Partial Identification of Probability Distributions
Statistical inference for fractal-based point processes is concentrated in the final three chapters. I found these chapters very readable and well presented. Although choosing a point process model that fits real data reasonably well is difficult, often due to the sparseness of the data and the lack of available statistical inference, the authors demonstrate how certain types of fractal behavior can be detected in real datasets by comparing several suitable statistics and/or by using resampling techniques. The authors show this skillfully on the seven real-data examples considered throughout the book. By means of simulations, they also nicely compare the performance of nonparametric estimators of point process measures and of the fractal exponent. Finally, the book’s last chapter considers the accompanying example of computer network traffic in greater detail. This phenomenon has been studied extensively in the literature, and the authors nicely incorporate the existing references into the established framework of fractal-based point processes. Previously considered statistical techniques are put into action, and several fractal-based point processes are fitted to a dataset comprising 1 million consecutive ethernet-packet arrivals. Pedagogically, each chapter is accompanied by well-selected problems that nicely complement the presentation and facilitate a deeper understanding of the subject. It is here that the reader’s mathematical background is put to the test; many may find it beneficial that solutions to selected problems are discussed in the Appendix. Practical aspects of data analysis and simulations can be accessed from the book’s website. Given the tremendous importance of the subject, Fractal-Based Point Processes will certainly prove valuable to scientists working in many distinct fields. Writing a good, practically oriented book on fractals and point processes suitable for a broad audience is not easy, especially considering the mathematical complexity of the concepts. It is apparent that the authors are truly at home in their field and succeed in presenting the topic in a comprehensive and systematic way. Along with the references listed in the presentation, I believe that Fractal-Based Point Processes will be of good use to anyone studying point processes with self-similarity characteristics.