Please ignore the bad appearance of some of the math symbols in the review.

**Foundations of Time Series Analysis and Prediction Theory.**
Mohsen POURAHMADI. New York: Wiley, 2001. ISBN 0-471-39434-3.
ix +414 pp. $89.95.

Analysis and modeling of time series data X, is a very important problem in many scientific fields. The 21st century will undoubtedly see the develop- ment of new methods of statistically fitting models to empirical time series. Statisticians desiring to practice time series analysis need a foundation in the elegant and applicable theories and methods of time series analysis that were developed in the 20th century. Statisticians are lucky to have Profes- sor Pourahmadi's textbook, which provides an excellent introduction to the remarkable developments during the 20th century in the theory of time series analysis.

Chapter 2, "Time Series Analysis: One Long Series," defines key concepts: stationary time series X, with constant mean ~.r and covariance y5 = cov[X,~5, X,~1 (X,.5, X,). using the notation of inner product in 1-lilbert space; transformation of nonstationary series to stationary and stationary to uncorrelated series; time series regression to estimate trend; spectral analy- sis; time domain (AR, MA) models; model-fitting strategies; forecasting; state space; statistical estimators; and prediction theory (Kolmogorov, Wiener).

Chapter 3, "Time Series Analysis: Many Short Series" provides a unique treatment of longitudinal data (a large number of short series of dependent variables measured over time). Measurement v~ on the ith unit at time t is modeled using y,, = x,13, ą e,,. One often assumes two- or three-parameter models of covariance matrices for e. This chapter presents statistical analysis techniques that are deeply rooted in time series analysis.

Chapter 4, "Stationary ARMA Models," presents linear prediction, recursive predictors, state-space representations that provide inference for Gaussian ARMA models by Kalman filter calculation of likelihood. Chapter, 5, "Stationary Processes," studies the shift operator B X = X_ , multivariate stationary processes, spectral representation, prediction, Wold decomposi- tion, and mixing conditions. Chapter 6, "Parametrizaton and Prediction," is an original development using linear regression of time-domain representation of stationary processes, including infinite-variance processes. Chapter 7, "Finite Prediction and Partial Correlations," introduces Cholesky decomposi- tion, the Durbin-Levinson algorithm, partial correlations, multiple and canon- ical correlations, convergence of predictors, and Szego orthogonal polynomi- als. Chapter 8. "Missing Values: Past and Future," is a unique introduction to interpolation, intervention analysis, and inverse correlations. Chapter 9, "Stationary Sequences in Hilbert Space," and Chapter 10, "Hardy Space," provide an excellent survey of the many roles of Hilbert spaces and Hardy spaces in time series analysis, including reproducing kernel Hilbert spaces and alternating projections.

Appendices discuss multivariate distributions and Bayesian forecasting. A unique feature of the book is that is explicitly calls our attention to "largely ignored results that deserve better recognition." I believe that this describes Pourahmadi's formula on page 69 expressing MA(o&l cocfficients bk in terms of the Fourier coefficients c5 of the log-spectral density. Pourahmadišs formula should be known and applied by all practicing time series analysts.

Emanuel PARZEN Texas A&M University