**and**

Some 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 realization 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 characterization of chaotic signals using invariants such as fractal dimensions or Liapunov exponents. Subsequently, attention turned to the possibility of predicting the future 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 realize 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 synchronization.

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:

- when will we learn how to predict the stock-market?
- can you suggest a good textbook for this course?

My answer to the first is
unprintable, and that to the second was a shrug of the shoulders:
despite the popularity of subject there was a complete lack of
suitable texts. This gap was at least partly filled a few years ago
by the excellent reprint collection *Coping with Chaos*, edited
by E. Ott, T. Sauer and J.A. Yorke. This consists of a well chosen
and well organized 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 contain 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 should be eminently accessible to just about anyone in a reasonably numerate discipline. It is organized 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,
Abarbanel’s *Analysis of Observed Chaotic Data*
unfortunately suffers somewhat in comparison. It is a perfectly
adequate book, and before the appearance of Kantz and
Schreiber’s 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 organized, 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.
This may I fear 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, let me just mention
that I shall resume teaching my 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?

Jaroslav Stark,

Wednesday, July 29, 1998.

*UK Nonlinear News* would like
to thank Cambridge University Press for providing a copy of
*Nonlinear Time Series Analysis* for review.