Fractal-based point processes / Steven Bradley Lowen, Malvin Carl Teich.

Format:

Book

Author/Creator:

Lowen, Steven Bradley, 1962-

Contributor:

Teich, Malvin Carl.

Benjamin Franklin Library Fund.

Language:

English

Subjects (All):

Point processes.

Fractals.

Physical Description:

xxiv, 594 pages : illustrations ; 24 cm

Place of Publication:

Hoboken, N.J. : Wiley-Interscience, 2005.

Summary:

This publication provides a complete and integrated presentation of the fields of fractals and point processes, from definitions and measures to analysis and estimation. The authors skillfully demonstrate how fractal-based point processes, established as the intersection of these two fields, are tremendously useful for representing and describing a wide variety of diverse phenomena in the physical and biological sciences. Topics range from information-packet arrivals on a computer network to action-potential occurrences in a neural preparation.

The authors begin with concrete and key examples of fractals and point processes, followed by an introduction to fractals and chaos. Point processes are defined, and a collection of characterizing measures are presented. With the concepts of fractals and point processes thoroughly explored, the authors move on to integrate the two fields of study. Mathematical formulations for several important fractal-based point-process families are provided, as well as an explanation of how various operations modify such processes. The authors also examine analysis and estimation techniques suitable for these processes. Finally, computer network traffic, an important application used to illustrate the various approaches and models set forth in earlier chapters, is discussed.

Throughout the presentation, readers are exposed to a number of important applications that are examined with the aid of a set of point processes drawn from biological signals and computer network traffic. Problems are provided at the end of each chapter allowing readers to put their newfound knowledge into practice, and all solutions are provided in an appendix. An accompanying Web site features links to supplementary materials and tools to assist with data analysis and simulation.

With its focus on applications and numerous solved problem sets, this is an excellent graduate-level text for courses in such diverse fields as statistics, physics, engineering, computer science, psychology, and neuroscience.

Contents:

1.1 Fractals 2

1.2 Point Processes 4

1.3 Fractal-Based Point Processes 4

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

3 Point 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

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 Levy-Dust Counterexample 95

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

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

7 Fractal Renewal Processes 153

7.1 Power-Law Distributed Interevent Intervals 155

7.2 Statistics of the Fractal Renewal Process 157

7.3 Nondegenerate Realization of a Zero-Rate Process 164

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

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

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

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

12 Analysis and Estimation 269

12.1 Identification of Fractal-Based Point Processes 271

12.2 Fractal Parameter Estimation 273

12.3 Performance of Various Measures 281

12.4 Comparison of Measures 309

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

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.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

C.1 Roman Symbols 506

C.2 Greek Symbols 510

C.3 Mathematical Symbols 511.

Notes:

Includes bibliographical references (pages 513-565) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Benjamin Franklin Library Fund.

ISBN:

0471383767

OCLC:

60188752

Publisher Number:

9780471383765