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 Preface Conventions and Notation Data, Software and Ancillary Material 1 Introduction to Spectral Analysis 1.0 Introduction 1.1 Some Aspects of Time Series Analysis Comments and Extensions to Section 1.1 1.2 Spectral Analysis for a Simple Time Series Model 1.3 Nonparametric Estimation of the Spectrum from Data 1.4 Parametric Estimation of the Spectrum from Data 1.5 Uses of Spectral Analysis 1.6 Exercises 2 Stationary Stochastic Processes 2.0 Introduction 2.1 Stochastic Processes 2.2 Notation 2.3 Basic Theory for Stochastic Processes Comments and Extensions to Section 2.3 2.4 Real-Valued Stationary Processes 2.5 Complex-Valued Stationary Processes Comments and Extensions to Section 2.5 2.6 Examples of Discrete Parameter Stationary Processes 2.7 Comments on Continuous Parameter Processes 2.8 Use of Stationary Processes as Models for Data 2.9 Exercises 3 Deterministic Spectral Analysis 3.0 Introduction 3.1 Fourier Theory – Continuous Time/Discrete Frequency Comments and Extensions to Section 3.1 3.2 Fourier Theory – Continuous Time and Frequency Comments and Extensions to Section 3.2 3.3 Band-Limited and Time-Limited Functions 3.4 Continuous/Continuous Reciprocity Relationships 3.5 Concentration Problem – Continuous/Continuous Case 3.6 Convolution Theorem – Continuous Time and Frequency 3.7 Autocorrelations and Widths – Continuous Time and Frequency 3.8 Fourier Theory – Discrete Time/Continuous Frequency 3.9 Aliasing Problem – Discrete Time/Continuous Frequency Comments and Extensions to Section 3.9 3.10 Concentration Problem – Discrete/Continuous Case 3.11 Fourier Theory – Discrete Time and Frequency Comments and Extensions to Section 3.11 3.12 Summary of Fourier Theory 3.13 Exercises 4 Foundations for Stochastic Spectral Analysis 4.0 Introduction 4.1 Spectral Representation of Stationary Processes Comments and Extensions to Section 4.1 4.2 Alternative Definitions for the Spectral Density Function 4.3 Basic Properties of the Spectrum Comments and Extensions to Section 4.3 4.4 Classification of Spectra 4.5 Sampling and Aliasing Comments and Extensions to Section 4.5 4.6 Comparison of SDFs and ACVSs as Characterizations 4.7 Summary of Foundations for Stochastic Spectral Analysis 4.8 Exercises 5 Linear Time-Invariant Filters 5.0 Introduction 5.1 Basic Theory of LTI Analog Filters Comments and Extensions to Section 5.1 5.2 Basic Theory of LTI Digital Filters Comments and Extensions to Section 5.2 5.3 Convolution as an LTI filter 5.4 Determination of SDFs by LTI Digital Filtering 5.5 Some Filter Terminology 5.6 Interpretation of Spectrum via Band-Pass Filtering 5.7 An Example of LTI Digital Filtering Comments and Extensions to Section 5.7 5.8 Least Squares Filter Design 5.9 Use of Slepian Sequences in Low-Pass Filter Design 5.10 Exercises 6 Periodogram and Other Direct Spectral Estimators 6.0 Introduction 6.1 Estimation of the Mean Comments and Extensions to Section 6.1 6.2 Estimation of the Autocovariance Sequence Comments and Extensions to Section 6.2 6.3 A Naive Spectral Estimator – the Periodogram Comments and Extensions to Section 6.3 6.4 Bias Reduction – Tapering Comments and Extensions to Section 6.4 6.5 Bias Reduction – Prewhitening Comments and Extensions to Section 6.5 6.6 Statistical Properties of Direct Spectral Estimators Comments and Extensions to Section 6.6 6.7 Computational Details 6.8 Examples of Periodogram and Other Direct Spectral Estimators Comments and Extensions to Section 6.8 6.9 Comments on Complex-Valued Time Series 6.10 Summary of Periodogram and Other Direct Spectral Estimators 6.11 Exercises 7 Lag Window Spectral Estimators 7.0 Introduction 7.1 Smoothing Direct Spectral Estimators Comments and Extensions to Section 7.1 7.2 First-Moment Properties of Lag Window Estimators Comments and Extensions to Section 7.2 7.3 Second-Moment Properties of Lag Window Estimators Comments and Extensions to Section 7.3 7.4 Asymptotic Distribution of Lag Window Estimators 7.5 Examples of Lag Windows Comments and Extensions to Section 7.5 7.6 Choice of Lag Window Comments and Extensions to Section 7.6 7.7 Choice of Lag Window Parameter Comments and Extensions to Section 7.7 7.8 Estimation of Spectral Bandwidth 7.9 Automatic Smoothing of Log Spectral Estimators Comments and Extensions to Section 7.9 7.10 Bandwidth Selection for Periodogram Smoothing Comments and Extensions to Section 7.10 7.11 Computational Details 7.12 Examples of Lag Window Spectral Estimators Comments and Extensions to Section 7.12 7.13 Summary of Lag Window Spectral Estimators 7.14 Exercises 8 Combining Direct Spectral Estimators 8.0 Introduction 8.1 Multitaper Spectral Estimators – Overview Comments and Extensions to Section 8.1 8.2 Slepian Multitaper Estimators Comments and Extensions to Section 8.2 8.3 Multitapering of Gaussian White Noise 8.4 Quadratic Spectral Estimators and Multitapering Comments and Extensions to Section 8.4 8.5 Regularization and Multitapering Comments and Extensions to Section 8.5 8.6 Sinusoidal Multitaper Estimators Comments and Extensions to Section 8.6 8.7 Improving Periodogram-Based Methodology via Multitapering Comments and Extensions to Section 8.7 8.8 Welch’s Overlapped Segment Averaging (WOSA) Comments and Extensions to Section 8.8 8.9 Examples of Multitaper and WOSA Spectral Estimators 8.10 Summary of Combining Direct Spectral Estimators 8.11 Exercises 9 Parametric Spectral Estimators 9.0 Introduction 9.1 Notation 9.2 The Autoregressive Model Comments and Extensions to Section 9.2 9.3 The Yule–Walker Equations Comments and Extensions to Section 9.3 9.4 The Levinson–Durbin Recursions Comments and Extensions to Section 9.4 9.5 Burg’s Algorithm Comments and Extensions to Section 9.5 9.6 The Maximum Entropy Argument 9.7 Least Squares Estimators Comments and Extensions to Section 9.7 9.8 Maximum Likelihood Estimators Comments and Extensions to Section 9.8 9.9 Confidence Intervals Using AR Spectral Estimators Comments and Extensions to Section 9.9 9.10 Prewhitened Spectral Estimators 9.11 Order Selection for AR(p) Processes Comments and Extensions to Section 9.11 9.12 Examples of Parametric Spectral Estimators 9.13 Comments on Complex-Valued Time Series 9.14 Use of Other Models for Parametric SDF Estimation 9.15 Summary of Parametric Spectral Estimators 9.16 Exercises 10 Harmonic Analysis 10.0 Introduction 10.1 Harmonic Processes – Purely Discrete Spectra 10.2 Harmonic Processes with Additive White Noise – Discrete Spectra Comments and Extensions to Section 10.2 10.3 Spectral Representation of Discrete and Mixed Spectra Comments and Extensions to Section 10.3 10.4 An Example from Tidal Analysis Comments and Extensions to Section 10.4 10.5 A Special Case of Unknown Frequencies Comments and Extensions to Section 10.5 10.6 General Case of Unknown Frequencies Comments and Extensions to Section 10.6 10.7 An Artificial Example from Kay and Marple Comments and Extensions to Section 10.7 10.8 Tapering and the Identification of Frequencies 10.9 Tests for Periodicity – White Noise Case Comments and Extensions to Section 10.9 10.10 Tests for Periodicity – Colored Noise Case Comments and Extensions to Section 10.10 10.11 Completing a Harmonic Analysis Comments and Extensions to Section 10.11 10.12 A Parametric Approach to Harmonic Analysis Comments and Extensions to Section 10.12 10.13 Problems with the Parametric Approach 10.14 Singular Value Decomposition Approach Comments and Extensions to Section 10.14 10.15 Examples of Harmonic Analysis Comments and Extensions to Section 10.15 10.16 Summary of Harmonic Analysis 10.17 Exercises 11 Simulation of Time Series 11.0 Introduction 11.1 Simulation of ARMA Processes and Harmonic Processes Comments and Extensions to Section 11.1 11.2 Simulation of Processes with a Known Autocovariance Sequence Comments and Extensions to Section 11.2 11.3 Simulation of Processes with a Known Spectral Density Function Comments and Extensions to Section 11.3 11.4 Simulating Time Series from Nonparametric Spectral Estimates Comments and Extensions to Section 11.4 11.5 Simulating Time Series from Parametric Spectral Estimates Comments and Extensions to Section 11.5 11.6 Examples of Simulation of Time Series Comments and Extensions to Section 11.6 11.7 Comments on Simulation of Non-Gaussian Time Series 11.8 Summary of Simulation of Time Series 11.9 Exercises References Author Index Subject Index "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