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.

Table of Contents:
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