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Wavelet-Based Estimation for Gaussian Time Series and Spatio-Temporal Processes

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Wavelet-Based Estimation for Gaussian Time Series and Spatio-Temporal Processes Wavelet-based estimation for Gaussian time series and spatio-temporal processes Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Wenjun Zheng, B.S. Graduate Program in Statistics The Ohio State University 2014 Dissertation Committee: Peter Craigmile, Advisor Mark Berliner Catherine Calder c Copyright by Wenjun Zheng 2014 Abstract Modern statistical analyses often require the modeling non-Markov time series and spatio-temporal dependencies. Traditional likelihood methods are computationally demanding for these models, leading us to consider approximate likelihood methods that are computationally efficient, while not overly compromising on the efficiency of the parameter estimates. In this dissertation various wavelet-based Whittle approxi- mations are investigated to model a certain class of nonstationary Gaussian time series and a class of Gaussian spatio-temporal processes. Wavelet transforms can help decor- relate processes across and within wavelet scales, allowing for the simplified modeling of time series and spatio-temporal processes. In addition to being computationally efficient, the proposed maximum wavelet-Whittle likelihood estimators of a Gaussian process are shown to be asymptotically normal. Asymptotic properties of the estima- tors are verified in simulation studies, demonstrating that the typical independence everywhere assumption assumed for wavelet-based estimation is not optimal. These methods are applied to the analyses of a Southern Oscillation Index climate series and the Irish wind speed data. Keywords: Approximate likelihood, between and within-scale decorrelation, dis- crete wavelet transform, Markov approximations, Matérn processes, spatio-temporal processes. ii Acknowledgments First and foremost, I would like to express my grateful thanks to my advisor, Dr. Peter Craigmile for his patient guidance during my PhD study. I am fortunate to have a great advisor who is enthusiastic about research, and cares about my work. I am grateful to my committee members, Dr. Mark Berliner and Dr. Catherine Calder, for their assistance throughout my dissertation. I acknowledge the NSF grant, DMS- 0906864, for support as a Graduate Research Assistant in 2012. Finally, I am deeply thankful to my family and all my friends for their support and encouragement over the years. I am grateful to my husband, Shuchi Wu, for sacrificing his valuable time to proofread my dissertation. I also thank my son Dylan for being a good baby, making it possible for me to write and finish my dissertation in his first few months of life. iii Vita 2008 ........................................B.S. Statistics, Wuhan University 2008-present ................................Graduate Teaching Associate, Depart- ment of Statistics, The Ohio State Uni- versity. Fields of Study Major Field: Statistics iv Table of Contents Page Abstract ....................................... ii Acknowledgments .................................. iii Vita ......................................... iv List of Tables .................................... viii List of Figures ................................... ix Chapters: 1. Introduction ................................. 1 1.1 Approximation models ......................... 2 1.2 The wavelet Whittle method ..................... 3 1.3 Extension to space-time processes ................... 5 1.4 Layout of the dissertation ....................... 6 2. Discrete wavelet transforms of time series processes ............ 8 2.1 The Gaussian processes of interest .................. 8 2.2 The discrete wavelet transforms of a time series ........... 10 2.3 Statistical properties of the wavelet coefficients ........... 19 2.3.1 Expected value of the DWT coefficients ........... 19 2.3.2 Covariance properties ..................... 20 2.3.3 Spectral representations of the DWTs ............ 23 2.4 An example ............................... 28 2.5 Conclusion ............................... 33 v 3. Asymptotic properties of the wavelet-Whittle estimators ......... 37 3.1 Wavelet-based Whittle approximations ................ 37 3.2 A white noise model .......................... 40 3.3 The proof of the white noise results .................. 43 3.4 An AR(1) model ............................ 53 3.4.1 The approximate likelihood .................. 53 3.4.2 Discussion about the MLEs under the AR(1) model ..... 54 3.5 Choices of J0 .............................. 58 3.6 Monte Carlo studies .......................... 59 3.6.1 The wavelet-based Whittle MLEs ............... 59 3.6.2 Comparison to other approximate methods ......... 63 3.6.3 The limiting variance ..................... 65 3.7 Southern Oscillation Index data .................... 67 3.8 Conclusion ............................... 73 4. The DWTs of space-time processes ..................... 76 4.1 A review of space-time processes ................... 77 4.1.1 The statistical properties ................... 77 4.1.2 The spectral representations .................. 80 4.2 The DWT of a space-time process .................. 82 4.2.1 Simplifying assumptions for the space-scale-time wavelet pro- cesses .............................. 86 4.2.2 Spectral representations of the DWT coefficients ...... 91 4.2.3 Examples of the whitening effects ............... 95 5. Wavelet Whittle models for space-time processes .............. 107 5.1 Wavelet Whittle approximations for space-time processes . 107 5.1.1 A white noise model ...................... 109 5.1.2 An AR(1) model ........................ 110 5.2 Asymptotic results of the wavelet-Whittle estimators ........ 112 5.3 Proof of Theorem 5.1 and Theorem 5.2 ............... 113 5.4 Monte Carlo studies .......................... 130 5.5 Application: Irish wind data ...................... 133 6. Discussion and future work ......................... 141 6.1 Conclusion and discussion ....................... 141 6.2 Future work ............................... 144 vi Appendix 147 A. Proof of the asymptotic normality of the approximate MLEs in the MODWT setting ..................................... 147 Bibliography .................................... 151 vii List of Tables Table Page 3.1 Matérn process: results for φˆ using wavelet coefficients on Levels 3 to J 61 3.2 Matérn process: results for σˆ2 using wavelet coefficients on Levels 3 to J 61 3.3 Matérn process: results for φˆ using wavelet coefficients on Levels 1 to J 62 3.4 Compare simulation time ......................... 62 3.5 Simulation results (FD process) ..................... 65 3.6 Limiting variance: all coefficients .................... 66 3.7 Limiting variance: non-boundary coefficients ............. 66 5.1 Simulation results of the temporal scaling parameter a. ........ 132 5.2 Simulation results of the spatial scaling parameter c. ......... 132 5.3 Simulation time for the space-time estimation (in seconds). 133 5.4 Estimation results for the wind speed covariance model. ........ 138 viii List of Figures Figure Page 2.1 DWT filters ................................ 16 2.2 DWT squared gain functions ....................... 17 2.3 LA(8) squared gain functions ...................... 18 2.4 DWT decorrelation example - SDF ................... 30 2.5 DWT decorrelation example (within-scale) with the LA(8) filter . 31 2.6 DWT decorrelation example (cross-scale) with the LA(8) filter . 32 2.7 DWT decorrelation example (within-scale) with the D(6) filter . 33 2.8 DWT decorrelation example (cross-scale) with the D(6) filter ..... 34 2.9 DWT decorrelation example (within-scale) with the D(4) filter . 35 2.10 DWT decorrelation example (cross-scale) with the D(4) filter ..... 36 3.1 Monthly SOI signal ............................ 68 3.2 SOI wavelet spectra ............................ 69 3.3 PACF of the standardized SOI ...................... 71 3.4 Fitted spectrum .............................. 74 3.5 Fitted spectrum - confidence bands ................... 75 ix 4.1 Space-time process between-scale covariances, with LA(8) filter and h =0 .................................. 98 || || 4.2 Space-time process between-scale covariances, with LA(8) filter and h =1 .................................. 99 || || 4.3 Space-time process between-scale covariances, with LA(8) filter and h =2 .................................. 100 || || 4.4 Space-time separable ACVS, LA(8) filter ................ 101 4.5 Space-time separable correlation, LA(8) filter. ............. 102 4.6 Space-time nonseparable ACVS, LA(8) filter .............. 103 4.7 Space-time nonseparable correlation, LA(8) filter. ........... 104 4.8 Space-time nonseparable ACVS with β =0.7, LA(8) filter. 105 4.9 Space-time process nonseparable ACVS with β =0.1, LA(8) filter. 106 5.1 Wind data wavelet spectra - 1 ...................... 135 5.2 Wind data wavelet spectra - 2 ...................... 136 5.3 Wind data wavelet spectra - 3 ...................... 137 5.4 Wind data space-time correlation .................... 139 5.5 Wind data cross (station) correlation .................. 140 x Chapter 1: Introduction Time series are collected in disciplines such as economics, environmental sciences, health sciences and geophysics. In time series analysis, researchers are interested in the statistical analysis of data observed over time. Modeling time series data helps people predict future outcomes and study the characteristics (statistical properties) of the process generating the data. One useful summary of a process, especially of a Gaussian process is its covariance function. Traditional statistical models fit
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