# Periodically Correlated Random Sequences: Spectral Theory and Practice

# Periodically Correlated Random Sequences: Spectral Theory and Practice

ISBN: 978-0-470-18282-6 November 2007 384 Pages

**E-Book**

$124.99

## Description

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

Acknowledgments.

Glossary.

**1. Introduction.**

1.1 Summary.

1.2 Historical Notes.

Problems.

**2. Examples, Models and Simulations.**

2.1 Examples and Models.

2.1.1 Random Periodic Sequences.

2.1.2 Sums of Periodic and Stationary Sequences.

2.1.3 Products of Scalar Periodic and Stationary Sequences.

2.1.4 Time Scale Modulation of Stationary Sequences.

2.1.5 Pulse Amplitude Modulation.

2.1.6 A More General Example.

2.1.7 Periodic Autoregressive Models.

2.1.8 Periodic Moving Average Models.

2.1.9 Periodically Perturbed Dynamical Systems.

2.2 Simulations.

2.2.1 Sums of Periodic and Stationary Sequences.

2.2.2 Products of Scalar Periodic and Stationary Sequences.

2.2.3 Time Scale Modulation of Stationary Sequences.

2.2.4 Pulse Amplitude Modulation.

2.2.5 Periodically Perturbed Logistic Maps.

2.2.6 Periodic Autoregressive Models.

2.2.7 Periodic Moving Average Models.

Problems.

**3. Review of Hilbert Spaces.**

3.1 Vector Spaces.

3.2 Inner Product Spaces.

3.3 Hilbert Spaces.

3.4 Operators.

3.5 Projection Operators.

3.6 Spectral Theory of Unitary Operators.

3.6.1 Spectral Measures.

3.6.2 Spectral Integrals.

3.6.3 Spectral Theorems.

Problems.

**4. Stationary Random Sequences.**

4.1 Univariate Spectral Theory.

4.1.1 Unitary Shift.

4.1.2 Spectral Representation.

4.1.3 Mean Ergodic Theorem.

4.1.4 Spectral Domain.

4.2 Univariate Prediction Theory.

4.2.1 Infinite Past, Regularity and Singularity.

4.2.2 Wold Decomposition.

4.2.3 Innovation Subspaces.

4.2.4 Spectral Theory and Prediction.

4.2.5 Finite Past Prediction.

4.3 Multivariate Spectral Theory.

4.3.1 Unitary Shift.

4.3.2 Spectral Representation.

4.3.3 Mean Ergodic Theorem.

4.3.4 Spectral Domain.

4.4 Multivariate Prediction Theory.

4.4.1 Infinite Past, Regularity and Singularity.

4.4.2 Wold Decomposition.

4.4.3 Innovations and Rank.

4.4.4 Regular Processes.

4.4.5 Infinite Past Prediction.

4.4.6 Spectral Theory and Rank.

4.4.7 Spectral Theory and Prediction.

4.4.8 Finite Past Prediction.

Problems.

**5. Harmonizable Sequences.**

5.1 Vector Measure Integration.

5.2 Harmonizable Sequences.

5.3 Limit of Ergodic Average.

5.4 Linear Time Invariant Filters.

Problems.

**6. Fourier Theory of the Covariance.**

6.1 Fourier Series Representation of the Covariance.

6.2 Harmonizability of *R(s; t)*.

6.2.1 Harmonizability of *Xt*.

6.4 Covariance and Spectra for Specific Cases.

6.4.1 PC White Noise.

6.4.2 Products of Scalar Periodic and Stationary Sequences.

6.5 Asymptotic Stationarity.

6.6 Lebesgue Decomposition of *F*.

6.7 The spectrum of *mt*.

6.8 Effects of Common Operations on PC Sequences.

6.8.1 Linear Time Invariant Filtering.

6.8.2 Differencing.

6.8.3 Random Shifts.

6.8.4 Sampling.

6.8.5 Bandshifting.

6.8.6 Periodically Time Varying (PTV) Filters.

Problems.

**7. Representations of PC Sequences.**

7.1 The Unitary Operator of a PC Sequence.

7.2 Representations Based on the Unitary Operator.

7.2.1 Gladyshev Representation.

7.2.2 Another Representation of Gladyshev Type.

7.2.3 Time-dependent Spectral Representation.

7.2.4 Harmonizability Again.

7.2.5 Representation Based on Principal Components.

7.3 Mean Ergodic Theorem.

7.4 PC Sequences as Projections of Stationary Sequences.

Problems.

**8. Prediction of PC Sequences.**

8.1 Wold Decomposition.

8.2 Innovations.

8.3 Periodic Autoregressions of Order 1.

8.4 Spectral Density of Regular PC Sequences.

8.4.1 Spectral Densities for PAR(1).

8.5 Least Mean Square Prediction.

8.5.1 Prediction Based on Infinite Past.

8.5.2 Prediction for a PAR(1) Sequence.

8.5.3 Finite Past Prediction.

Problems.

**9. Estimation of Mean and Covariance.**

9.1 Estimation of *mt* : Theory.

9.2 Estimation of *mt* : Practice.

Problems.

**10. Spectral Estimation.**

10.1 The Shifted Periodogram.

10.2 Consistent Estimators.

10.3 Asymptotic Normality.

10.4 Spectral Coherence 363.

10.4.1 Spectral Coherence for Known *T*.

10.4.2 Spectral Coherence for Unknown *T*.

10.5 Spectral Estimation : Practice.

10.5.1 Confidence Intervals.

10.5.2 Examples.

10.6 Effects of Discrete Spectral Components.

10.6.1 Removal of the Periodic Mean.

10.6.2 Testing for Additive Discrete Spectral Components.

10.6.3 Removal of Detected Components.

Problems.

**11. A Paradigm for Nonparametric Analysis of PC Time Series.**

11.1 The Period *T* is Known.

11.2 The Period *T* is Unknown.

References.

Index.

*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 practioners in areas of probability and statistics such as time series analysis, stochastic processes, and prediction theory." (

*Mathematical Review*, Issue 2009e)