Uniquely combining theory, application, and computing, this book explores the spectral approach to time series analysis

The use of periodically correlated (or cyclostationary) processes has become increasingly popular in a range of research areas such as meteorology, climate, communications, economics, and machine diagnostics. Periodically Correlated Random Sequences presents the main ideas of these processes through the use of basic definitions along with motivating, insightful, and illustrative examples. Extensive coverage of key concepts is provided, including second-order theory, Hilbert spaces, Fourier theory, and the spectral theory of harmonizable sequences. The authors also provide a paradigm for nonparametric time series analysis including tests for the presence of PC structures.

Features of the book include:

• An emphasis on the link between the spectral theory of unitary operators and the correlation structure of PC sequences

• A discussion of the issues relating to nonparametric time series analysis for PC sequences, including estimation of the mean, correlation, and spectrum

• A balanced blend of historical background with modern application-specific references to periodically correlated processes

• An accompanying Web site that features additional exercises as well as data sets and programs written in MATLAB(R) for performing time series analysis on data that may have a PC structure

Periodically Correlated Random Sequences is an ideal text on time series analysis for graduate-level statistics and engineering students who have previous experience in second-order stochastic processes (Hilbert space), vector spaces, random processes, and probability. This book also serves as a valuable reference for research statisticians and practitioners in areas of probability and statistics such as time series analysis, stochastic processes, and prediction theory.

Harry L. Hurd, PhD, is Adjunct Professor of Statistics at TheUniversity of North Carolina at Chapel Hill. He is the founder ofHurd Associates, Inc., a research and development firmconcentrating in the areas of signal processing and stochasticprocesses. Dr. Hurd has published extensively on the topics ofnonstationary random processes, periodically correlated processes,and nonparametric time series.

Abolghassem Miamee, PhD, is Professor of Mathematics at HamptonUniversity in Virginia. His research interests include stochasticprocesses, time series analysis, and harmonic and functionalanalysis.

Preface xiii

Acknowledgments xv

Glossary xvii

1 Introduction

2 Examples, Models, and Simulations 19

3 Review of Hilbert Spaces 45

4 Stationary Random Sequences 67

5 Harmonizable Sequence 133

6 Fourier Theory of the Covariance 151

7 Representations of PC Sequences 199

8 Prediction of Sequences 215

9 Estimation of Mean and Covariance 249

10 Spectral Estimation 297

11 A Paradigm for Nonparametric Analysis of PC Time Series 331

References 337

Index 351