Spatiotemporal Data Analysis

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