Fractal-Based Point ProcessesAn integrated approach to fractals and point processes 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 | |
9 | |
3 Point Processes Definition and Measures | 49 |
4 Point Processes Examples | 81 |
5 Fractal and FractalRate Point Processes | 101 |
6 Processes Based on Fractional Brownian Motion | 135 |
7 Fractal Renewal Processes | 153 |
8 Processes Based on the Alternating Fractal Renewal Process | 171 |
11 Operations | 225 |
12 Analysis and Estimation | 269 |
13 Computer Network Traffic | 313 |
Appendix A Derivations | 355 |
Appendix B Problem Solutions | 397 |
Appendix C List of Symbols | 505 |
Bibliography | 513 |
Author Index | 567 |
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analysis autocorrelation Cantor set cascaded point Chapter coincidence rate computer network traffic counting distribution cutoff data set deletion detrended fluctuation detrended fluctuation analysis dimension discussed in Sec displayed in Fig dN(t dN₁(t dNR(t doubly stochastic Poisson duration Equation example exhibit exponential Fourier transform fractal behavior fractal exponent fractal renewal point fractal renewal process fractal shot noise fractal-based point processes fractal-rate point process fractional Brownian motion frequency Gaussian process homogeneous Poisson process impulse response function integral integrate-and-reset interevent intervals interevent-interval density interval-based long-range dependence Lowen & Teich Mandelbrot measures nonfractal normalized Haar-wavelet variance number of events obtain parameters periodogram Poisson point process power-law power-law exponent Prob probability density function provided in Eq queue random variables rate spectrum renewal point process rescaled range Saleh scaling sequence shot noise shuffling simulation solid curve spectra statistics unity values wavelet variance yields zero
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