Fractal-Based Point Processes

As its title suggests, this book deals with the marriage of two well-known topics. Scaled versions of fractal geometrical objects possess certain congruence properties, usually called self-similarities. Point processes are mathematical models of the occurrence of discrete events in time, space, or some other arena. Due to a degree of incompatibility between the partners, this is not an easy marriage. Points, by definition, have no extension and are invariant under shift or scale transformations, so fractality can reside only in the distribution of the points. The approach of this book is to exploit scaling behavior of such distributions and the concomitant power-law behavior of their statistical summaries. The unnaturalness of continuous mathematical descriptors of discrete processes is most evident when one is dealing with real-world measurements. It is probably for this reason that there are not many effective data analysis techniques. Chapter 2 is a succinct, clear introduction to fractals. A reader completely new to the concepts might need to turn to other books, such as those meticulously referenced by the authors. My favorite is the no-nonsense book of Feder (1988). The next two chapters nicely describe several representations of point processes (based on event times, intervals between events, and counts of events in bins), a number of statistical quantities that characterize the processes, and numerous examples: homogeneous Poisson, renewal, doubly stochastic Poisson, integrate-and-reset, cascaded, and branching point processes. Of special interest for subsequent discussions, and probably new to many readers, is the normalized Haar-wavelet variance. Apparently first introduced by Allan in 1966, I as well as others have used this measure to characterize time series data that appear to be self-scaling, and modeled as “1/f noise processes” (Scargle et al. 1993). The terms scalegram or wavelet spectrum is often used (see Abry, Goncalves, and Flandrin 1995) and Flandrin (1999) for excellent mathematical discussions. Chapter 5 leads off the main content of the book, introducing the concept of fractal point processes. As mentioned above, the approach is indirect: the scaling properties of the statistical measures introduced in Chapter 2 define fractal behavior in point processes. Elaboration of this idea relies on the strong connection between power-law behavior of the measures, as functions of time, and the appropriate scaling behavior. This key chapter and its exercises make connections among a whole collection of measures of point process fractal behavior. For example, the wavelet variance and power spectrum express similar information about how variability depends on scale: they capture the self-similarity of fractal processes through the well-known “1/f α ” power-law dependence (on time scale or frequency, resp.). They also have similar ways of revealing what is sometimes called the noise floor at short time scales or high frequencies. The authors also follow a canonical set of fractal-based point processes through all of the measures (and transformations in Chap. 11) to help identify similarities and general behavior. Subsequent chapters treat special classes: fractal Brownian motion, renewal, alternating renewal, shot noise, and shot-noise-driven processes, all of considerable interest is a number of the sciences. Chapter 11 discusses a number of interesting ways that one point process can be transformed to another. These procedures are useful in randomization procedures (e.g., bootstrap methods) of various kinds. After a sad-but-true litany of the reasons that identification of fractal-based point processes is difficult or impossible, Chapter 12 studies estimation of parameters in the representations introduced earlier in the book, and the performance of the estimators. The final chapter, Computer Network Traffic, can be considered as a case study for the general reader, and presumably is of more specialized interest to computer scientists and telecommunications technologists. Each chapter ends with numerous problems suitable for students at various levels. There are three appendices containing derivations, problem solutions, and a list of symbols. Errata and addenda, C program source code, and data sets can be found at the authors’ website http://cordelia.mclean.org/~lowen/fbpp.html, and the book is also available for purchase from the publishers in e-book form. It is hard to find fault with this carefully crafted book. My only significant disappointment was with the paucity of practical methods for data analysis. Presumably in a course, students will learn computational methods from problems and computer laboratory assignments. Frankly, I did not pay much attention to the exercises included. The authors’ point of view is that the most powerful data analytic tools are the normalized Haar wavelet variance and the power spectrum. A reader interested in computing estimates of these descriptors should consult the code provided at the above website. Furthermore, one should take seriously the fact that there are important limitations in the current state of knowledge and, as pointed out by the authors, there are inherent mathematical difficulties, especially in the area of identification. The comment “Count-number statistics also provide the only systematic analysis approach available for spaces of dimension greater than one” in Section 3.4 short-changes recent progress using generalizations of point-based representations and their statistics in the fields of stochastic geometry (Stoyan, Kendall, and Mecke 1995) and computational geometry. For example, Voronoi cells of data points are slick estimators of local density and its gradient, and have been used in segmentation analysis of 3D data on the distribution of galaxies in the Universe (Scargle, Jackson, and Norris 2003; see also Scargle and Babu 2003) and no doubt will play a role in research into scaling properties of this distribution. These objections are quite minor. All in all, this is an excellent exposition of a cutting-edge topic, and will be extremely valuable as a textbook and for scientists in diverse fields, including astronomy.