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MR1878355 (2003c:37118)

Galka, Andreas(D-KIEL)
Topics in nonlinear time series analysis. (English. English summary)
With implications for EEG analysis. Advanced Series in Nonlinear Dynamics, 14.
World Scientific Publishing Co., Inc., River Edge, NJ, 2000. xvi+342 pp. ISBN 981-02-4148-8

This book is a review of recently developed chaos-related methods for analyzing nonlinear time series. The specific and explicit interest of the author is to establish the relevance of these methods---if any---for electroencephalogram (EEG) analysis. The underlying hope is that even a partially positive indication of such relevance would have enormous implications for the medical diagnostics, prognosis, treatment, and possibly cure of certain brain disorders. This hope is warranted by an "educated expectation" that seems to become more and more accepted, namely, that the brain dynamics---despite its undisputed complexity---shares essential features with the dynamics of sufficiently low-dimensional nonlinear systems.

The author approaches the subject in a didactic and informed, but light way. After a short general introduction on time series, Chapter 2 reviews the basic properties and descriptors of deterministic and stochastic, discrete and continuous systems. Linear methods based on Fourier decomposition and correlation functions are briefly recalled in Chapter 3. Chapters 4 and 5 are devoted to a few theoretical and practical aspects, respectively, of state space reconstruction from nonlinear time series. In real-world applications, when more often than not one is dealing with noisy data, a critical issue is the determination of the best embedding parameters. Chapter 6 reviews various notions of dimension (e.g., topological, Hausdorff, information, capacity, etc.). The definitions are somewhat "bare-bone" and addressed to the physicist, medical doctor, or otherwise technical person intended to apply them. Given the aim of the book, this is not a serious deficiency. In Chapter 7 the Lyapunov exponents are introduced and their relationship with the Kolmogorov entropy is mentioned, via Pesin's theorem. Different numerical methods for estimating the correlation dimension (CD) are reviewed in Chapter 8, while Chapter 9 is devoted to various sources of errors introduced by data length and quality in the estimates of the CD. Chapter 11 reviews the surrogate data technique and Chapter 12 summarizes recent applications of the nonlinear measures and methods to EEG time series. A few special topics, such as Monte Carlo or interspike analysis, usually not treated in standard books on nonlinear dynamics, are mentioned in Chapters 10 and 13. The bibliography is judiciously selected and up-to-date.

The book is pleasantly written and makes for easy reading. It is informative for anyone with a sufficiently deep knowledge of nonlinear dynamics. However, a less skilled reader could hardly be expected, after reading this book, to actually implement any of the techniques and algorithms described.

Reviewed by V. Protopopescu