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Spectral Analysis and Time Series. Volume 1: Univariate Series. (Probability and Mathematical Statistics): M. B. Priestley

A principal feature of this book is the substantial care and attention devoted to explaining the basic ideas of the subject. Whenever a new theoretical concept is introduced it is carefully explained by reference to practical examples drawn mainly from the physical sciences. Subjects covered include: spectral analysis which is closely intertwined with the "time domain" approach, elementary notions of Hilbert Space Theory, basic probability theory, and practical analysis of time series data. The inclusion of material on "kalman filtering", state-space filtering", "non-linear models" and continuous time" models completes the impressive list of unique and detailed features which will give this book a prominent position among related literature. The first sectionVolume 1deals with single (univariate) series, while the secondVolume 2treats the analysis of several (multivariate) series and the problems of prediction, forecasting and control.

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.

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