Spatiotemporal Data Analysis

"A severe thunderstorm morphs into a tornado that cuts a swath of destruction through Oklahoma. How do we study the storm's mutation into a deadly twister? Avian flu cases are reported in China. How do we characterize the spread of the flu, potentially preventing an epidemic? The way to answer important questions like these is to analyze the spatial and temporal characteristics--origin, rates, and frequencies--of these phenomena. This comprehensive text introduces advanced undergraduate students, graduate students, and researchers to the statistical and algebraic methods used to analyze spatiotemporal data in a range of fields, including climate science, geophysics, ecology, astrophysics, and medicine. Gidon Eshel begins with a concise yet detailed primer on linear algebra, providing readers with the mathematical foundations needed for data analysis. He then fully explains the theory and methods for analyzing spatiotemporal data, guiding readers from the basics to the most advanced applications. This self-contained, practical guide to the analysis of multidimensional data sets features a wealth of real-world examples as well as sample homework exercises and suggested exams"-- Provided by publisher Cover ......Page 1 Spatiotemporal Data Analysis......Page 2 Title ......Page 4 Copyright......Page 5 Dedication......Page 6 Contents......Page 8 Preface......Page 12 Acknowledgments......Page 16 PART 1. FOUNDATIONS......Page 18 ONE Introduction and Motivation......Page 20 TWO Notation and Basic Operations......Page 22 3.1 Vector Spaces......Page 31 3.2 Matrix Rank......Page 37 3.3 Fundamental Spaces Associated with AÎR M×N......Page 42 3.4 Gram-Schmidt Orthogonalization......Page 60 3.5 Summary......Page 64 4.1 Preface......Page 66 4.2 Eigenanalysis Introduced......Page 67 4.3 Eigenanalysis as Spectral Representation......Page 76 4.4 Summary......Page 92 5.1 SVD Introduced......Page 94 5.2 Some Examples......Page 99 5.3 SVD Applications......Page 105 5.4 Summary......Page 109 PART 2. METHODS OF DATA ANALYSIS......Page 112 SIX The Gray World of Practical Data Analysis: An Introduction to Part 2......Page 114 SEVEN Statistics in Deterministic Sciences: An Introduction......Page 115 7.1 Probability Distributions......Page 118 7.2 Degrees of Freedom......Page 123 EIGHT Autocorrelation......Page 128 8.1 Theoretical Autocovariance and Autocorrelation Functions of AR(1) and AR(2)......Page 137 8.2 Acf-Derived Timescale......Page 142 8.3 Summary of Chapters 7 and 8......Page 144 9.2 Setting Up the Problem......Page 145 9.3 The Linear System Ax = b......Page 149 9.4 Least Squares: The SVD View......Page 163 9.5 Some Special Problems Giving Rise to Linear Systems......Page 168 9.6 Statistical Issues in Regression Analysis......Page 184 9.7 Multidimensional Regression and Linear Model Identification......Page 204 9.8 Summary......Page 214 10.2 The Forward Problem......Page 216 10.3 The Inverse Problem......Page 217 11.1 Introduction......Page 219 11.3 Reshaping Multidimensional Data Sets for EOF Analysis......Page 220 11.4 Forming Anomalies and Removing Time Mean......Page 223 11.5 Missing Values, Take 1......Page 224 11.6 Choosing and Interpreting the Covariability Matrix......Page 227 11.7 Calculating the EOFs......Page 237 11.8 Missing Values, Take 2......Page 244 11.9 Projection Time Series, the Principal Components......Page 247 11.10 A Final Realistic and Slightly Elaborate Example: Southern New York State Land Surface Temperature......Page 253 11.11 Extended EOF Analysis, EEOF......Page 263 11.12 Summary......Page 279 TWELVE. THE SVD ANALYSIS OF TWO FIELDS......Page 280 12.1 A Synthetic Example......Page 284 12.2 A Second Synthetic Example......Page 287 12.3 A Real Data Example......Page 290 12.4 EOFs as a Prefilter to SVD......Page 292 12.5 summary......Page 293 13.1 Homework 1, Corresponding to Chapter 3......Page 295 13.2 Homework 2, Corresponding to Chapter 3......Page 302 13.3 Homework 3, Corresponding to Chapter 3......Page 309 13.4 Homework 4, Corresponding to Chapter 4......Page 311 13.5 Homework 5, Corresponding to Chapter 5......Page 315 13.6 Homework 6, Corresponding to Chapter 8......Page 319 13.7 A Suggested Midterm Exam......Page 322 13.8 A Suggested Final Exam......Page 330 Index......Page 332 Cover 1 Spatiotemporal Data Analysis 2 Title 4 Copyright 5 Dedication 6 Contents 8 Preface 12 Acknowledgments 16 PART 1. FOUNDATIONS 18 ONE Introduction and Motivation 20 TWO Notation and Basic Operations 22 THREE Matrix Properties, Fundamental Spaces, Orthogonality 31 3.1 Vector Spaces 31 3.2 Matrix Rank 37 3.3 Fundamental Spaces Associated with AÎR M×N 42 3.4 Gram-Schmidt Orthogonalization 60 3.5 Summary 64 FOUR Introduction to Eigenanalysis 66 4.1 Preface 66 4.2 Eigenanalysis Introduced 67 4.3 Eigenanalysis as Spectral Representation 76 4.4 Summary 92 FIVE The Algebraic Operation of SVD 94 5.1 SVD Introduced 94 5.2 Some Examples 99 5.3 SVD Applications 105 5.4 Summary 109 PART 2. METHODS OF DATA ANALYSIS 112 SIX The Gray World of Practical Data Analysis: An Introduction to Part 2 114 SEVEN Statistics in Deterministic Sciences: An Introduction 115 7.1 Probability Distributions 118 7.2 Degrees of Freedom 123 EIGHT Autocorrelation 128 8.1 Theoretical Autocovariance and Autocorrelation Functions of AR(1) and AR(2) 137 8.2 Acf-Derived Timescale 142 8.3 Summary of Chapters 7 and 8 144 NINE Regression and Least Squares 145 9.1 Prologue 145 9.2 Setting Up the Problem 145 9.3 The Linear System Ax = b 149 9.4 Least Squares: The SVD View 163 9.5 Some Special Problems Giving Rise to Linear Systems 168 9.6 Statistical Issues in Regression Analysis 184 9.7 Multidimensional Regression and Linear Model Identification 204 9.8 Summary 214 TEN. THE FUNDAMENTAL THEOREM OF LINEAR ALGEBRA 216 10.1 Introduction 216 10.2 The Forward Problem 216 10.3 The Inverse Problem 217 ELEVEN. EMPIRICAL ORTHOGONAL FUNCTIONS 219 11.1 Introduction 219 11.2 Data Matrix Structure Convention 220 11.3 Reshaping Multidimensional Data Sets for EOF Analysis 220 11.4 Forming Anomalies and Removing Time Mean 223 11.5 Missing Values, Take 1 224 11.6 Choosing and Interpreting the Covariability Matrix 227 11.7 Calculating the EOFs 237 11.8 Missing Values, Take 2 244 11.9 Projection Time Series, the Principal Components 247 11.10 A Final Realistic and Slightly Elaborate Example: Southern New York State Land Surface Temperature 253 11.11 Extended EOF Analysis, EEOF 263 11.12 Summary 279 TWELVE. THE SVD ANALYSIS OF TWO FIELDS 280 12.1 A Synthetic Example 284 12.2 A Second Synthetic Example 287 12.3 A Real Data Example 290 12.4 EOFs as a Prefilter to SVD 292 12.5 summary 293 THIRTEEN. SUGGESTED HOMEWORK 295 13.1 Homework 1, Corresponding to Chapter 3 295 13.2 Homework 2, Corresponding to Chapter 3 302 13.3 Homework 3, Corresponding to Chapter 3 309 13.4 Homework 4, Corresponding to Chapter 4 311 13.5 Homework 5, Corresponding to Chapter 5 315 13.6 Homework 6, Corresponding to Chapter 8 319 13.7 A Suggested Midterm Exam 322 13.8 A Suggested Final Exam 330 Index 332

A severe thunderstorm morphs into a tornado that cuts a swath of destruction through Oklahoma. How do we study the storm's mutation into a deadly twister? Avian flu cases are reported in China. How do we characterize the spread of the flu, potentially preventing an epidemic? The way to answer important questions like these is to analyze the spatial and temporal characteristics-origin, rates, and frequencies-of these phenomena. This comprehensive text introduces advanced undergraduate students, graduate students, and researchers to the statistical and algebraic methods used to analyze spatiotemporal data in a range of fields, including climate science, geophysics, ecology, astrophysics, and medicine.

Gidon Eshel begins with a concise yet detailed primer on linear algebra, providing readers with the mathematical foundations needed for data analysis. He then fully explains the theory and methods for analyzing spatiotemporal data, guiding readers from the basics to the most advanced applications. This self-contained, practical guide to the analysis of multidimensional data sets features a wealth of real-world examples as well as sample homework exercises and suggested exams.

A severe thunderstorm morphs into a tornado that cuts a swath of destruction through Oklahoma. How do we study the storm’s mutation into a deadly twister? Avian flu cases are reported in China. How do we characterize the spread of the flu, potentially preventing an epidemic? The way to answer important questions like these is to analyze the spatial and temporal characteristics—origin, rates, and frequencies—of these phenomena. This book introduces advanced undergraduate students, graduate students, and researchers to the statistical and algebraic methods used to analyze spatiotemporal data in a range of fields, including climate science, geophysics, ecology, astrophysics, and medicine. The book begins with a concise yet detailed primer on linear algebra, providing readers with the mathematical foundations needed for data analysis. It then fully explains the theory and methods for analyzing spatiotemporal data, guiding readers from the basics to the most advanced applications. This self-contained, practical guide to the analysis of multidimensional data sets features a wealth of real-world examples as well as sample homework exercises and suggested exams.