Periodically correlated random sequences : spectral theory and practice

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

The use of periodically correlated (or cyclostationary)processes has become increasingly popular in a range of researchareas such as meteorology, climate, communications, economics, andmachine diagnostics. Periodically Correlated Random Sequencespresents the main ideas of these processes through the use of basicdefinitions along with motivating, insightful, and illustrativeexamples. Extensive coverage of key concepts is provided, includingsecond-order theory, Hilbert spaces, Fourier theory, and thespectral theory of harmonizable sequences. The authors also providea paradigm for nonparametric time series analysis including testsfor the presence of PC structures.

Features of the book include:
* An emphasis on the link between the spectral theory of unitaryoperators and the correlation structure of PC sequences
* A discussion of the issues relating to nonparametric time seriesanalysis for PC sequences, including estimation of the mean,correlation, and spectrum
* A balanced blend of historical background with modernapplication-specific references to periodically correlatedprocesses
* An accompanying Web site that features additional exercises aswell as data sets and programs written in MATLAB® forperforming time series analysis on data that may have a PCstructure

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

Author Notes

Harry L. Hurd is Adjunct Professor of Statistics at The University of North Carolina at Chapel Hill.

Preface	p. xiii
Acknowledgments	p. xv
Glossary	p. xvii
1 Introduction	p. 1
1.1 Summary	p. 6
1.2 Historical Notes	p. 14
Problems and Supplements	p. 16
2 Examples, Models, and Simulations	p. 19
2.1 Examples and Models	p. 20
2.1.1 Random Periodic Sequences	p. 20
2.1.2 Sums of Periodic and Stationary Sequences	p. 21
2.1.3 Products of Scalar Periodic and Stationary Sequences	p. 21
2.1.4 Time Scale Modulation of Stationary Sequences	p. 22
2.1.5 Pulse Amplitude Modulation	p. 23
2.1.6 A More General Example	p. 24
2.1.7 Periodic Autoregressive Models	p. 25
2.1.8 Periodic Moving Average Models	p. 27
2.1.9 Periodically Perturbed Dynamical Systems	p. 28
2.2 Simulations	p. 29
2.2.1 Sums of Periodic and Stationary Sequences	p. 29
2.2.2 Products of Scalar Periodic and Stationary Sequences	p. 30
2.2.3 Time Scale Modulation of Stationary Sequences	p. 32
2.2.4 Pulse Amplitude Modulation	p. 33
2.2.5 Periodically Perturbed Logistic Maps	p. 35
2.2.6 Periodic Autoregressive Models	p. 38
2.2.7 Periodic Moving Average Models	p. 40
Problems and Supplements	p. 42
3 Review of Hilbert Spaces	p. 45
3.1 Vector Spaces	p. 45
3.2 Inner Product Spaces	p. 47
3.3 Hilbert Spaces	p. 49
3.4 Operators	p. 51
3.5 Projection Operators	p. 53
3.6 Spectral Theory of Unitary Operators	p. 60
3.6.1 Spectral Measures	p. 60
3.6.2 Spectral Integrals	p. 61
3.6.3 Spectral Theorems	p. 64
Problems and Supplements	p. 65
4 Stationary Random Sequences	p. 67
4.1 Univariate Spectral Theory	p. 68
4.1.1 Unitary Shift	p. 68
4.1.2 Spectral Representation	p. 70
4.1.3 Mean Ergodic Theorem	p. 72
4.1.4 Spectral Domain	p. 74
4.2 Univariate Prediction Theory	p. 75
4.2.1 Infinite Past, Regularity and Singularity	p. 75
4.2.2 Wold Decomposition	p. 76
4.2.3 Innovation Subspaces	p. 78
4.2.4 Spectral Theory and Prediction	p. 84
4.2.5 Finite Past Prediction	p. 91
4.3 Multivariate Spectral Theory	p. 99
4.3.1 Unitary Shift	p. 100
4.3.2 Spectral Representation	p. 101
4.3.3 Mean Ergodic Theorem	p. 102
4.3.4 Spectral Domain	p. 102
4.4 Multivariate Prediction Theory	p. 107
4.4.1 Infinite Past, Regularity and Singularity	p. 107
4.4.2 Wold Decomposition	p. 108
4.4.3 Innovations and Rank	p. 109
4.4.4 Regular Processes	p. 116
4.4.5 Infinite Past Prediction	p. 119
4.4.6 Spectral Theory and Rank	p. 121
4.4.7 Spectral Theory and Prediction	p. 123
4.4.8 Finite Past Prediction	p. 125
Problems and Supplements	p. 129
5 Harmonizable Sequences	p. 133
5.1 Vector Measure Integration	p. 134
5.2 Harmonizable Sequences	p. 141
5.3 Limit of Ergodic Average	p. 145
5.4 Linear Time Invariant Filters	p. 146
Problems and Supplements	p. 149
6 Fourier Theory of the Covariance	p. 151
6.1 Fourier Series Representation of the Covariance	p. 152
6.2 Harmonizability of PC Sequences	p. 160
6.3 Some Properties of B[subscript k]([tau]), F[subscript k], and F	p. 168
6.4 Covariance and Spectra for Specific Cases	p. 170
6.4.1 PC White Noise	p. 170
6.4.2 Products of Scalar Periodic and Stationary Sequences	p. 171
6.5 Asymptotic Stationarity	p. 172
6.6 Lebesgue Decomposition of F	p. 173
6.7 The Spectrum of m[subscript t]	p. 174
6.8 Effects of Common Operations on PC Sequences	p. 176
6.8.1 Linear Time Invariant Filtering	p. 176
6.8.2 Differencing	p. 181
6.8.3 Random Shifts	p. 182
6.8.4 Sampling	p. 187
6.8.5 Bandshifting	p. 191
6.8.6 Periodically Time Varying (PTV) Filters	p. 192
Problems and Supplements	p. 194
7 Representations of PC Sequences	p. 199
7.1 The Unitary Operator of a PC Sequence	p. 200
7.2 Representations Based on the Unitary Operator	p. 201
7.2.1 Gladyshev Representation	p. 201
7.2.2 Another Representation of Gladyshev Type	p. 203
7.2.3 Time-Dependent Spectral Representation	p. 203
7.2.4 Harmonizability Again	p. 205
7.2.5 Representation Based on Principal Components	p. 207
7.3 Mean Ergodic Theorem	p. 210
7.4 PC Sequences as Projections of Stationary Sequences	p. 212
Problems and Supplements	p. 213
8 Prediction of PC Sequences	p. 215
8.1 Wold Decomposition	p. 218
8.2 Innovations	p. 220
8.3 Periodic Autoregressions of Order 1	p. 226
8.4 Spectral Density of Regular PC Sequences	p. 229
8.4.1 Spectral Densities for PAR(1)	p. 231
8.5 Least Mean-Square Prediction	p. 235
8.5.1 Prediction Based on Infinite Past	p. 235
8.5.2 Prediction for a PAR(1) Sequence	p. 236
8.5.3 Finite Past Prediction	p. 237
Problems and Supplements	p. 246
9 Estimation of Mean and Covariance	p. 249
9.1 Estimation of m[subscript t]: Theory	p. 250
9.2 Estimation of m[subscript t]: Practice	p. 261
9.2.1 Computation of [Characters not reproducible], [subscript N]	p. 262
9.2.2 Computation of [Characters not reproducible], [subscript N]	p. 263
9.3 Estimation of R(t + [tau], t): Theory	p. 264
9.3.1 Estimation of R(t + [tau], t)	p. 265
9.3.2 Estimation of B[subscript k]([tau])	p. 272
9.4 Estimation of R(t + [tau], t): Practice	p. 282
9.4.1 Computation of [Characters not reproducible] (t + [tau], t)	p. 283
9.4.2 Computation of [Characters not reproducible], [subscript NT]([tau])	p. 288
Problems and Supplements	p. 292
10 Spectral Estimation	p. 297
10.1 The Shifted Periodogram	p. 299
10.2 Consistent Estimators	p. 302
10.3 Asymptotic Normality	p. 306
10.4 Spectral Coherence	p. 308
10.4.1 Spectral Coherence for Known T	p. 308
10.4.2 Spectral Coherence for Unknown T	p. 310
10.5 Spectral Estimation: Practice	p. 312
10.5.1 Confidence Intervals	p. 312
10.5.2 Examples	p. 313
10.6 Effects of Discrete Spectral Components	p. 322
10.6.1 Removal of the Periodic Mean	p. 323
10.6.2 Testing for Additive Discrete Spectral Components	p. 323
10.6.3 Removal of Detected Components	p. 327
Problems and Supplements	p. 328
11 A Paradigm for Nonparametric Analysis of PC Time Series	p. 331
11.1 The Period T is Known	p. 332
11.2 The Period T is Unknown	p. 334
References	p. 337
Index	p. 351

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Table of Contents