Customer Reviews

Spectral Analysis and Time Series. Volumes I and II in 1 book. (Probability and Mathematical Statistics)

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The author has assembled a wonderfully accessible study of time series analysis from the point of view of spectral theory. This book really bridges the gap between Brockwell & Davis' elementary text Introduction to Time Series and Forecasting and their advanced text Time Series: Theory and Methods.

The book is logically partitioned into two volumes: Volume I (Chapters 1-8) considers spectral methods for time series, while Volume II (Chapters 9-11) extends the results to multivariate time series.

Priestly tries to keep the prerequisites to a minimum, but the reader is well advised to do a little background preparation before jumping in to this book. For the required material in mathematical analysis of Fourier series, I recommend Rudin's Real and Complex Analysis. Although Priestly provide a brief introduction to probability theory, I'd recommend a more solid grounding, as can be found in Chung's A Course in Probability Theory. The elementary text by Brockwell & Davis Introduction to Time Series and Forecasting presents the needed material on time series analysis.

In Chapter 1, Priestly sets up the motivation for considering spectral analysis of stationary time series, and gives four practical reasons for the use of spectral methods.

The reader will find a brief, 70 page overview of probability theory in Chapter 2. If the terms don't look familiar on a quick scan of this chapter, you'll want to get more detail from Chung's text before proceeding with Priestly.

Chapter 3 introduces stochastic processes and time series. Stationary time series are defined, as is the auto-covariance and autocorrelation function. ARMA(p,q) models are introduced and some basic results are established about these models.

The core results from spectral analysis are given in Chapter 4. The two main results are the Wiener-Khintchine Theorem (characterized those functions which can be the autocorrelation function of a stationary process), and the Spectral Representation Theorem for Stationary Processes.

Chapter 5 gives a really nice treatment of ARMA(p,q) model specification and estimation. The author motivates the well-known conditional maximum likelihood techniques for estimating coefficients, and gives really insight into the development of methods of order estimation using the information criterion ala Akaike (i.e. AIC) and Schwartz.

The next section consists discuss spectral estimation and consists of Chapters 6, 7, and 8. Chapter 6 tackles the theoretical issues surrounding estimated the spectral density of a stationary process. The author does a good job explaining the shortcomings of the periodogram as an estimator, as well as the need for tapering or 'windowing'. Chapter 7 continues along this theme by giving empirical guidance for selecting windowing schemes. Chapter 8 discusses the thorny problem of posed by processes containing both a continuous and a discrete spectrum.

The last part of the book comprised Volume II and extends the results of the first volume to cover the case of multivariate time series. Applications considered in this volume include problems of filtering and prediction. In the last chapter of the book, Priestly presents some of his own research on "evolutionary spectra" which is an attempt to extend the analysis to non-stationary processes.

The book is written in monograph style; as such there are no formal exercises. However, the author gives lots of examples using real-world datasets. Working through the examples serves to reinforce the reading. The author states several theorems, but usually prefers to justify these results with a heuristic argument. On occasion, a formal proof is given, but there are no end-of-proof markers (e.g. QED). The reader must take care to determine where the proof ends and the discussion resumes.

I thought James D. Hamiltons book Time Serives Aanlysis was better. It was easier to understand and covered more material, including VAR models and State Space. Still this was and is an excellent book, and it goes into details about multivariate statistics that are not contained in Hamilton's book. I have the same complaint about this book as Hamilton's. Not enough examples. I compare these two books to those of Hosmer and Lemeshow's Applied Logistic Regression where there were nurmerous examples and problems to solve based on data they had provided.

Michael Quigley Director, Statistical Model and Data Mining Wells Fargo Bank

This book covers almost all possible aspects of spectral analysis of time series. The problem is that it is almost exclusively theoretical. It should not be used for learning spectral analysis but rather as a reference book. There are very few practical examples but when looking for a proof or an abstract presentation of a particular concept, this book should allow you to understand the theory that lies behind... However, a very good treatment of spectral analysis and very broad coverage of the subject...

The book covers a good deal of spectral analysis with all the details and results and applications. It starts with probability theory and random processes, definition of Fourier series, spectrums, estimation using spectral methods and more advanced topics. The book focuses on frequency domain methods, though in parts it discusses time-series models like AR, ARMA, ... but in a frequency domain context. The book is rather mathematical and urges the reader to have some math background or familiarity with system theory and filters. It discusses the results for both continuous and discrete time processes, though the emphasis is more on the continuous time. The discussions are done neatly but can be cumbersome or unnecessary for people who just want to use a method. It is more a book for those who want to get to the bottom of the methods with a deep understanding of concepts.

I found this book useful and rather complete and I am using that in a lot of my own projects.