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Stochastic modeling of scientific data. (English) Zbl 0862.60034

London: Chapman & Hall (ISBN 978-0-412-99281-0/hbk; 978-0-367-44900-1/pbk; 978-0-203-73825-2/ebook). xii, 372 p. (1995).
When it comes to introducing Markov chains, everyone talks about the weather but nobody does anything about getting real data. In this book, though, we get not only the pattern of rainfall in Snoqualmie Falls, Washington, but wind directions in South Africa, and interarrival times of cyclones in the Bay of Bengal. The object is to provide an introduction to stochastic processes suitable to those who, while not necessarily shy of mathematics, are primarily interested in problems with the flavor of real life. The proofs here, even when they require abstract concepts, are anchored in the concrete, and grow more sparse as the book progresses through more technically demanding subjects. Still, even hard-bitten mathematical probabilists may find new insights in this insistently realistic approach.
Chapter 1 is a brief introduction which introduces the conceptual underpinnings of stochastic modelling in the sciences. Chapter 2 covers discrete-time Markov chains, including Bienaymé-Galton-Watson processes, hidden Markov models, and Markov-chain Monte Carlo methods. Chapter 3 extends the above to continuous time. Chapter 4 provides a short account of Markov random fields, emphasizing the Ising model and applications to image processing. Chapter 5 discusses point processes. Traffic intensities are as usual a primary example, but the hard data, this time from Swedish country roads, change the emphasis significantly: parameter estimation and mixed processes come to the fore. Chapter 6 concludes with diffusions, especially Brownian motion and Ornstein-Uhlenbeck processes, and an elementary introduction to stochastic integrals. The discussion of physical Brownian motion, and its relation to the mathematical model, is unusual and enlightening.
The examples range widely, over meteorology, geology, economics, physics, biology, and a good deal more. Almost all are genuine applications which at least seemed reasonable to the experts in the relevant field, from whose journals they have been culled. The general statistical method is to fit models to data via maximum-likelihood estimators; this is a reasonable approach, though some might object to the lack of any mention of the limitations of this method, or of other possibilities. Each chapter concludes with an assortment of exercises, divided into theoretical, computational, and data exercises. Data sets for these last are available by anonymous ftp.

MSC:

60G35 Signal detection and filtering (aspects of stochastic processes)
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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