Spatiotemporal Data Analysis

"A perfect and much-needed book for students and professionals tackling the complexities of data analysis in space and time. In my twenty-five years teaching in the environmental sciences, I've not encountered such a comprehensive and well-structured book that eloquently lays out the mathematical basis and data-analysis tools required to understand real-world environmental data structures with practical examples, applications, and problem assignments."--Alfredo Huete, University of Technology Sydney, Australia

"Spatiotemporal Data Analysis is accessible and applicable without sacrificing rigor. The key is a steady stream of well-chosen examples and, most unusual in any textbook, a distinctive narrative voice that guides readers through the material, explaining the details while making sure the big picture is always in view. It will become an essential text for earth scientists and many others who analyze spatiotemporal data."--Mark Cane, Columbia University

"This book offers an excellent survey of the mathematical aspects of spatiotemporal data analysis and will be useful to geoscientists in such applications as collecting, archiving, and interpreting satellite data. While treating some rather abstract matters, Eshel does not drown the reader in overly opaque notation but instead derives results and illustrates them with interesting examples, both numerical and conceptual."--Gerald R. North, Texas A&M University

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 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.

"Spatiotemporal Data Analysis is based on a lecture course that influenced the thesis work of every graduate student I knew in the department. This is due to the applicability of the material to a broad range of topics, and to Eshel's clear and insightful presentation."--David Archer, University of Chicago

"A perfect and much-needed book for students and professionals tackling the complexities of data analysis in space and time. In my twenty-five years teaching in the environmental sciences, I've not encountered such a comprehensive and well-structured book that eloquently lays out the mathematical basis and data-analysis tools required to understand real-world environmental data structures with practical examples, applications, and problem assignments."--Alfredo Huete, University of Technology Sydney, Australia

"Spatiotemporal Data Analysis is accessible and applicable without sacrificing rigor. The key is a steady stream of well-chosen examples and, most unusual in any textbook, a distinctive narrative voice that guides readers through the material, explaining the details while making sure the big picture is always in view. It will become an essential text for earth scientists and many others who analyze spatiotemporal data."--Mark Cane, Columbia University

"This book offers an excellent survey of the mathematical aspects of spatiotemporal data analysis and will be useful to geoscientists in such applications as collecting, archiving, and interpreting satellite data. While treating some rather abstract matters, Eshel does not drown the reader in overly opaque notation but instead derives results and illustrates them with interesting examples, both numerical and conceptual."--Gerald R. North, Texas A&M University

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 perfect and much-needed book for students and professionals tackling the complexities of data analysis in space and time. In my twenty-five years teaching in the environmental sciences, I've not encountered such a comprehensive and well-structured book that eloquently lays out the mathematical basis and data-analysis tools required to understand real-world environmental data structures with practical examples, applications, and problem assignments."--Alfredo Huete, University of Technology Sydney, Australia

"Spatiotemporal Data Analysis is accessible and applicable without sacrificing rigor. The key is a steady stream of well-chosen examples and, most unusual in any textbook, a distinctive narrative voice that guides readers through the material, explaining the details while making sure the big picture is always in view. It will become an essential text for earth scientists and many others who analyze spatiotemporal data."--Mark Cane, Columbia University

"This book offers an excellent survey of the mathematical aspects of spatiotemporal data analysis and will be useful to geoscientists in such applications as collecting, archiving, and interpreting satellite data. While treating some rather abstract matters, Eshel does not drown the reader in overly opaque notation but instead derives results and illustrates them with interesting examples, both numerical and conceptual."--Gerald R. North, TexasA&M University

Preface xi

Acknowledgments xv>

Part 1. Foundations

Chapter One: Introduction and Motivation 1

Chapter Two: Notation and Basic Operations 3

Chapter Three: Matrix Properties, Fundamental Spaces, Orthogonality 12

3.1 Vector Spaces 12

3.2 Matrix Rank 18

3.3 Fundamental Spaces Associated with A d R M # N 23

3.4 Gram-Schmidt Orthogonalization 41

3.5 Summary 45

Chapter Four: Introduction to Eigenanalysis 47

4.1 Preface 47

4.2 Eigenanalysis Introduced 48

4.3 Eigenanalysis as Spectral Representation 57

4.4 Summary 73

Chapter Five: The Algebraic Operation of SVD 75

5.1 SVD Introduced 75

5.2 Some Examples 80

5.3 SVD Applications 86

5.4 Summary 90

Part 2. Methods of Data Analysis

Chapter Six: The Gray World of Practical Data Analysis: An Introduction to Part 2 95

Chapter Seven Statistics in Deterministic Sciences: An Introduction 96

7.1 Probability Distributions 99

7.2 Degrees of Freedom 104

Chapter Eight: Autocorrelation 109

8.1 Theoretical Autocovariance and Autocorrelation Functions of AR(1) and AR(2) 118

8.2 Acf-derived Timescale 123

8.3 Summary of Chapters 7 and 8 125

Chapter Nine: Regression and Least Squares 126

9.1 Prologue 126

9.2 Setting Up the Problem 126

9.3 The Linear System Ax = b 130

9.4 Least Squares: The SVD View 144

9.5 Some Special Problems Giving Rise to Linear Systems 149

9.6 Statistical Issues in Regression Analysis 165

9.7 Multidimensional Regression and Linear Model Identification 185

9.8 Summary 195

Chapter Ten:. The Fundamental Theorem of Linear Algebra 197

10.1 Introduction 197

10.2 The Forward Problem 197

10.3 The Inverse Problem 198

Chapter Eleven:. Empirical Orthogonal Functions 200

11.1 Introduction 200

11.2 Data Matrix Structure Convention 201

11.3 Reshaping Multidimensional Data Sets for EOF Analysis 201

11.4 Forming Anomalies and Removing Time Mean 204

11.5 Missing Values, Take 1 205

11.6 Choosing and Interpreting the Covariability Matrix 208

11.7 Calculating the EOFs 218

11.8 Missing Values, Take 2 225

11.9 Projection Time Series, the Principal Components 228

11.10 A Final Realistic and Slightly Elaborate Example: Southern New York State Land Surface Temperature 234

11.11 Extended EOF Analysis, EEOF 244

11.12 Summary 260

Chapter Twelve:. The SVD Analysis of Two Fields 261

12.1 A Synthetic Example 265

12.2 A Second Synthetic Example 268

12.3 A Real Data Example 271

12.4 EOFs as a Prefilter to SVD 273

12.5 Summary 274

Chapter Thirteen:. Suggested Homework 276

13.1 Homework 1, Corresponding to Chapter 3 276

13.2 Homework 2, Corresponding to Chapter 3 283

13.3 Homework 3, Corresponding to Chapter 3 290

13.4 Homework 4, Corresponding to Chapter 4 292

13.5 Homework 5, Corresponding to Chapter 5 296

13.6 Homework 6, Corresponding to Chapter 8 300

13.7 A Suggested Midterm Exam 303

13.8 A Suggested Final Exam 311

Index 313