Spectral Analysis for Univariate Time Series (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 51)

Spectral Analysis Is Widely Used To Interpret Time Series Collected In Diverse Areas. This Book Covers The Statistical Theory Behind Spectral Analysis And Provides Data Analysts With The Tools Needed To Transition Theory Into Practice. Actual Time Series From Oceanography, Metrology, Atmospheric Science And Other Areas Are Used In Running Examples Throughout, To Allow Clear Comparison Of How The Various Methods Address Questions Of Interest. All Major Nonparametric And Parametric Spectral Analysis Techniques Are Discussed, With Emphasis On The Multitaper Method, Both In Its Original Formulation Involving Slepian Tapers And In A Popular Alternative Using Sinusoidal Tapers. The Authors Take A Unified Approach To Quantifying The Bandwidth Of Different Nonparametric Spectral Estimates. An Extensive Set Of Exercises Allows Readers To Test Their Understanding Of Theory And Practical Analysis. The Time Series Used As Examples And R Language Code For Recreating The Analyses Of The Series Are Available From The Book's Website. 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 "This chapter provides a quick introduction to the subject of spectral analysis. Except for some later references to the exercises of Section 1.6, this material is independent of the rest of the book and can be skipped without loss of continuity. Our intent is to use some simple examples to motivate the key ideas. Since our purpose is to view the forest before we get lost in the trees, the particular analysis techniques we use here have been chosen for their simplicity rather than their appropriateness"-- Provided by publisher Spectral analysis is an important technique for interpreting time series data. This book uses the R language and real world examples to show data analysts interested in time series in the environmental, engineering and physical sciences how to bridge the gap between the statistical theory behind spectral analysis and its application to actual data