Some Advances in the Multitaper Method of Spectrum Estimation by Kyle Quentin Lepage A thesis submitted to the Department of Mathematics and Statistics in conformity with the requirements for the degree of Doctor of Philosophy Queen’s University Kingston, Ontario, Canada February 2009 Copyright c Kyle Quentin Lepage, 2009 Abstract Four contributions to the multitaper method of applied spectrum estimation are pre- sented. These are a generalization of the multitaper method of spectrum estimation to time-series possessing irregularly spaced samples, a robust spectrum estimate suit- able for cyclostationary, or quasi-cyclostationary time-series, an improvement over the standard, multitaper spectrum estimates using quadratic inverse theory, and finally a method of scan-free spectrum estimation using a rotational shear-interferometer. Each of these topics forms a chapter in this thesis. i Co-Authorship Chapters 2, 3, 4 are co-authored with my supervisor, David J. Thomson. Chapter 5, involving scan-free spectrum estimation with a rotational shear interferometer is co-authored with Shawn Kraut, David J. Brady, and David J. Thomson. The work is a continuation of work begun by David J. Brady, worked on by Shawn Kraut while a post-doctoral researcher at Duke University, and finished by myself and David J. Thomson. ii Acknowledgments I thank David J. Thomson, Shawn Kraut and Mark Halpern for invaluable mentor- ship. I would like to thank Ryan Graham, Karim Rahim, and Azadeh Moghtaderi for numerous enlightening discussions on time-series analysis and statistics, my loving parents Edmond Lepage and Irene Makar, and the rest of the Makar clan for their continual support, and finally my beautiful partner and companion, Elsa Hansen, who has, during these past four years, made the finer things sweeter and the lows not quite as deep. iii Statement of Originality The robust estimate, presented in this thesis, of the spectrum of a cyclostation- ary time-series (Chapter 2) is originally the author’s idea. David J. Thomson pro- vided valuable reference material, corrections to some of the development, and proof- reading. The reduced mean-square error quadratic inverse spectrum estimator, presented in Chapter 3, is the author’s idea. Inspiration for the idea stems from empirical experimentation, and weighting given in [46]. David J. Thomson provided valuable reference material, corrections to some of the development, a closed-form solution for the weights, and proof-reading. The generalization of the multitaper method of spectrum estimation presented in Chapter 4 has been independently discovered by the present author and by David J. Thomson (see last sentence of Section 36.6 of his unpublished notes). Inspiration for the idea stems from the derivation of the multitaper method of spectrum estimation presented in [46]. David J. Thomson provided valuable reference material, corrections to some of the development, and proof-reading. The central idea presented in the paper titled, “Scan-free spectrum estimation with a rotational-shear interferometer”, see ([21]), is Shawn Kraut’s. The work was begun by David J. Brady. David J. Thomson kindly proof-read the manuscript. The iv remaining work was performed by the author of this thesis. v Table of Contents Abstract i Co-Authorship ii Acknowledgments iii Statement of Originality iv Table of Contents vi List of Figures viii Chapter 1: The Multitaper Method of Spectrum Estimation . 1 1.1 Spectral Analysis of Time-series . .. 2 1.2 Spectral Analysis of Time-Series with the Multitaper Method .... 6 1.3 Optical Interferometry and the Rotational Shear Interferometer . 11 Chapter 2: Spectral Analysis of Cyclostationary Time-Series: A Ro- bustMethod .......................... 15 2.1 Introduction................................ 15 2.2 StatisticalRobustness .......................... 18 2.3 CyclostationaryTime-Series . 20 2.4 Method .................................. 26 2.5 Quasi-CyclostationaryAdaptation. ... 29 2.6 SeismicData ............................... 39 2.7 Discussion................................. 45 2.8 Acknowledgments............................. 47 vi Chapter 3: Reduced Mean-Square Error Quadratic Inverse Spec- trumEstimator ........................ 49 3.1 Introduction................................ 49 3.2 QuadraticInverseTheory . 51 3.3 Weighted Quadratic Inverse Estimator . .. 54 3.4 Simulations ................................ 56 3.5 Discussion................................. 68 Chapter 4: Multitaper Spectrum Estimation for Irregularly Spaced Data............................... 70 4.1 Introduction................................ 70 4.2 Irregularly Spaced Data: Discrete Fourier Transform . ....... 73 4.3 Irregularly Spaced Data: Fundamental Equation of Spectrum Estimation 76 4.4 SomeSamplingProperties . 82 4.5 Simulations ................................ 85 4.6 Discussion................................. 94 Chapter 5: Multitaper Scan-free Spectrum Estimation with a Rota- tional Shear Interferometer . 95 5.1 Multitaper Scan-free Spectrum Estimation Using a Rotational Shear Interferometer............................... 95 Chapter 6: General Discussion . 97 Chapter 7: Summary & Conclusion . 99 Appendix A: Weighted Average of the High-resolution Spectrum Es- timator: Mean-Square Error . 107 vii List of Figures 2.1 High-pass filtered, seismic velocity data from station NNA, situated in Na˜na,Peru.˜ ................................ 16 2.2 Synthetic Data formed by modulating a low-pass filtered, Gaussian, iid data sequence with a periodic sequence of period T = 100. ... 28 2.3 Uncontaminated data (thick, solid curve) prior to the addition of events, (thin, solid curve) and the cleaned data (thick, dashed curve). .... 36 2.4 Fake Data: Upper-left, stationary, vector-valued representation of the synthetic time-series depicted in Fig. (2.2). ... 37 2.5 Simulation (continued), from top to bottom: contaminated data, orig- inal data, cleaned data, and the difference between the original and thecleaneddata. ............................. 38 2.6 Synthetic Data Spectrum Estimates: Four multi-section, multitaper spectrum estimates are compared. These estimates are all computed using a dimensionless time-bandwidth product of 2, and 3 Slepian ta- pers. Each estimate is computed using ... 40 2.7 Synthetic Data Spectra Estimates: Same spectra estimates as in Fig. (2.6), but plotted as the ratio of the estimate computed using the uncontaminated data to the robust estimates. 41 2.8 NNA North Seismic Velocity: Raw data (top-half upper-left and bottom- half upper-left). Data cleaned using the method described in this chap- ter (upper-half upper-right, top-half bottom, bottom-half upper-right, bottom-half bottom). Events have been replaced by a relatively low- variance interpolation computed using the method described in Section (2.5.3). .................................. 42 2.9 NNA North: Stationary, vector-valued process representation. Clock- wise from upper-left: Raw data. Event detection, detected events are white. Cleaned data. Original data minus cleaned data. Events visible by eye in the original data are detected and replaced; often by streaks. These... ................................. 44 viii 2.10 Top: Ratio of multi-section spectrum estimate (mean and median) computed from the original data to a multi-section spectrum estimate (mean) computed from the cleaned data. Spectrum estimates are com- puted using 5, 20 day, disjoint data sections with start times occuring between day 2 to day 142 of the north component of the high-pass filtered seismic velocity data from station NNA, during the year 1999. 46 3.1 Top: Single realization of the ARMA(4,2) time-series in the 100 it- eration simulation. Bottom: Single realization of the non-adaptive, eigenvalue weighted, multitaper spectrum estimate, the proposed esti- mate, and the actual ARMA(4,2) spectrum. ... 57 3.2 In all plots the number of data points used in the spectrum estimates is 1000, the time-bandwidth parameter is 4, giving a bandwidth, W of 4 10−3 cycles/sample. Three curves are plotted in all plots, these correspond× to estimates using 4 (solid line), 6 (dashed line), or 8 data tapers (dash-dot). Upper left: Ratio of the absolute bias of the pro- posed estimator (QMT), to the theoretical bias of the QMT estimator. Upper right: Ratio of variance of the QMT estimator to the theoretical variance.... ................................ 58 3.3 Mean-square error of the non-adaptive, eigenvalue weighted multitaper estimator (MT) and the weighted quadratic inverse estimator (QMT), computed using the 4, 6, and 8 most in-band concentrated Slepian tapers. Themean-squareerrorof... 60 3.4 Top: Single realization of an ARMA(5,2) time-series in the 100 itera- tion simulation. Bottom: Single realization of the non-adaptive, eigen- value weighted, multitaper spectrum estimate, the proposed estimate, and the actual ARMA(4,2) spectrum. The ARMA ... 61 3.5 Quantities plotted identical to those plotted in Fig. (3.2) are plotted for the ARMA(5,2) process shown in Fig. (3.4). Theoretical values ... 62 3.6 See Fig. (3.3) for a description of the quantities plotted. The mean- square error of the proposed quadratic inverse spectrum estimator (QMT) is less than the non-adaptive, eigenvalue weighted multitaper estimator (MT) for the plotted numbers of used Slepian data tapers. 63 3.7 Ratio of the mean-square error of the non-adaptive eigenvalue weighted spectrum estimator (MT) to the mean-square error of the proposed quadratic inverse estimator (QMT) for three time-bandwidth param- eters,