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:

- 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 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, 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
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?

- A listing of books reviewed in
`UK Nonlinear News`is available at:`http://www.amsta.leeds.ac.uk/Applied/news.dir/uknonl-books.html`. - A listing of Publisher's Announcements distributed through
`UK Nonlinear News`is available at`http://www.amsta.leeds.ac.uk/Applied/news.dir/publishers.html`.

`UK Nonlinear News` thanks
Cambridge
University Press for providing a review copy of
`Nonlinear Time Series Analysis`.

<< Move to `UK Nonlinear News` Issue Thirteen Index Page
(August 1998).

Last Updated: 7th August 1998.

`uknonl@amsta.leeds.ac.uk`.