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.

Preface xiAcknowledgments 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