Spectral Analysis for Univariate Time Series

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