Spectral analysis for univariate time series

✍ Scribed by Percival D.B., Walden A.T

• Publisher: Cambridge University Press
• Year: 2020
• Tongue: English
• Leaves: 717
• Category: Library

✦ Table of Contents

Contents......Page 8
Preface......Page 14
Conventions and Notation......Page 17
Data, Software and Ancillary Material......Page 25
1.1 Some Aspects of Time Series Analysis......Page 26
1.2 Spectral Analysis for a Simple Time Series Model......Page 30
1.3 Nonparametric Estimation of the Spectrum from Data......Page 36
1.4 Parametric Estimation of the Spectrum from Data......Page 39
1.5 Uses of Spectral Analysis......Page 40
1.6 Exercises......Page 42
2.1 Stochastic Processes......Page 46
2.2 Notation......Page 47
2.3 Basic Theory for Stochastic Processes......Page 48
Comments and Extensions to Section 2.3......Page 50
2.4 Real-Valued Stationary Processes......Page 51
2.5 Complex-Valued Stationary Processes......Page 54
2.6 Examples of Discrete Parameter Stationary Processes......Page 56
2.8 Use of Stationary Processes as Models for Data......Page 63
2.9 Exercises......Page 66
3.0 Introduction......Page 72
3.1 Fourier Theory – Continuous Time/Discrete Frequency......Page 73
Comments and Extensions to Section 3.1......Page 77
3.2 Fourier Theory – Continuous Time and Frequency......Page 78
Comments and Extensions to Section 3.2......Page 80
3.3 Band-Limited and Time-Limited Functions......Page 82
3.4 Continuous/Continuous Reciprocity Relationships......Page 83
3.5 Concentration Problem – Continuous/Continuous Case......Page 87
3.6 Convolution Theorem – Continuous Time and Frequency......Page 92
3.7 Autocorrelations and Widths – Continuous Time and Frequency......Page 97
3.8 Fourier Theory – Discrete Time/Continuous Frequency......Page 99
3.9 Aliasing Problem – Discrete Time/Continuous Frequency......Page 106
Comments and Extensions to Section 3.9......Page 109
3.10 Concentration Problem – Discrete/Continuous Case......Page 110
3.11 Fourier Theory – Discrete Time and Frequency......Page 116
Comments and Extensions to Section 3.11......Page 118
3.12 Summary of Fourier Theory......Page 120
3.13 Exercises......Page 127
4.0 Introduction......Page 132
4.1 Spectral Representation of Stationary Processes......Page 133
Comments and Extensions to Section 4.1......Page 138
4.2 Alternative Definitions for the Spectral Density Function......Page 139
4.3 Basic Properties of the Spectrum......Page 141
Comments and Extensions to Section 4.3......Page 143
4.4 Classification of Spectra......Page 145
4.5 Sampling and Aliasing......Page 147
Comments and Extensions to Section 4.5......Page 148
4.6 Comparison of SDFs and ACVSs as Characterizations......Page 149
4.7 Summary of Foundations for Stochastic Spectral Analysis......Page 150
4.8 Exercises......Page 152
5.0 Introduction......Page 157
5.1 Basic Theory of LTI Analog Filters......Page 158
Comments and Extensions to Section 5.1......Page 162
5.2 Basic Theory of LTI Digital Filters......Page 165
5.3 Convolution as an LTI filter......Page 167
5.4 Determination of SDFs by LTI Digital Filtering......Page 169
5.5 Some Filter Terminology......Page 170
5.6 Interpretation of Spectrum via Band-Pass Filtering......Page 172
5.7 An Example of LTI Digital Filtering......Page 173
Comments and Extensions to Section 5.7......Page 176
5.8 Least Squares Filter Design......Page 177
5.9 Use of Slepian Sequences in Low-Pass Filter Design......Page 180
5.10 Exercises......Page 182
6.0 Introduction......Page 188
6.1 Estimation of the Mean......Page 189
Comments and Extensions to Section 6.1......Page 190
6.2 Estimation of the Autocovariance Sequence......Page 191
Comments and Extensions to Section 6.2......Page 194
6.3 A Naive Spectral Estimator – the Periodogram......Page 195
Comments and Extensions to Section 6.3......Page 204
6.4 Bias Reduction – Tapering......Page 210
Comments and Extensions to Section 6.4......Page 219
6.5 Bias Reduction – Prewhitening......Page 222
6.6 Statistical Properties of Direct Spectral Estimators......Page 226
Comments and Extensions to Section 6.6......Page 234
6.7 Computational Details......Page 244
6.8 Examples of Periodogram and Other Direct Spectral Estimators......Page 249
Comments and Extensions to Section 6.8......Page 255
6.9 Comments on Complex-Valued Time Series......Page 256
6.10 Summary of Periodogram and Other Direct Spectral Estimators......Page 257
6.11 Exercises......Page 260
7.0 Introduction......Page 270
7.1 Smoothing Direct Spectral Estimators......Page 271
Comments and Extensions to Section 7.1......Page 277
7.2 First-Moment Properties of Lag Window Estimators......Page 280
Comments and Extensions to Section 7.2......Page 282
7.3 Second-Moment Properties of Lag Window Estimators......Page 283
Comments and Extensions to Section 7.3......Page 286
7.4 Asymptotic Distribution of Lag Window Estimators......Page 289
7.5 Examples of Lag Windows......Page 293
Comments and Extensions to Section 7.5......Page 303
7.6 Choice of Lag Window......Page 312
Comments and Extensions to Section 7.6......Page 315
7.7 Choice of Lag Window Parameter......Page 316
Comments and Extensions to Section 7.7......Page 321
7.8 Estimation of Spectral Bandwidth......Page 322
7.9 Automatic Smoothing of Log Spectral Estimators......Page 326
Comments and Extensions to Section 7.9......Page 331
7.10 Bandwidth Selection for Periodogram Smoothing......Page 332
Comments and Extensions to Section 7.10......Page 337
7.11 Computational Details......Page 339
7.12 Examples of Lag Window Spectral Estimators......Page 341
Comments and Extensions to Section 7.12......Page 361
7.13 Summary of Lag Window Spectral Estimators......Page 365
7.14 Exercises......Page 368
8.0 Introduction......Page 376
8.1 Multitaper Spectral Estimators – Overview......Page 377
Comments and Extensions to Section 8.1......Page 380
8.2 Slepian Multitaper Estimators......Page 382
Comments and Extensions to Section 8.2......Page 391
8.3 Multitapering of Gaussian White Noise......Page 395
8.4 Quadratic Spectral Estimators and Multitapering......Page 399
8.5 Regularization and Multitapering......Page 407
Comments and Extensions to Section 8.5......Page 415
8.6 Sinusoidal Multitaper Estimators......Page 416
Comments and Extensions to Section 8.6......Page 425
8.7 Improving Periodogram-Based Methodology via Multitapering......Page 428
8.8 Welch’s Overlapped Segment Averaging (WOSA)......Page 437
Comments and Extensions to Section 8.8......Page 444
8.9 Examples of Multitaper and WOSA Spectral Estimators......Page 450
8.10 Summary of Combining Direct Spectral Estimators......Page 457
8.11 Exercises......Page 461
9.1 Notation......Page 470
9.2 The Autoregressive Model......Page 471
Comments and Extensions to Section 9.2......Page 472
9.3 The Yule–Walker Equations......Page 474
9.4 The Levinson–Durbin Recursions......Page 477
Comments and Extensions to Section 9.4......Page 485
9.5 Burg’s Algorithm......Page 491
Comments and Extensions to Section 9.5......Page 494
9.6 The Maximum Entropy Argument......Page 496
9.7 Least Squares Estimators......Page 500
Comments and Extensions to Section 9.7......Page 503
9.8 Maximum Likelihood Estimators......Page 505
Comments and Extensions to Section 9.8......Page 508
9.9 Confidence Intervals Using AR Spectral Estimators......Page 510
Comments and Extensions to Section 9.9......Page 515
9.10 Prewhitened Spectral Estimators......Page 516
9.11 Order Selection for AR(p) Processes......Page 517
Comments and Extensions to Section 9.11......Page 520
9.12 Examples of Parametric Spectral Estimators......Page 521
9.13 Comments on Complex-Valued Time Series......Page 526
9.14 Use of Other Models for Parametric SDF Estimation......Page 528
9.15 Summary of Parametric Spectral Estimators......Page 530
9.16 Exercises......Page 531
10.1 Harmonic Processes – Purely Discrete Spectra......Page 536
10.2 Harmonic Processes with Additive White Noise – Discrete Spectra......Page 537
Comments and Extensions to Section 10.2......Page 542
10.3 Spectral Representation of Discrete and Mixed Spectra......Page 543
Comments and Extensions to Section 10.3......Page 544
10.4 An Example from Tidal Analysis......Page 545
10.5 A Special Case of Unknown Frequencies......Page 548
10.6 General Case of Unknown Frequencies......Page 549
Comments and Extensions to Section 10.6......Page 552
10.7 An Artificial Example from Kay and Marple......Page 555
Comments and Extensions to Section 10.7......Page 559
10.8 Tapering and the Identification of Frequencies......Page 560
10.9 Tests for Periodicity – White Noise Case......Page 563
Comments and Extensions to Section 10.9......Page 568
10.10 Tests for Periodicity – Colored Noise Case......Page 569
Comments and Extensions to Section 10.10......Page 573
10.11 Completing a Harmonic Analysis......Page 574
Comments and Extensions to Section 10.11......Page 577
10.12 A Parametric Approach to Harmonic Analysis......Page 578
Comments and Extensions to Section 10.12......Page 582
10.13 Problems with the Parametric Approach......Page 583
10.14 Singular Value Decomposition Approach......Page 588
10.15 Examples of Harmonic Analysis......Page 592
Comments and Extensions to Section 10.15......Page 608
10.16 Summary of Harmonic Analysis......Page 609
10.17 Exercises......Page 612
11.0 Introduction......Page 618
11.1 Simulation of ARMA Processes and Harmonic Processes......Page 619
Comments and Extensions to Section 11.1......Page 624
11.2 Simulation of Processes with a Known Autocovariance Sequence......Page 626
Comments and Extensions to Section 11.2......Page 628
11.3 Simulation of Processes with a Known Spectral Density Function......Page 629
Comments and Extensions to Section 11.3......Page 634
11.4 Simulating Time Series from Nonparametric Spectral Estimates......Page 636
Comments and Extensions to Section 11.4......Page 638
11.5 Simulating Time Series from Parametric Spectral Estimates......Page 642
Comments and Extensions to Section 11.5......Page 643
11.6 Examples of Simulation of Time Series......Page 644
11.7 Comments on Simulation of Non-Gaussian Time Series......Page 656
11.8 Summary of Simulation of Time Series......Page 662
11.9 Exercises......Page 663
References......Page 668
Author Index......Page 686
Subject Index......Page 692