Über die Autorin bzw. den Autor

Gidon Eshel is Bard Center Fellow at Bard College.

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"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

Aus dem Klappentext

"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

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Spatiotemporal Data Analysis

PRINCETON UNIVERSITY PRESS

Contents

Preface......................................................................................xiAcknowledgments..............................................................................xvONE Introduction and Motivation..............................................................1TWO Notation and Basic Operations............................................................3THREE Matrix Properties, Fundamental Spaces, Orthogonality...................................12FOUR Introduction to Eigenanalysis...........................................................47FIVE The Algebraic Operation of SVD..........................................................75SIX The Gray World of Practical Data Analysis: An Introduction to Part 2.....................95SEVEN Statistics in Deterministic Sciences: An Introduction..................................96EIGHT Autocorrelation........................................................................109NINE Regression and Least Squares............................................................126TEN. THE FUNDAMENTAL THEOREM OF LINEAR ALGEBRA...............................................197ELEVEN. EMPIRICAL ORTHOGONAL FUNCTIONS.......................................................200TWELVE. THE SVD ANALYSIS OF TWO FIELDS.......................................................261THIRTEEN. SUGGESTED HOMEWORK.................................................................276Index........................................................................................313

Chapter One

Before you start working your way through this book, you may ask yourself—Why analyze data? This is an important, basic question, and it has several compelling answers.

The simplest need for data analysis arises most naturally in disciplines addressing phenomena that are, in all likelihood, inherently nondeterministic (e.g., feelings and psychology or stock market behavior). Since such fields of knowledge are not governed by known fundamental equations, the only way to generalize disparate observations into expanded knowledge is to analyze those observations. In addition, in such fields predictions are entirely dependent on empirical models of the types discussed in chapter 9 that contain parameters not fundamentally constrained by theory. Finding these models' numerical values most suitable for a particular application is another important role of data analysis.

A more general rationale for analyzing data stems from the complementary relationship of empirical and theoretical science and dominates contexts and disciplines in which the studied phenomena have, at least in principle, fully knowable and usable fundamental governing dynamics (see chapter 7). In these contexts, best exemplified by physics, theory and observations both vie for the helm. Indeed, throughout the history of physics, theoretical predictions of yet unobserved phenomena and empirical observations of yet theoretically unexplained ones have alternately fixed physics' ropes. When theory leads, its predictions must be tested against experimental or observational data. When empiricism is at the helm, coherent, reproducible knowledge is systematically and carefully gleaned from noisy, messy observations. At the core of both, of course, is data analysis.

Empiricism's biggest triumph, affording it (ever so fleetingly) the leadership role, arises when novel data analysis-based knowledge—fully acquired and processed—proves at odds with relevant existing theories (i.e., equations previously thought to govern the studied phenomenon fail to explain and reproduce the new observations). In such cases, relatively rare but game changing, the need for a new theory becomes apparent. When a new theory emerges, it either generalizes existing ones (rendering previously reigning equations a limiting special case, as in, e.g., Newtonian vs. relativistic gravity), or introduces an entirely new set of equations. In either case, at the root of the progress thus achieved is data analysis.

Once a new theory matures and its equation set becomes complete and closed, one of its uses is model-mediated predictions. In this application of theory, another rationale for data analysis sometimes emerges. It involves phenomena (e.g., fluid turbulence) for which governing equations may exist in principle, but their applications to most realistic situations is impossibly complex and high-dimensional. Such phenomena can thus be reasonably characterized as fundamentally deterministic yet practically stochastic. As such, practical research and modeling of such phenomena fall into the first category above, that addressing inherently nondeterministic phenomena, in which better mechanistic understanding requires better data and better data analysis.

Data analysis is thus essential for scientific progress. But is the level of algebraic rigor characteristic of some of this book's chapters necessary? After all, in some cases we can use some off-the-shelf spreadsheet-type black box for some rudimentary data analysis without any algebraic foundation. How you answer this question is a subjective matter. My view is that while in a few cases some progress can be made without substantial understanding of the underlying algebraic machinery and assumptions, such analyses are inherently dead ends in that they can be neither generalized nor extended beyond the very narrow, specific question they address. To seriously contribute to any of the progress routes described above, in the modular, expandable manner required for your work to potentially serve as the foundation of subsequent analyses, there is no alternative to thorough, deep knowledge of the underlying linear algebra.

Chapter Two

While algebraic basics can be found in countless texts, I really want to make this book as self contained as reasonably possible. Consequently, in this chapter I introduce some of the basic players of the algebraic drama about the unfold, and the uniform notation I have done my best to adhere to in this book. While chapter 3 is a more formal introduction to linear algebra, in this introductory chapter I also present some of the most basic elements, and permitted manipulations and operations, of linear algebra.

1. Scalar variables: Scalars are given in lowercase, slanted, Roman or Greek letters, as in a, b, x, a, β, θ.

2. Stochastic processes and variables: A stochastic variable is denoted by an italicized uppercase X. A particular value, or realization, of the process X is denoted by x.

3. Matrix variables: Matrices are the most fundamental building block of linear algebra. They arise in many, highly diverse situations, which we will get to later. A matrix is a rectangular array of numbers, e.g.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.1)

A matrix is said to be M x N (M by N) when it comprises M rows and N columns. A vector is a special case of matrix for which either M or N equals 1. By convention, unless otherwise stated, we will treat vectors as column vectors.

4. Fields: Fields are sets of elements satisfying the addition and multiplication field axioms (associativity, commutativity, distributivity, identity, and...