Time series analysis and its applications. With R examples. 3rd revised ed.

*(English)*Zbl 1276.62054
Springer Texts in Statistics. New York, NY: Springer (ISBN 978-1-4419-7864-6/hbk; 978-1-4419-7865-3/ebook; 978-1-4614-2759-9/pbk). xi, 596 p. (2011).

[For the review of the 2nd ed. from 2006 see Zbl 1096.62088.]

This book reaches a balance between theory and examples, between theoretical aspects such as autocorrelations and cross correlations, parametric and nonparametric spectral estimation, and practical ones like to model global warming, how to better understand New York stock exchange or how to discriminate between waveforms generated by earthquakes or explosions. The book is organised in 7 chapters and 4 appendices.

The first chapter presents in detail the theoretical description of time series offering details for a good understanding of the nature of time series and reviewing the known and commonly used time series statistical models. Also measures of dependence such as autocorrelations and cross correlations are discussed in detail. To support the theoretical concepts, examples on stationary time series and estimation of correlations are presented. A description of vector valued and multidimensional series is also included. The second chapter covers time series regression and exploratory data analysis. Classical regression and smoothing are presented in the time series context. The chapter also contains a section with examples of exploratory data analysis varying from detecting global temperature to discovering signal in noise. These examples are used to introduce approaches such as scatter plots and periodogram analysis. The third chapter focuses on ARIMA models. It commences with autoregressive moving average models on which autocorrelations and partial correlations, forecasting and estimation are presented. Next, integrated models for non-stationary data are introduced, followed by an in-depth description or ARIMA models. The fourth covers spectral analysis and filtering. Analysis of critical behaviour and periodicity are followed by a detailed description of periodograms and discrete Fourier transforms. Next parametric and nonparametric spectral estimations are discussed. The second half of the chapter presents concepts of filtering, linear filters, signal extraction and optimum filtering. The chapter concludes with spectral analysis of multidimensional series. The fifth chapter introduces additional time domain topics. GARCH and threshold models are presented in detail together with long memory ARMA, fractional differencing and unit root testing. The chapter ends with concepts on regression with auto correlated errors and lagged regression. The sixth chapter presents state- space models with details on filtering, smoothing, forecasting and maximum likelihood estimation. It also presents methods to handle missing data and errors. The ARMAX models and multivariate regression with autocorrelated errors are presented in detail. The chapter continues with a description of bootstrapping state space models, dynamic linear models with switching and nonlinear and non-normal state-space models using Monte Carlo methods. The seventh chapter discusses statistical methods in the frequency domain. First spectral matrices and likelihood functions are introduced. These are followed by regression for jointly stationary series with deterministic input or with random coefficients. The chapter concludes with discrimination and cluster analyses and principal components and factor analyses.

The first appendix covers large sample theory and includes a detailed description of convergence models and central limit theorems. In this appendix the mean and autocorrelation are also discussed and linked to the chapter 1 section on measures of dependence. The second appendix presents time domain theory with notions on Hilbert spaces and the projection theorem and on causal conditions for ARMA models. It also includes in detail proofs for large sample distributions for the \(AR(p)\) conditional least squares estimators and a brief overview of the world decomposition theorem. Appendix C presents notions of spectral domain theory, focussing on the spectral representation theorem, on the large sample distribution of the DFT and smoothed periodograms and complex multivariate normal distributions. The last appendix offers a brief introduction to the R programming environment and commands. It concludes with time series functions used throughout the book.

Due to its balanced content of theory, examples and problems (which can be found at the end of each chapter), the book is a valuable resource for students at undergraduate and graduate levels and researchers. The R code for almost all the numerical examples, and the appendices with tutorials containing basic R and R time series commands, are helpful for a better understanding of the theoretical concepts by bringing the theory into a more practical context.

This book reaches a balance between theory and examples, between theoretical aspects such as autocorrelations and cross correlations, parametric and nonparametric spectral estimation, and practical ones like to model global warming, how to better understand New York stock exchange or how to discriminate between waveforms generated by earthquakes or explosions. The book is organised in 7 chapters and 4 appendices.

The first chapter presents in detail the theoretical description of time series offering details for a good understanding of the nature of time series and reviewing the known and commonly used time series statistical models. Also measures of dependence such as autocorrelations and cross correlations are discussed in detail. To support the theoretical concepts, examples on stationary time series and estimation of correlations are presented. A description of vector valued and multidimensional series is also included. The second chapter covers time series regression and exploratory data analysis. Classical regression and smoothing are presented in the time series context. The chapter also contains a section with examples of exploratory data analysis varying from detecting global temperature to discovering signal in noise. These examples are used to introduce approaches such as scatter plots and periodogram analysis. The third chapter focuses on ARIMA models. It commences with autoregressive moving average models on which autocorrelations and partial correlations, forecasting and estimation are presented. Next, integrated models for non-stationary data are introduced, followed by an in-depth description or ARIMA models. The fourth covers spectral analysis and filtering. Analysis of critical behaviour and periodicity are followed by a detailed description of periodograms and discrete Fourier transforms. Next parametric and nonparametric spectral estimations are discussed. The second half of the chapter presents concepts of filtering, linear filters, signal extraction and optimum filtering. The chapter concludes with spectral analysis of multidimensional series. The fifth chapter introduces additional time domain topics. GARCH and threshold models are presented in detail together with long memory ARMA, fractional differencing and unit root testing. The chapter ends with concepts on regression with auto correlated errors and lagged regression. The sixth chapter presents state- space models with details on filtering, smoothing, forecasting and maximum likelihood estimation. It also presents methods to handle missing data and errors. The ARMAX models and multivariate regression with autocorrelated errors are presented in detail. The chapter continues with a description of bootstrapping state space models, dynamic linear models with switching and nonlinear and non-normal state-space models using Monte Carlo methods. The seventh chapter discusses statistical methods in the frequency domain. First spectral matrices and likelihood functions are introduced. These are followed by regression for jointly stationary series with deterministic input or with random coefficients. The chapter concludes with discrimination and cluster analyses and principal components and factor analyses.

The first appendix covers large sample theory and includes a detailed description of convergence models and central limit theorems. In this appendix the mean and autocorrelation are also discussed and linked to the chapter 1 section on measures of dependence. The second appendix presents time domain theory with notions on Hilbert spaces and the projection theorem and on causal conditions for ARMA models. It also includes in detail proofs for large sample distributions for the \(AR(p)\) conditional least squares estimators and a brief overview of the world decomposition theorem. Appendix C presents notions of spectral domain theory, focussing on the spectral representation theorem, on the large sample distribution of the DFT and smoothed periodograms and complex multivariate normal distributions. The last appendix offers a brief introduction to the R programming environment and commands. It concludes with time series functions used throughout the book.

Due to its balanced content of theory, examples and problems (which can be found at the end of each chapter), the book is a valuable resource for students at undergraduate and graduate levels and researchers. The R code for almost all the numerical examples, and the appendices with tutorials containing basic R and R time series commands, are helpful for a better understanding of the theoretical concepts by bringing the theory into a more practical context.

Reviewer: Irina Ioana Mohorianu (Norwich)

##### MSC:

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

62-04 | Software, source code, etc. for problems pertaining to statistics |

62M20 | Inference from stochastic processes and prediction |

62-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics |

62-07 | Data analysis (statistics) (MSC2010) |

62Pxx | Applications of statistics |

65C05 | Monte Carlo methods |