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Spatiotemporal Data Analysis

Gidon Eshel
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

This is a book about multidimensional data analysis. Although the word “spatiotemporal” in the title makes the book sound more specialized, many of the techniques described are broadly relevant to all kinds of data analysis. The author mainly focuses on data that possess a well defined covariance matrix and have at least one dimension in which order is important. A vector time series is the most natural example.

The author bases his book on class notes for an upper undergraduate-beginning graduate course in data analysis for students in fields ranging from astronomy to oceanography with widely varying mathematical backgrounds. He writes, “Multidimensional data analysis almost universally boils down to linear algebra.” Yes! Accordingly he devotes about the first third of the book to that subject. Depending on the class, an instructor would probably pick and choose from the topics here. The most important idea from this part is Singular Value Decomposition because it is far and away the most valuable tool and is used extensively throughout the second part of the book.

That second part concentrates on methods of data analysis. Its short introduction hits just the right tone with the title: “The Gray World of Practical Data Analysis”, where vectors may be only sort of orthogonal, matrices almost singular, and quality of the data often at least a little bit doubtful. The topics begin with a little statistics and then go on to autocorrelation, regression and least squares, and finally empirical orthogonal functions.

The author’s terminology is troublesome. He is not careful about definitions, and the meaning of some of the terms is rather slippery. Early on, for example, he discusses eigenvalues and eigenvectors as part of a chapter on “eigenanalysis”. At the end of one of the sections he then writes, “Eigenanalysis is a form of a spectrum”, a strange mangling of process and outcome. He’s not too definite about what he means by “spectrum” either, and says instead that there are various definitions. At another point he refers to a “singular spectrum”, also undefined, that seems to mean a collection of eigenvalues — some of which are zero or nearly zero. “Empirical orthogonal functions” were also a puzzle to me until I realized that the author meant new bases for data vectors, usually eigenvectors of a covariance matrix.

The author prevents examples in image compression, noise filtering, meteorology and climatology. The raw material of the examples is great, the execution less so. The author tends to go through the examples too quickly for a text at this level. I think that the real business of learning data analysis comes in working with real data, and carefully worked-through examples are perhaps the best teaching tools.

I wanted very much to like this book and was progressively disappointed as looked at it more carefully. The author is definitely aimed in the right direction for an introductory course, and he provides a good mixture of theory, technique and street-smarts about data. Nonetheless, the execution still needs a lot of work.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

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