Modeling Spatial Covariance Functions Inkyung Choi Purdue University
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Purdue University Purdue e-Pubs Open Access Dissertations Theses and Dissertations Summer 2014 Modeling spatial covariance functions InKyung Choi Purdue University Follow this and additional works at: https://docs.lib.purdue.edu/open_access_dissertations Part of the Applied Statistics Commons Recommended Citation Choi, InKyung, "Modeling spatial covariance functions" (2014). Open Access Dissertations. 246. https://docs.lib.purdue.edu/open_access_dissertations/246 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. *UDGXDWH6FKRRO(7')RUP 5HYLVHG 0114 PURDUE UNIVERSITY GRADUATE SCHOOL Thesis/Dissertation Acceptance 7KLVLVWRFHUWLI\WKDWWKHWKHVLVGLVVHUWDWLRQSUHSDUHG %\ InKyung Choi (QWLWOHG MODELING SPATIAL COVARIANCE FUNCTIONS Doctor of Philosophy )RUWKHGHJUHHRI ,VDSSURYHGE\WKHILQDOH[DPLQLQJFRPPLWWHH Hao Zhang Bo Li Xiao Wang Tonglin Zhang To the best of my knowledge and as understood by the student in the Thesis/Dissertation Agreement, Publication Delay, and Certification/Disclaimer (Graduate School Form 32), this thesis/dissertation adheres to the provisions of Purdue University’s “Policy on Integrity in Research” and the use of copyrighted material. Hao Zhang $SSURYHGE\0DMRU3URIHVVRU V BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB $SSURYHGE\ Rebecca W. Doerge 08/29/2014 +HDGRIWKHDepartment *UDGXDWH3URJUDP 'DWH MODELING SPATIAL COVARIANCE FUNCTIONS A Dissertation Submitted to the Faculty of Purdue University by InKyung Choi In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2014 Purdue University West Lafayette, Indiana ii ACKNOWLEDGMENTS First of all, I want to express my sincere gratitude to my advisor Professor Hao Zhang. He encouraged independent thinking and supported my ideas, which allowed me to freely pursue and explore topics that interested me, and whenever I went astray, his insightful comments put me back on the right track. I am also greatly indebted to Professor Bo Li who had been my advisor for the first three years of the Ph.D. program. I cannot overstate her influence at the beginning of my research in the field of Spatial Statistics. Her extensive knowledge about the field and passion for research had been an inspiration to me. Even after moving to the University of Illinois at Urbana-Champaign, she has always taken good care of me and put herself out of the way to support me. I appreciate my committee members; Professor Xiao Wang and Professor Tonglin Zhang for their helpful input and warm encouragement. The support of the Depart- ment of Statistics and the staff members should not go unmentioned for it was only through their consistent help and assistance that I could complete my study. I was fortunate to study at university that hosts a great consulting center like Statistical Consulting Service center. While working at the center, I learnt a great deal about how to apply statistics in practice, knowledge that can not be easily obtained in a classroom. The invaluable experience that I had obtained through working with clients from various fields will be of a great asset for the rest of my professional career and I am thankful for all the professors, colleagues and staff members of the center, and especially Professor Bruce Craig for his guidance and mentoring. In a seemingly endless fight with oneself and facing the battle all alone in a foreign country, I often found facets of myself that I had never truly realized before. Hence the last five years has not only been about the pursuit of academic knowledge but also a journey to understand myself, and perhaps the world around me as well. My friends iii both inside and outside the department have been wonderful companions along the journey and I'm grateful to them for being good friends and teachers during the important time of my life. Lastly, I want to thank my family in Korea. Although separated by continents, knowing that I have people who always love and support me without condition gave me strength and courage to go on even in the darkest night. I'm forever indebted for their unwavering devotion which is the true force that has shaped who I am now. iv TABLE OF CONTENTS Page LIST OF TABLES :::::::::::::::::::::::::::::::: vi LIST OF FIGURES ::::::::::::::::::::::::::::::: vii ABSTRACT ::::::::::::::::::::::::::::::::::: viii 1 INTRODUCTION :::::::::::::::::::::::::::::: 1 1.1 Motivation :::::::::::::::::::::::::::::::: 1 1.2 Preliminaries :::::::::::::::::::::::::::::: 2 1.2.1 Covariance function of random fields ::::::::::::: 2 1.2.2 Spectral representation of stationary covariance functions :: 4 1.2.3 Some nonstationary covariance functions ::::::::::: 5 1.2.4 Spatial random effect model :::::::::::::::::: 6 1.3 Overview ::::::::::::::::::::::::::::::::: 8 2 NONPARAMETRIC COVARIANCE ESTIMATION ::::::::::: 9 2.1 Introduction ::::::::::::::::::::::::::::::: 9 2.2 Nonparametric Covariance Estimation :::::::::::::::: 12 2.2.1 Construction of completely monotone function :::::::: 12 2.2.2 Estimation :::::::::::::::::::::::::::: 14 2.3 Extension to Space-Time Covariance Function :::::::::::: 17 2.4 Simulation Study :::::::::::::::::::::::::::: 19 2.4.1 Simulation set up :::::::::::::::::::::::: 20 2.4.2 Number of knots and order of B-splines :::::::::::: 21 2.4.3 Comparison with parametric models ::::::::::::: 22 2.4.4 Simulation with space-time covariance function ::::::: 25 2.5 Two Data Examples :::::::::::::::::::::::::: 27 2.5.1 Indiana July mean temperature data ::::::::::::: 27 2.5.2 Irish wind data ::::::::::::::::::::::::: 29 2.6 Conclusion :::::::::::::::::::::::::::::::: 33 3 ANALYSIS OF TELECONNECTION ::::::::::::::::::: 35 3.1 Introduction ::::::::::::::::::::::::::::::: 35 3.2 ENSO Teleconnection to North America ::::::::::::::: 38 3.3 Model for Time-varying Teleconnection :::::::::::::::: 42 3.3.1 Model :::::::::::::::::::::::::::::: 43 3.3.2 Estimation :::::::::::::::::::::::::::: 45 v Page 3.3.3 Empirical confidence interval for covariance of spatial random effect ::::::::::::::::::::::::::::::: 48 3.4 Data Analysis :::::::::::::::::::::::::::::: 50 3.4.1 Teleconnection analysis ::::::::::::::::::::: 50 3.4.2 Spatial prediction :::::::::::::::::::::::: 57 3.5 Conclusion :::::::::::::::::::::::::::::::: 63 REFERENCES :::::::::::::::::::::::::::::::::: 65 VITA ::::::::::::::::::::::::::::::::::::::: 70 vi LIST OF TABLES Table Page 2.1 Mean and standard error (in parenthesis) of integrated squared error (ISE) and mean squared prediction error (MSPE) of nonparametric model at different knot number m and order of B-splines p (unit: 10−2). :::: 23 2.2 Mean and standard error (in parenthesis) of integrated squared error (ISE) and mean squared prediction error (MSPE) of three parametric models (spherical, Mat´ernand Gaussian) and the nonparametric model (with or- der of B-splines p = 3 and knot number m = 5), together with the MSPE of true covariance model (unit: 10−2). :::::::::::::::::: 24 2.3 Mean and standard error (in parenthesis) of integrated squared error (ISE) and mean squared prediction error (MSPE) of the nonparametric space- time covariance model at different knot number (mφ; m ) and order of B-splines (pφ; p ) and of the parametric models (unit: 10−2). :::::: 26 2.4 Mean squared prediction error (MSPE) of one-day ahead prediction of Irish wind data. ::::::::::::::::::::::::::::::: 32 3.1 Mean of mean squared prediction error (MSPE) and logarithm score (LogS) of prediction based 1) Two regions : uses both North America data and Ni~no3region data; and 2) One region : uses data from the same region as the region of the dropped-site ::::::::::::::::::::::: 63 vii LIST OF FIGURES Figure Page [q] 2.1 Example of functions ffj : j = 1; : : : ; m + pg for m = 2 and p = 3, [q] and two covariance functions constructed based on those fj : C1(h) = [2] 2 [2] 2 [2] 2 [2] 2 5f1 (h ) + 0:5f5 (h ) and C2(h) = 0:7f3 (h ) + 0:1f4 (h ). ::::::: 15 2.2 86 weather stations in Indiana with July 2011 mean temperatures, together with temperature surface interpolated using the fitted nonparametric co- variance function :::::::::::::::::::::::::::::: 28 2.3 Emprical covariance estimates (dots) and fitted covariance functions for Indiana July 2011 temperature data (distance unit: 10 km) ::::::: 29 2.4 Empirical covariance estimates (dots) of Irish wind data and fitted non- parametric model (solid line) at four different sets of space-time lags (dis- tance unit: 100 km) :::::::::::::::::::::::::::: 31 3.1 Nino3 region and North America region ::::::::::::::::: 41 3.2 Temporal variation of residual variance (top: Ni~no3region; bottom: North America) before standardization (blue lines) and after the standardization (red lines) :::::::::::::::::::::::::::::::::: 51 3.3 Location of centers of spatial basis functions :::::::::::::: 52 3.4 Estimated covariancesσ ^ij(t) from 1871 to 1990 ::::::::::::: 53 3.5 Estimated covariancesσ ^ij(t) from 1871 to 1990 ::::::::::::: 54 3.6 Estimated covariance σbij(t) for spatial random effects α1; : : : ; α6 :::: 55 3.7 Estimated covariance σbij(t) for spatial random effects α7; : : : ; α12 ::: 56 3.8 Mean and empirical confidence interval of covariances associated with α1; : : : ; α4 :::::::::::::::::::::::::::::::::: 58 3.9 Mean and empirical confidence interval of covariances associated with α6; α7; α8 and α10 :::::::::::::::::::::::::::::: 59 3.10 Squared Frobenius norm of estimated cross-region covariance matrix :: 60 3.11 Location of centers of spatial basis functions