» Multiscale Analysis of Complex Time Series: Integration of Chaos and Random Fractal Theory, and Beyond

Multiscale Analysis of Complex Time Series: Integration of Chaos and Random Fractal Theory, and Beyond Reviews

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Summary: Multiscale Analysis of Complex Time Series
Comment: The need to find order in data is the scientist's corollary to the more catholic need to find order in the universe which scientists sample. In the 1970s Mandelbrot drew attention to a kind of geometric order, which he called fractal, that was more complex and often more useful than the Euclidean geometry that had dominated mathematics for more than two millennia. A side effect of this fractal geometry was a set of insights into analyzing complex time series. Part of Mandelbrot's charm is that his fractal mathematics are as powerful in dealing with stock prices and water accumulation behind dams as they were in describing the shape of a coastline or a mountain range. These insights dovetailed beautifully and synergistically with developments in nonlinear dynamics, leading to what is now known as complexity theory. Many good books on fractals and complexity theory have followed, but until recently it was difficult to find books that increased insight on understanding complex time series while never losing sight of the bigger picture: the complex universe that spawned the time series. Gao et al.'s new book "Multiscale Analysis of Complex Time Series" meets this standard.

One disclosure: Although I've never met Jianbo Gao in person, I once collaborated by email with Gao on a study of 1/f^b processes in visual perception. 1/f^b phenomena are becoming important in vision and are found in many different circumstances, ranging from natural images and visual textures (a focus of my own work) to time series analysis of multistable perception. I was therefore not surprised by the special focus in this book on 1/f^b processes. One thing that distinguishes this book from several others on complex time series is the care in which the authors weave the 1/f story from so many different theoretical and experimental strands. The only comparable treatment I can think of is Schroeder's "Fractals, Chaos and Power Laws", which is an easier and more delightful read, but much less useful for a working scientist trying to get his hands around real world data. If I'm enthusiastic about the book, it is perhaps for selfish reasons; reading it gave me a deeper understanding of some of the specialized problems I'm considering.

To develop understanding of multiscale 1/f-like time series, Gao et al. review fractals, chaos, probability theory, stochastic systems and the use of wavelets to probe at many different scales. They then delve more deeply into Levy motions, multifractals, multiplicative processes and bifurcation theory. The book is suitable as a textbook, with problems sets after each chapter, and datasets and software available on the Wiley website, but I suspect it will find more use as a monograph, for specialists pondering meaning in the multi-scaled datasets that their laboratories have just produced. The authors show sensitivity to a beginner's sensibilities, with "warm-up" exercises provided before plunging into more difficult material.

One topic I would like to have seen treated in depth in the book is stochastic resonance - a qualitative change in the behavior of a system when driven by noise. Stochastic resonance is a natural fit to this material: some kinds of SR are best driven by 1/f noise and some kinds of SR create 1/f behaviors. I suspect that SR's relative immaturity as a discipline makes it harder to treat authoritatively. This is a problem that time should solve, perhaps in time for a second edition.

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