Spectral analysis and time series. Volume 1: Univariate series. Volume 2: Multivariate series, prediction and control.

*(English)*Zbl 0537.62075
Probability and Mathematical Statistics. London etc.: Academic Press. A Subsidiary of Harcourt Brace Jovanovich, Publishers. XVIII, 890, AIX, RXXI, IXV p. $ 119.50; $ 55.00 (1981).

This bi-volume work is written by one of the leading experts: the author has contributed constantly of over twenty years to theory and practice of spectral analysis and time series. For that reason and by the size of about 900 pages the reader will get a distinguished treatment of that topic. The book is written for post-graduate mathematicians and statisticians specializing in time series analysis and research workers in applied fields such as physics, engineering and economics. Therefore much space is devoted to discussion of essential ideas. These are done in such an excellent way, that the experienced reader gets new insights, too.

For the most part, the style is of the applied mathematician. Certain basic mathematical results are presented precisely, especially those which are crucial for a proper understanding. Altogether, the text can be understood by readers who have basic knowledge of statistical inference when they skip over the developments using more advanced mathematics. On the other hand, the author’s experience leads to much practical advice throughout. So theory and practice balance quite right.

Exercises are not included. But use of computers is essential in modern applied time series analysis. Therefore exercises could be theoretical only and would not match the aim of the book.

The first volume is devoted to univariate series and its major content is concentrated on spectral-analysis of such series. Chapter 1 gives an introduction to spectral analysis. It surveys the whole work in an intuitive manner. Chapter 2 gives the basics of statistical distribution theory on 75 pages. Together with the first part of chapter 5 - basic ideas of statistical inference - it makes the book self-contained.

Chapter 3 is an introduction into the theory of stationary random processes in the time domain. The standard concepts are introduced: autocovariance - and autocorrelation function, white noise, normal-, AR-, MA-, and ARMA-processes. Also stochastic limiting operations are explained and standard continuous parameter models - continuous ARMA- processes and filtered Poisson-processes - are discussed.

Chapter 4 is on the foundations of spectral analysis. The mathematical basis of Fourier-series and -integrals is given before spectral theory of stationary processes is discussed. The discussion includes: relationship between spectral density and autocovariance function, decomposition of the integrated spectrum, spectral representation, linear transformations, and filters.

Chapter 5 gives a survey of estimation in the time domain on about 90 pages. However, the survey is really complete and contains estimation of parameters in ARMA-models. The completeness may be seen by 33 references on randomly selected three pages. For determination of the order of an ARMA-model the author recommends Akaike’s AIC-criterion. But in the mean- while there is some evidence in favor of Akaike’s BIC-criterion, which is also mentioned in the text.

Chapters 6, 7 and 8 deal with the main topic of the book, statistical analysis of time series in the frequency domain. They give an encyclopedic but very readable treatment. Especially chapter 7 on spectral analysis in practice gives a unique exposition. Also it contains all besides computer software and series to arm the reader to do spectral analysis. Chapter 6 is on estimation in the frequency domain and chapter 8 on analysis of processes with mixed spectra.

The first chapter of the second volume is on multivariate and multidimensional processes, the latter are treated at the end of the chapter. At the beginning, the bivariate case is used to transpose and generalize concepts from univariate to multivariate processes. Besides correlation and spectra it contains multivariate AR-, MA-, ARMA- and transferfunction models.

Chapter 10 (prediction, filtering, and control) gives a carefully written account of the Kolmogorov approach, the Wiener approach, and the Box- Jenkins approach to forecasting. The latter part includes seasonal ARIMA- models and exponentially wheighted MA-predictors (”exponential smoothing”). The state space approach and Kalman filtering is discussed, too. This needs multivariate processes and may be one reason for arranging the prediction problem after multivariate processes. The other prediction approaches are given mainly for univariate series.

Chapter 11 (non-stationarity and non-linearity) contains the theory of evolutionary spectra which is in great parts developed by the author himself. Also some non-linear models are introduced. These are now special cases of the author’s recent state-dependent non-linear models [J. Time Ser. Anal. 1, 47-71 (1980; Zbl 0496.62076)]. At last it should be mentioned, that a cheaper paperback edition is now available.

For the most part, the style is of the applied mathematician. Certain basic mathematical results are presented precisely, especially those which are crucial for a proper understanding. Altogether, the text can be understood by readers who have basic knowledge of statistical inference when they skip over the developments using more advanced mathematics. On the other hand, the author’s experience leads to much practical advice throughout. So theory and practice balance quite right.

Exercises are not included. But use of computers is essential in modern applied time series analysis. Therefore exercises could be theoretical only and would not match the aim of the book.

The first volume is devoted to univariate series and its major content is concentrated on spectral-analysis of such series. Chapter 1 gives an introduction to spectral analysis. It surveys the whole work in an intuitive manner. Chapter 2 gives the basics of statistical distribution theory on 75 pages. Together with the first part of chapter 5 - basic ideas of statistical inference - it makes the book self-contained.

Chapter 3 is an introduction into the theory of stationary random processes in the time domain. The standard concepts are introduced: autocovariance - and autocorrelation function, white noise, normal-, AR-, MA-, and ARMA-processes. Also stochastic limiting operations are explained and standard continuous parameter models - continuous ARMA- processes and filtered Poisson-processes - are discussed.

Chapter 4 is on the foundations of spectral analysis. The mathematical basis of Fourier-series and -integrals is given before spectral theory of stationary processes is discussed. The discussion includes: relationship between spectral density and autocovariance function, decomposition of the integrated spectrum, spectral representation, linear transformations, and filters.

Chapter 5 gives a survey of estimation in the time domain on about 90 pages. However, the survey is really complete and contains estimation of parameters in ARMA-models. The completeness may be seen by 33 references on randomly selected three pages. For determination of the order of an ARMA-model the author recommends Akaike’s AIC-criterion. But in the mean- while there is some evidence in favor of Akaike’s BIC-criterion, which is also mentioned in the text.

Chapters 6, 7 and 8 deal with the main topic of the book, statistical analysis of time series in the frequency domain. They give an encyclopedic but very readable treatment. Especially chapter 7 on spectral analysis in practice gives a unique exposition. Also it contains all besides computer software and series to arm the reader to do spectral analysis. Chapter 6 is on estimation in the frequency domain and chapter 8 on analysis of processes with mixed spectra.

The first chapter of the second volume is on multivariate and multidimensional processes, the latter are treated at the end of the chapter. At the beginning, the bivariate case is used to transpose and generalize concepts from univariate to multivariate processes. Besides correlation and spectra it contains multivariate AR-, MA-, ARMA- and transferfunction models.

Chapter 10 (prediction, filtering, and control) gives a carefully written account of the Kolmogorov approach, the Wiener approach, and the Box- Jenkins approach to forecasting. The latter part includes seasonal ARIMA- models and exponentially wheighted MA-predictors (”exponential smoothing”). The state space approach and Kalman filtering is discussed, too. This needs multivariate processes and may be one reason for arranging the prediction problem after multivariate processes. The other prediction approaches are given mainly for univariate series.

Chapter 11 (non-stationarity and non-linearity) contains the theory of evolutionary spectra which is in great parts developed by the author himself. Also some non-linear models are introduced. These are now special cases of the author’s recent state-dependent non-linear models [J. Time Ser. Anal. 1, 47-71 (1980; Zbl 0496.62076)]. At last it should be mentioned, that a cheaper paperback edition is now available.

Reviewer: R.Schlittgen

##### MSC:

62M15 | Inference from stochastic processes and spectral analysis |

62M20 | Inference from stochastic processes and prediction |

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |