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Spectral-Based Tests for Periodicities

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Spectral-Based Tests for Periodicities SPECTRAL-BASED TESTS FOR PERIODICITIES DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Lai Wei, M.S. ***** The Ohio State University 2008 Dissertation Committee: Approved by Professor Peter F. Craigmile, Adviser Professor Thomas J. Santner Adviser Professor Wayne M. King Graduate Program in Statistics c Copyright by Lai Wei 2008 ABSTRACT In this thesis, tests for periodicity are investigated based on a spectral analysis of a time series. Some important fundamental theories of spectral analysis of sta- tionary and harmonic processes are reviewed. A regression model is developed in the frequency domain based on the Fourier transformation. We present some of the peri- odogram based tests, which are the global test and three local tests, i.e., the hearing test, local F test and Thomson’s multitaper test. We will show that most of the tests can be derived from our regression model with the error term having an approxi- mately diagonal covariance matrix. The distribution of the error term of the spectral regression model is based on the asymptotic distribution of the tapered Fourier trans- form of the error process. This asymptotic distribution has approximately a diagonal covariance matrix when the sample size is large and the spectral density functions (SDFs) of error processes have small dynamic range. We contrast the F test in the time domain and the local F test in the frequency domain as well as the global and local spectral-based tests. The global test uses the spectral estimates at all sample points, whereas a local test uses a subset of the sample points available. Standard global tests for periodicity are often based on the assumption of a Gaussian IID error process. Using a smoothing spline approach, we extend the global test to the non-IID case. We compare this approach to a number of local tests for periodicity such as the local F test, the test commonly used in hearing sciences, and Thomson’s multitaper F ii test. Using regression-based F tests, we demonstrate that asymptotic size and power calculations can be made for some of these tests. We compare the size and power at finite sample sizes, under a number of different experimental conditions. According to the exploratory data analysis, we applied the local F test to hearing data, Distortion Product Otoacoustic Emissions (DPOAEs), collected in the Depart- ment of Speech and Hearing Sciences, OSU. The logistic regression model and the noncentral F mixed effects regression models are explored to capture the important features of hearing data. In particular, noncentral-F mixed effects regression mod- els capture within-subject-variability of the distortion products of healthy hearing subjects. This is key to understanding the underlying processes inherent in DPOAE- based hearing tests. The Penalized Quasi-Likelihood method is used to estimate the model parameters and we demonstrate how to do the model selection and diagnosing. iii To my family iv ACKNOWLEDGMENTS I would like to express my sincere gratitude toward my advisor, Professor Peter F. Craigmile, for his support and guidance throughout my research. I owed a great deal to him for the knowledge that I have gained in the spectral analysis in time series as well as the experience in it’s applications. I am also grateful for the DPOAE dataset provided by Professor Wayne King. I want to thank Professor Thomas Santner and Professor Wayne King for taking time to serve in my dissertation committee. Finally, I would like to thank my parents, my husband Sijin, and my son Kevin for their constant and unselfish love and encouragement. v VITA 1994-1998 ..................................B.S. Electrical Engineering, Electronic Science and Technology of China, Chengdu, China 1998-2000 ..................................Graduate Research Associate, Elec- tronic Science and Technology of China, Chengdu, China 2000-2003 ..................................M.S. Electrical Engineering, The Ohio State University, Columbus, Ohio, USA 2003-2004 ..................................M.S.Statistics, The Ohio State Univer- sity, Columbus, Ohio, USA 2000-present ................................Graduate Research/Teaching Asso- ciate, The Ohio State University, Columbus, Ohio, USA FIELDS OF STUDY Major Field: Statistics Studies in Spectral-based Tests for Periodicity: Professor Peter F. Craigmile vi TABLE OF CONTENTS Page Abstract....................................... ii Dedication...................................... iv Acknowledgments.................................. v Vita ......................................... vi ListofTables.................................... ix ListofFigures ................................... x Chapters: 1. Introduction.................................. 1 1.1 Spectral-based Tests for Periodicity of Time Series . ...... 1 1.2 Distortion Product Otoacoustic Emissions(DPOAEs) . ..... 2 1.3 GeneralizedLinearModels. 4 1.4 OutlineoftheThesis.......................... 5 2. A Review of The Spectral Analysis of Stationary and Harmonic Processes 7 2.1 StationaryProcesses .......................... 7 2.2 Spectral Analysis of Stationary Processes . ... 11 2.3 Introduction of Harmonic Processes . 19 3. Spectral-based Tests for Periodicity . ..... 23 3.1 Analysis of the Tapered Fourier Transform of Stationary Process . 32 3.2 RegressionModelintheSpectralDomain . 41 vii 3.3 Global F test.............................. 44 3.3.1 Global F test with one periodicity (L =1).......... 44 3.3.2 Global F Test with More Than One Periodicities (L> 1) . 50 3.4 LocalTests ............................... 57 3.4.1 TestfromtheHearingScience. 58 3.4.2 Local F Test .......................... 58 3.4.3 Thomson’s F Test ....................... 66 4. SizeandPowerComparisons......................... 71 4.1 TheSizeoftheTest .......................... 73 4.2 ThePoweroftheTest ......................... 82 5. Analysis of Distortion Product Otoacoustic Emissions (DPOAEs) . 96 5.1 Introduction of Distortion Product Otoacoustic Emissions (DPOAEs) 96 5.2 ExploratoryDataAnalysis. 99 5.3 LogisticRegressionModels . 102 5.3.1 Review of The Logistic Regression Model . 103 5.3.2 Logistic Regression Model for DPOAE-based Healthy Cochlear Status.............................. 105 5.4 Generalized Linear Mixed Effects Model for the DPOAE Periodicity F Statistics ............................... 114 5.4.1 Parameter Estimation in the Generalized Linear Mixed Ef- fectsModel ........................... 115 5.4.2 Noncentral-F Linear Mixed Effects Model . 120 5.4.3 Generalized Linear Mixed Effects Model for the DPOAE data 125 6. Discussion................................... 142 Appendices: A. Noncentral-Fdistribution . 146 B. Notation.................................... 147 Bibliography .................................... 151 viii LIST OF TABLES Table Page 5.1 AIC for different links using the 20% trimmed mean of the times series ofall100trials. .............................. 110 5.2 AIC for different links using the 10% trimmed mean of the times series ofevery20trials. ............................. 113 5.3 Pearson residual analysis table for independent model . ........ 127 5.4 randomeffectmodel ........................... 128 5.5 Cross validation results for the log-Gaussian and noncentral F mixed effectsmodel ............................... 140 ix LIST OF FIGURES Figure Page 2.1 Plots of rectangular taper and 50% cosine taper in time domain and frequency domain, and the direct spectral estimates of the AR(4) noise process using rectangular taper and 50% cosine taper respectively . 14 3.1 Time series plots and ACVS plots of three periodicities, AR(2) noise process, and a time series with three periodicities plus AR(2) noise process. .................................. 24 3.2 Plot of diagram of transformation between the time domain and fre- quencydomain .............................. 28 3.3 Plot of periodogram of the time series with three periodicities and AR(2) noise process in frequency domain . 29 3.4 Plots of variance and correlation for AR(2) with sample size N = 16 and N = 256, AR(4) with N = 256, AR(4) with 20% cosine taper and N = 256 respectively. Correlations are calculated between frequency 1 2 η and η′ = η as well as η and η′ = η .............. 39 ± N ± N 3.5 Plots of absolute correlation for AR(2) with sample size N = 16 and N = 256, AR(4) with N = 256, AR(4) with 20% cosine taper and N =256respectively............................ 40 4.1 SDFs of the IID (solid line), AR(2) (dashed line), and AR(4) (dotted line)errorprocesses. ........................... 72 4.2 Plots of the estimated size as a function of the test frequency, ζ1, of the local F test, the Gorga et al. test and the global test for IID and AR(2) error processes with one periodicity (L = 1). In each case, the estimate at each frequency is based on a simulation of 1,000 time series with sample size N = 512. The local tests use bin size B =4. .... 74 x 4.3 Plots of the estimated size as a function of the nuisance frequency, ζ2, of the local F test, the Gorga et al. test and the global test for IID and AR(2) error processes with two periodicities (L = 2). In each simulation ζ1 = 144/512. In each case, the estimate at each frequency is based on a simulation of 1,000 time series with sample size N = 512. The local tests use bin size B =4..................... 75 4.4 Plots of the estimated size as a function of the test frequency, ζ1, of the local F test without taper, the local F tests with 5% cosine taper and
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