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Reviewer: Luminita State

Time series analysis is one of the most exciting areas of mathematical statistics, and there is a great deal of literature on it. The main objective of time series analysis is to develop mathematical models that provide plausible descriptions for sample data, yielding suitable statistical settings to explain the random fluctuations over time of the sample data. The main types of time series analysis are frequency domain analysis and time domain analysis. The frequency domain can be viewed as a regression on independent variables that isolate the frequencies of the cyclic behavior, the regression variables being cosines and sines evaluated at known frequencies and times. This is to say that the frequency domain approach assumes that the characteristics of interest in time series analyses relate to periodic or sinusoidal variations found in most data. The time domain approach is based of the assumption that correlation between adjacent points in time is best explained in terms of a dependence of the current value on past values, and, consequently, it focuses on modeling some future value of a time series as a parametric function of the current and past values. The content of the book is organized into five chapters. The first chapter is focused on presenting the basics of time series analysis and several related mathematical tools. The various examples supplied here are useful, and they provide substantial help in understanding the concepts. Chapter 2 shows several regression techniques for time series that are related to classical ordinary and weighted or correlated least squares. A detailed presentation is given of the autoregressive (AR) and autoregressive moving average (ARMA), autoregressive integrated moving average (ARIMA), and the Box-Jenkins method for identifying a plausible ARIMA model. Techniques for parameter estimation and forecasting are also described. This chapter also presents the long memory ARMA, threshold autoregressive models, regression with ARMA errors, and an extension of the Box-Jenkins method for predicting a single output from a collection of possible input series. Chapter 3 is devoted to spectral analysis and filtering. In this chapter, it is pointed out that the concept of regularity of a series can be best expressed in terms of periodic variations of the underlying phenomenon that produced the series. Expressed in these terms, the distinction between a time domain approach and a frequency domain approach becomes one between regression on the past, as favored in time domain approaches, or regressions on periodic sines and cosines, as embodied in the frequency domain approaches. Chapter 4 supplies a detailed presentation of the dynamic linear model (DLM) based on multivariate time series regression techniques. The first sections of chapter 4 give a detailed presentation of the properties of the discrete Fourier transform (DFT) of a multivariate time series, and various approximations of the likelihood function based on the large-sample properties and the properties of the complex multivariate normal distribution enabling the extension of the classical techniques to the case of multivariate time series. The final chapter of the book is focused on statistical methods in the frequency domain. It covers the extension of frequency domain methodology to multivariate discrimination and clustering, the analysis of the structure of the spectral matrix, and the use of the principal component technique for decomposing the spectral matrix into a smaller subset of component factors that explain decreasing amounts of power. Since analyzing time series is a rather difficult subject, it is worth emphasizing that one important merit of this book is that it provides a comprehensible treatment that considers both theory and practice. Enough theory is given to introduce the concepts of time series analysis and make the book mathematically attractive. In addition, various practical examples are considered so as to help the reader tackle the analysis of real data. The book can be used as a text for an undergraduate or postgraduate course in time series analysis, or it can be used by researchers for self-study. The book assumes some knowledge of basic probability theory and elementary statistical inference. Online Computing Reviews Service