Chaos: A Statistical Perspective: (Springer Series in Statistics)

This book discusses dynamical systems that are typically driven by stochastic dynamic noise. It is written by two statisticians essentially for the statistically inclined readers, although readers whose primary interests are in determinate systems will find some of the methodology explained in this book of interest. The statistical approach adopted in this book differs in many ways from the deterministic approach to dynamical systems. Even the very basic notion of initial-value sensitivity requires careful development in the new setting provided. This book covers, in varying depth, many of the contributions made by the statisticians in the past twenty years or so towards our understanding of estimation, the Lyapunov-like index, the nonparametric regression, and many others, many of which are motivated by their dynamical system counterparts but have now acquired a distinct statistical flavour. Kung-Sik Chan is a professor at the University of Iowa, Department of Statistics and Actuarial Science. He is an elected member of the International Statistical Institute. He has served on the editorial boards of the Journal of Business and Economic Statistics and Statistica Sinica. He received a Faculty Scholar Award from the University of Iowa in 1996. Howell Tong holds the Chair of Statistics at the London School of Economics and the University of Hong Kong. He is a foreign member of the Norwegian Academy of Science and Letters, an elected member of the International Statistical Institute and a Council member of its Bernoulli Society, an elected fellow of the Institute of Mathematical Statistics, and an honorary fellow of the Institute of Actuaries (London). He was the Founding Dean of the Graduate School and sometimes the Acting Pro-Vice Chancellor (Research) at the University of Hong Kong.

Table of Contents:
1 Introduction and Case Studies.- 2 Deterministic Chaos.- 3 Chaos and Stochastic Systems.- 4 Statistical Analysis I.- 5 Statistical Analysis II.- 6 Nonlinear Least-Square Prediction.- 7 Miscellaneous Topics.- Appendix A Deterministic Chaos.- A.1 Introduction.- A.2 Attracting Sets.- A.3 Another Look At the Logistic Maps.- A.4 Attractors.- A.5 Two Approaches to Studying Chaos.- A.6 Invariant and Ergodic Distributions.- A.7 Lyapunov Exponents.- A.8 Natural Measures.- A.9 Dimensions of an Attractor.- A.9.1 Box-Counting Dimension.- A.9.2 Correlation Dimension.- A.10 Map Reconstruction.- A. 11 Some Elements of Differentiable Manifolds.- A.12 Hyperbolic Sets.- A.13 Notes.- Appendix B Supplements to Chapter 3.- B.1 Criteria for Ergodicity.- B.1.1 Notes.- B.2 Proofs of Two Theorems in §3.3.2.- B.3 Shadowing and Hyperbolic Attractors.- Appendix C Data Sets and Software.- References.- Author Index.

Review :
From the reviews: SHORT BOOK REVIEWS "The authors have done an excellent job, providing an overview of known results with detailed references to the literature, as well as pointing out some open problems. In general, the book serves to 'encourage more statisticians to join in with the fun of chaos'." "The book fills a gap in the need to overview the present state of statistics and to point into the right direction for research. It seems to me that this has been achieved by the authors in an excellent way. Chan and Tong's book certainly deserves recommendation to anyone who is interested in dynamics, either as a statistician or as a researcher in the theory of dynamical systems, ergodic theory or differential equations." (Manfred Denker, Metrika, September, 2003) "The authors fully attain their aim stated in the introduction. Their style is very friendly and they take much care to prevent technical details from obscuring the essential issues. The book requires careful reading but the profit is well worth the effort. A truly enjoyable and recommendable book!" (Ricardo Maronna, Statistical Papers, Vol. 44 (1), 2003)