Nonlinear and Nonstationary Signal Processing

Citation

Andrew, A.M. (2002), "Nonlinear and Nonstationary Signal Processing", Kybernetes, Vol. 31 No. 2. https://doi.org/10.1108/k.2002.06731bae.002

Nonlinear and Nonstationary Signal Processing

W.J. Fitzgerald, R.L. Smith, A.T. Walden and P.C. Young (Editors)Cambridge University Press,Cambridge2000ISBN 0-521-80044-7xii + 471 pp.hardback, £60.00

This is a collection of thirteen papers arising from a research "programme" held in the Isaac Newton Institute of Mathematical Sciences of the University of Cambridge from July to December 1998. The Institute, according to a note in the book, exists to stimulate research in all branches of the mathematical sciences, and the research programmes it runs each year bring together leading mathematical scientists from all over the world to exchange ideas through seminars, teaching and informal interaction. Each of the chapters of the book was either presented as a talk at one of the workshops of the programme, or written as a research paper by one of the long-stay members.

In the Introduction it is explained that this programme was motivated by the observation that the whole field of signal processing and time series analysis has by now moved far beyond its roots in the theory of linear stationary processes, but many of the new techniques to handle nonlinear and nonstationary processes have developed in individual areas of statistics, engineering or more specialised fields such as environmental science or mathematical finance, with limited interaction between different groups. The established theory of linear stationary processes is of course associated with Norbert Wiener and with the origins of Cybernetics.

As part of the programme activities, five open workshops were held, as well as numerous informal meetings. The topics of the five workshops were: (a) Bayesian statistics in signal processing, (b) environmental modelling, (c) the interaction between time series analysis and dynamical systems, (d) statistical methods in finance, and (e) data analysis with a particular emphasis on wavelet methods.

The proceedings are summarised in a useful four-page Introduction, in which it is noted that the particularly noteworthy achievements of the programme were new methodological developments and applications of wavelets, a wider appreciation of Bayesian methods, the interaction between nonlinear time series analysts and dynamical systems experts, and the development of new areas of application such as risk management in insurance and finance.

The idea of wavelets as an analytical tool dates hack to a paper by Dennis Gabor as early as 1954, a paper that aroused much interest in the context of Cybernetics, not least in further developments by Donald MacKay. The final chapter of the present book discusses a development called "wavestrapping", also described as adaptive wavelet-based bootstrapping, used to examine the statistical properties of time series from either long or short memory processes. In an earlier paper a set of approaches denoted by the term "multitaper" is described. These overcome limitations of wavelet and spectrogram approaches by the use of a Loève or dual-frequency transformation.

Some of the new methods reported under the heading of Bayesian inference have become feasible because of the availability of fast computers and sophisticated Monte Carlo simulation methods. Some intriguing display methods, for example one showing temperature distribution over a two- dimensional area as a three-dimensional perspective figure, appear in a paper on spatial statistics in environmental science.

The material reviewed in each chapter reflects the current "state of the art" and a fairly high level of mathematical sophistication is assumed. The density of equations is not particularly high, as mathematical texts go, and the discussion is lucidly presented. At the same time, a fair amount of background knowledge is assumed since clearly this is not a book for beginners. It is well produced, with numerous graphs and tables in the tidy style that is made possible by modern computer grapbics, and on a few pages there are illustrations in colour. In some of these, colours are associated with a numerical range so as to allow effective plotting of values in an extra dimension.

The book will certainly be welcomed by those wishing to understand and apply the latest techniques in this area.

Alex M. Andrew