Ten years ago, I developed and began teaching a MSc course at Imperial College on the analysis of time series generated by chaotic dynamical systems. This is a branch of nonlinear dynamics which has exploded in popularity in the last two decades, largely due to the realisation that even simple deterministic non-linear systems can give rise to complex, random looking behaviour. This has raised the possibility that we may be able to describe apparently complex phenomena using simple non-linear models and has led to the development of a variety of novel techniques for the analysis and manipulation of nonlinear time series.
Initially, these methods concentrated on the characterisation of chaotic signals using invariants such as fractal dimensions or Liapunov exponents. Subsequently, attention turned to the possibility of predicting the short term behaviour of such signals, and this in turn led to algorithms for signal separation and noise reduction in time series having a chaotic component. In appropriate circumstances, such algorithms are capable of achieving levels of performance which are far superior to those obtained using classical linear signal processing techniques. Finally, most recently we have come to realise that in certain situations the presence of chaos can in fact be advantageous and the last few years have seen a flood of papers taking advantage of chaotic behaviour in areas such as control and synchronisation.
When I came to UCL, I took the time series course with me, and subsequently handed it over to my colleague Mike Davies (who did a far better job of teaching it). Throughout all these changes there has been a consistent refrain from the students, consisting of two questions:
My answer to the first is unprintable, and that to the second was a shrug of the shoulders: despite the popularity of the subject there was a complete lack of suitable texts. This gap was partly filled a few years ago by the excellent reprint collection Coping with Chaos. This consists of a well chosen and well organised selection of the most significant papers in the subject since it early beginnings. This is complemented by a brief introduction and suggestions for further reading for each of the main topics, and more significantly, a 60 page introduction by the editors which covers the fundamental ideas required to follow the reprinted articles. Together, these give a coherent overview of the subject comprehensible to a wide audience, and yet contains sufficient detail to take the interested reader right up to the current frontiers of knowledge.
However, no matter how good, a reprint collection is no substitute for a proper textbook, and I am therefore delighted that two comprehensive volumes on chaotic times series have recently been published: Nonlinear Time Series Analysis by H. Kantz and T. Schreiber and Analysis of Observed Chaotic Data by H.D.I Abarbanel, both by well known experts in the field.
The first of these is an absolute joy. It is one of the best written books on nonlinear systems that I have read for some time and is eminently accessible to just about anyone in a reasonably numerate discipline. It is organised in two sections: the first of these introduces ideas at their simplest level and gives recipes for implementing them. The second section then returns to many of these and treats them in greater theoretical detail. Everything is explained in a very clear and concise way and superbly illustrated by a variety of real data sets. This is one of the main strengths of the book, and something of which the authors are justifiably proud. In my opinion, its other great strength is its critical approach and its refusal to apply favourite techniques in an indiscriminate fashion. This has long been one of the most serious problems in the field and hopefully the appearance of this book will go a long way towards redressing the balance. Indeed, it must be pointed out that the book is not just about time series derived from chaotic systems, but rather about nonlinear time series more generally. It thus contains strong warnings about the dangers of applying algorithms which assume chaotic behaviour without first obtaining unambiguous evidence of chaos in the data. Such a critical and level-headed attitude is unfortunately surprisingly rare in the literature, and hence is extremely welcome.
Given such a target, Abarbanels Analysis of Observed Chaotic Data unfortunately suffers somewhat in comparison. It is a perfectly adequate book, and before the appearance of Kantz and Schreibers volume I would have been happy to recommend it. Now however, in my opinion, it is both less accessible to the uninitiated reader and less useful to the active user of nonlinear time series methods. This may be partly due to its ancestry as a Rev. Mod. Phys. review paper. This was obviously intended for a more advanced audience and for such readers it may still form a useful reference. However, it is somewhat less well organised, so that many ideas such as Liapunov exponents or fractal dimensions are mentioned very early without explanation, but only discussed several chapters later. Relatively, I also feel that it is less well written, so that some sections are not entirely easy to follow (e.g. the one dealing with Takens embedding theorem). However, this may perhaps be an unfair criticism, Kantz and Schreiber write so exceptionally well that they make many complex ideas appear straightforward. Furthermore, they do so without any sacrifice in technical accuracy. This is a truly superb achievement, and almost any other text will by contrast appear mediocre.
Abarbanel is also more careless with detail, so that for instance the fundamental paper of Sauer et al (T. Sauer, J.A. Yorke and M. Casdagli, 1991, Embedology, J. Stat. Phys., 65, 579-616) is referred to as Casdagli, Sauer and Yorke. Both books of course give more emphasis to those methods developed by their authors and their collaborators, but this seems more pronounced in Analysis of Observed Chaotic Data. Finally, and probably most significantly, Abarbanel seems less critical about the applicability of various algorithms to real data. Indeed, the whole book is based on the assumption that one knows that the time series that one is analysing originated in a low dimensional deterministic system. There is too little discussion of how one might test for this, or what one might do if this in fact is not the case. I fear this will tempt a non-expert reader into applying some of these techniques in inappropriate ways, and deriving unwarranted conclusions from them. Overall, I therefore prefer the approach taken by Kantz and Schreiber.
To conclude, I must mention that I shall resume teaching the time series course next spring. Frankly, I am inclined to simply hold up a copy of Nonlinear Time Series Analysis and tell my students to go away and read it, and then walk out of the lecture hall. Who needs a lecturer when a book this good is available?
UK Nonlinear News thanks Cambridge University Press for providing a review copy of Nonlinear Time Series Analysis.
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Last Updated: 7th August 1998.