operations
on point processes
ion channels, See alternating fractal re-
newal process(es)
Isham, Valerie, vi
Kenrick, Gleason W., 172
knockout mice, 227
Kolmogorov, Andrei, 135–136
Lapicque, Louis, 81, 91
lateral geniculate nucleus, See actionpo-
tentials
laws of physics, See power-law behavior
Leyden-jar discharge, 33, 39
light scattering, See fractals
logistic equation, 26, 37
logistic map, 26, 28, 30, 46
lognormal distribution, 36, 57, 147
long-range dependence, 15
L´
evy, Paul, 35, 36, 171, 174
L´
evy dust, See point process(es)
L´
evy-stable distributions, See stable dis-
tributions
Mandelbrot, Benoit, viii, 4, 135, 136,
153, 154
markedpointprocess, Seepoint process(es)
medulla, See action potentials
mixedPoissonprocess, Seedoubly stochas-
tic Poisson process(es)
molecular evolution, See fractal renewal
process(es)
monofractals, 15–16, 274, 331

590 SUBJECT INDEX
mood fluctuations, 43
multidimensional point process, Seepoint
process(es)
multifractals, 15–16, 75, 188, 331–332,
353–354
multivariatepointprocess, See pointpro-
cess(es)
Newton’s Law, 34
Neyman, Jerzy, 94, 201, 202
Neyman Type-A distribution, See count-
ing statistics
Neyman–Scottprocess, See cascadedpro-
cess(es)
Nile river flow patterns, 59, 116, 269
noncentral limit theorem, 174
nonextended dead time, See event dele-
tion
nonfractal(s)
Euclidian shapes, 6–7, 23
examples of, 14, 23–24, 26–28, 46
generalized dimensions, 75, 96
heart rate variability measures, 43,
275
homogeneous Poisson process, See
homogeneousPoisson process
influences, 279
orbits in a two-body system, 24
pointprocesses, Seepointprocess(es)
radioactive decay, 24, 50
nonparalyzable dead time, Seeeventdele-
tion
nonstationary point process, See point
process(es)
normalization, See operations on point
processes
normalized Haar-wavelet covariance, See
counting statistics
normalized Haar-wavelet variance, See
counting statistics
normalizedvariance, See countingstatis-
tics
normalizedwaveletcross-correlationfunc-
tion, 77
normalizing transformation, See opera-
tions on point processes
Omori’s Law, 33
on–off process, See alternating renewal
process(es)
operations on point processes
block shuffling, 255, 262
bootstrap method, 255
event deletion, See event deletion
event-timedisplacement,145, 226,
242–247, 251, 254, 255, 262
hypothesis testing, 126, 247, 253,
254, 271
imposed by experimenter, 227
imposed by measurement system,
227
interval displacement, 242
intervalexponentialization,226, 227,
249–251,255, 261, 264–267,
326, 327, 337, 343, 345
interval normalization, 249
interval shuffling, 226–227, 252–
256,261–262, 265–267, 271,
326, 327, 337, 342–344, 351,
352
interval transformation, 226, 247–
251, 261–262
intrinsicto underlying process,227
phase randomization, 227
point-process identification, 255–
256
superposition, See superposition
surrogate data, 15, 126, 227, 247,
253, 265–267, 271
time dilation, 226, 228–229, 262
Palm, Conny, 50, 82, 257, 313, 315, 316
paralyzable dead time, See event dele-
tion
Pareto, Vilfredo, 33, 153–154
Pareto distribution, 33, 138, 154–155,
165, 198, 346
Pareto’s Law, 138, 154
Penck, Albrecht, 1–2
periodogram,See countingstatistics, See
interval statistics
phase randomization, See operations on
point processes
photon statistics
betaluminescence, 202
cathodoluminescence, 202

SUBJECT INDEX 591
ˇ
Cerenkov radiation, 34, 40, 204,
220–222
in presence of atmospheric turbu-
lence, 36
in presence of dead time, 263
radioluminescence, 202
scattered light, 40
superposed coherent and thermal
light, 147
thermal light, 146
photonic materials
diffractals, 40
fractal reflectance, 40
fractal transmittance, 40
group-velocity reduction, 40
light scattering, 40
multilayer structures, 40
phase screen, 40
pseudo-bandgaps, 40
Poincar´
e, Henri, 9, 25, 174
point process(es), 4–5, 50–80, 82–99
Bartlett–Lewis,See cascaded point
process(es)
bivariate, 77
branching,See cascadedprocess(es)
capacity dimension, 164
cascaded,See cascadedprocess(es)
coincidence rate, 70–72, 74, 80,
105–106,110–111, 133, 159–
160, 214
computer network traffic, See com-
puter network traffic
correlation in a bivariate process,
77–78
count-based measures, 63–70
deleted, See event deletion
doubly stochastic Poisson, Seedou-
blystochastic Poisson process(es)
early work, 50
estimation of, See fractal parame-
ter estimation
examples of, 4, 79, 82–99
filtered general, 197–198
fractal, 76, 121–123, 131, 255
fractal-based, 4–5, 120–124, 130
fractal behavior in, 115–120
fractal parameter estimation, See
fractal parameter estimation
fractal-rate,76, 123–124, 251, 255,
345
fractal renewal process, See fractal
renewal process(es)
fractal-shot-noise-driven,See fractal-
shot-noise-driven point pro-
cess(es)
from Brownian motion, 149–150
from fractal binomial noise, 182–
183
general measures of, 70–76
Hawkes, 217
homogeneous Poisson, See homo-
geneous Poisson process
identificationof, 125–131,255–256,
270–273, 337–351
infinitelydivisiblecascade, 123,331
integrate-and-reset, See integrate-
and-reset process(es)
intermittency, 122
interval-based measures, 54–62
limitations of measures, 282
L´
evy dust, 15, 95–96, 138
marked, 54, 77, 176, 222, 334
measures of fractal behavior, 103–
107
modulated integrate-and-reset, See
integrate-and-resetpointpro-
cess(es)
monofractal, 131
multidimensional, 82
multifractal, 123
multivariate, 77
Neyman–Scott,See cascaded point
process(es)
nonfractal, 96, 124–125, 131
nonstationary, 71, 110, 112, 133–
134
operations on, See operations on
point processes
orderly, 52–54, 66, 75, 79, 95, 96,
110, 174, 197, 206, 220
periodic, 91, 232–236
renewal, See renewal process(es)
right-continuous, 51
self-exciting, 217
sinusoidally modulated, 98
spectrum, 72–74, 80, 88, 91, 97,
103, 111, 115, 121, 122, 124,

592 SUBJECT INDEX
125, 133, 149, 151, 157–159,
166,183, 215–216, 218–220,
230–232, 261, 262, 311, 312
spectrum, normalized form, 72
superposed, See superposition
Poisson, Sim´
eon Denis, 49, 63
Poissonprocess, SeehomogeneousPois-
son process
power-law behavior
anharmonic-oscillator energy, 34
ˇ
Cerenkov radiation, 34
computer file sizes, 33, 323, 329–
330, 335, 336
Coulomb’s Law, 34
diffusionprocesses, 34–35, 204, 223
dipole field, 34
expansion-modificationsystems, 37
fractal exponent, See fractal expo-
nent(s)
fractals, connection to, 14, 32–39
Gutenberg–Richter Law, 33
harmonic-oscillator energy, 34
highly optimized tolerance, 37
Hooke’s Law, 34
hydrogen-atom energy, 34
infinite-quantum-well energy, 34
interval distribution, See fractal re-
newal process(es)
Kepler’s Third Law, 34
Langmuir–Childs Law, 34
laws of physics, 33–34
line of charge, 34
logistic equation, 37
lognormal distribution, 36
mass distributions, 198–199
Newton’s Law, 34
Omori’s Law, 33
Pareto’s Law, 33, 138, 154–155,
165, 198, 346
preservation of, 103
quadrupole field, 34
quantum number, 34
relationshipsamong measures, 114–
115
Richardson’s Law, 3
rigid-rotor energy, 34
scale-free networks, 37–38
scaling functions, 3, 12–13
self-organized criticality, 37
stable distributions, 35–36
superposedrelaxation processes, 38–
39
time functions, 34
van der Waals force, 34
queueing theory, See computer network
traffic
random deletion, See event deletion
random telegraph signal, See alternating
renewal process(es)
rate spectrum, See counting statistics
recovery function, See event deletion
refractoriness, See event deletion
relative dead time, See event deletion
relative refractoriness, See event dele-
tion
renewal process(es), 85–87
alternating,See alternatingrenewal
process(es)
coincidence rate, 85–86
decimatedPoisson process, 97, 263–
264
doubly stochastic Poisson version
of, 90
event deletion, See event deletion
exponential density, 97
factorial moments, 87
fractal,See fractalrenewalprocess(es)
gamma density, 97
gamma density for computer net-
work traffic, 336
history of, 85
invariance to shuffling, 271
operations on, See operations on
point processes
random deletion of, 236, 263
relationbetween interval and count-
ing statistics, 87
spectrum, 86–87, 97
superpositionof, Seesuperposition
R´
enyi dimension, See dimension
rescaledrange analysis, See interval statis-
tics
reticular formation, See actionpotentials
retinal ganglion cell, See action poten-
tials
Rice, Steven O., 147, 185, 186

SUBJECT INDEX 593
Richardson, Lewis Fry, 1–3
Rudin–Shapiro sequences, Seesemicon-
ductors
scale-freenetworks, 37–38, 40, 321–322
scaling, See fractals
scaling cutoffs, See fractal exponent(s)
scalingexponents, See fractalexponent(s)
Schottky, Walter, 185, 186
Scott, Elizabeth, 201, 202
self-organized criticality, 37
semiconductors
fractional scaling exponents, 34
multilayer structures, 40
noise in, 39–40, 116, 169, 172, 173
particle detectors, 223–224
range of time constants, 39
trapping in, 169, 224
semi-experiments,See operations onpoint
processes
shot noise, 186–187
amplitude, 186–187
as a rate function, 90
filteredgeneral point process,197–
198
fractal, See fractal shot noise
Gaussian limit, 186, 191
generalized, 187
impulse response function, 202
shuffling, See operations on point pro-
cesses
sick time, See event deletion
somatosensory cortex, See action poten-
tials
spectral smoothing, 117, 128, 326, 339
spectrum, See counting statistics, See in-
tervalstatistics, See pointpro-
cess(es)
spike trains, See action potentials
stable distributions, 35–36, 79, 157, 174,
192, 193
stochastic dead time, See event deletion
striate cortex, See action potentials
superposition
alternating renewal processes, 41,
173, 179–182
doublystochastic Poisson processes,
258–259
fractal-basedand homogeneous Pois-
son processes, 273
fractal-based point processes, 258,
261, 310
fractal content, 262
fractal Gaussian process and mod-
ulating stimulus, 145
fractal ion-channel transitions, 42
fractalrenewalprocesses, 260–261
harmonic functions, 101
packet arrival times, 323
periodic series of events, 81
point processes, 84–85, 227, 256–
261
Poisson-process limit, 85
relaxation processes, 38–39
renewal processes, 259–260, 334
secondary events comprising, 218
surrogate data, See operations on point
processes
survivor function, See interval statistics
synapse, See vesicular exocytosis
telephone network traffic, 40, 84, 97,
154,166–167, 313, 315–320,
324
tent map, 46
thalamus, See action potentials
thinning, See event deletion
Thomasdistribution, See countingstatis-
tics
Thomasprocess, Seecascadedprocess(es)
Thue–Morse sequences, Seephotonicma-
terials, See semiconductors
time dilation, See operations on point
processes
time series, 4
topological dimension, See dimension
translation, See operations on point pro-
cesses
triadic Cantor set, See Cantor set
type-pdead time, See event deletion
unblocked counter, See event deletion
Van Ness, John W., viii, 135, 136
variance-to-meanratio, See countingstatis-
tics

594 SUBJECT INDEX
vesicular exocytosis, 41–43, 117–119,
128–132,149, 151–152, 233–
235, 244–246, 250, 251, 253,
254, 344, 345
visual-systeminterneuron, Seeactionpo-
tentials
Wald’s Lemma, 164, 237
wavelet(s)
computer-network-trafficanalysis,
336
Daubechies, 120
estimating the generalized dimen-
sion, 75
generatingfractional Brownian mo-
tion, 139
Haar, 101, 269
higher-order moments, 332
interval wavelet variance, See in-
terval statistics
normalizedDaubechies-waveletvari-
ance, See counting statistics
normalizedgeneral-waveletvariance,
See counting statistics
normalizedHaar-waveletcovariance,
See counting statistics
normalizedHaar-waveletvariance,
See counting statistics
removing trends, 62, 113
transform, 58, 67, 74
Weibull distribution, 57, 328
Wiener–Khintchine theorem, 73
Yule, G. Udny, 49, 64, 95, 147
Yule–Furry branching process, See cas-
caded process(es)
zeta distribution, 33, 38