# Spatiotemporal Data Analysis

Gidon Eshel
Edition: STU - Student edition
Pages: 368
https://www.jstor.org/stable/j.ctt7zthhv

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xiv)
4. Acknowledgments
(pp. xv-xvi)
5. ### PART 1. FOUNDATIONS

• ONE Introduction and Motivation
(pp. 1-2)

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...

• TWO Notation and Basic Operations
(pp. 3-11)

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...

• THREE Matrix Properties, Fundamental Spaces, Orthogonality
(pp. 12-46)

For our purposes, it is sufficient to think of a vector space as the set of all vectors of a certain type. While the vectors need not be actual vectors (they can also be functions, matrices, etc.), in this book “vectors” are literally column vectors of real number elements, which means we consider vector spaces over$\mathbb{R}$.

The lowest dimensional vector space is${\mathbb{R}^{0}}$, comprising a single point, 0; not too interesting. In$\mathbb{R}$, the real line, one and only one kind of inhabitant is found: 1-vectors (scalars) whose single element is any one of the real...

• FOUR Introduction to Eigenanalysis
(pp. 47-74)

Eigenanalysis and its numerous offsprings form the suite of algebraic operations most important and relevant to data analysis, as well as to dynamical systems, modeling, numerical analysis, and related key branches of applied mathematics. This chapter introduces, and places in a broader context, the algebraic operation of eigen-decomposition.

To have eigen-decomposition, a matrix must be square. Yet data matrices are very rarely square. The direct relevance of eigen-decomposition to data analysis is therefore limited. Indirectly, however, generalized eigenanalysis is enormously important to studying data matrices, as we will see later. Because of the centrality of generalized eigenanalysis to data matrices,...

• FIVE The Algebraic Operation of SVD
(pp. 75-92)

In the preceding chapter we discussed the eigenvalue/eigenvector diagonalization of a matrix. Perhaps the biggest problem for this to be very useful in data analysis is the restriction to square matrices. We have already emphasized time and again that data matrices, unlike dynamical operators, are rarely square. The algebraic operation of the singular value decomposition, SVD, is the answer. Note the distinction between the data analysis method widely known as SVD and the actual algebraic machinery. The former uses the latter, but isn’t the latter! In this chapter, I describe the method, postponing the discussion of the analytic tool to...

6. ### PART 2. METHODS OF DATA ANALYSIS

• SIX The Gray World of Practical Data Analysis: An Introduction to Part 2
(pp. 95-95)

This second part of the book is the crux of the matter: how to analyze actual data. While part 2 builds on part 1, especially on linear algebra fundamentals covered in part 1, the two are not redundant. The main distinguishing characteristic of part 2 is its nuanced grayness.

In the ideal world of algebra (and thus in most of part 1), things are black or white: two vectors are either mutually orthogonal or not, real numbers are either zero or not, a vector either solves a linear system or does not. By contrast, realistic data analysis, the province of...

• SEVEN Statistics in Deterministic Sciences: An Introduction
(pp. 96-108)

Data analysis is a branch of statistical science. Perhaps the first question that comes to mind about statistics in the natural sciences, especially those—like physics or engineering—that are based, at least in principle, on well-defined, closed governing equations, is “who needs statistics!?” After all, most physical phenomena of concern to such sciences are governed by fundamental, mostly known, physics. It follows, then, that we should be able to write down the governing equations of the system under investigation and use the dynamical system thus constructed to study the system and predict its future states. The answer to this...

• EIGHT Autocorrelation
(pp. 109-125)

To estimate the degrees of freedom (df) of a given sample, one tool at our disposal is the autocorrelation function (acf),ρ. Recall that the common thread of the above examples aboutdfwas that when neighboring data points are not entirely independent of one another, there is some redundancy in the time series. Linearly quantifying this redundancy is one of the jobs of the acf.

Let’s first be sure you remember the correlation coefficient of two random variablesXandY,${{\rho }_{X,Y}}=\text{cor}(X,Y)=\frac{\text{cov}(X,Y)}{{{\sigma }_{X}}{{\sigma }_{Y}}}=\frac{E\left[ \left( X-{{\mu }_{X}} \right)\left( Y-{{\mu }_{Y}} \right) \right]}{{{\sigma }_{X}}{{\sigma }_{Y}}},\caption {(8.1)}$where “cor” and “cov” denote correlation and covariance,$\sigma _{X}^{2}$and$\sigma _{Y}^{2}$areX’s andY’s expected values...

• NINE Regression and Least Squares
(pp. 126-196)

The focus of this chapter, linear regression, is the process of identifying the unique model that best explains a set of observed data among a specified class of general models. Regression thus occupies a uniquely important position at the very interface of modeling and data analysis. Regression arises very often, in various guises, in handling and analyzing data. Since it is one of the most basic, useful, and frequently employed data analysis tools, and since we will need some understanding of regression in later sections, below we discuss regression in some detail. However, the topic is very broad, and its...

• TEN The Fundamental Theorem of Linear Algebra
(pp. 197-199)

This brief chapter, somewhat of a return to fundamentals, had to wait until we fully mastered regression. The chapter summarizes pictorially some of the linear algebraic foundations we have discussed thus far (most notably, section 3.3.1 and chapter 9) by revisiting the fundamental theorem of linear algebra, the unifying view of matrices, vectors, and their interactions. To make our discussion helpful and informal yet rigorous, and to complement the slightly more formal introduction of the basic ideas in section 3.3.1, here we emphasize the theorem’s pictorial representation.

Figure 10.1 shows schematically what happens when$\bold{A}\in {{\mathbb{R}}^{M\times N}}$maps an$\bold x \in {{\mathbb{R}}^{N}}$from A’s...

• ELEVEN Empirical Orthogonal Functions
(pp. 200-260)

One of the most useful and common eigen-techniques in data analysis is the construction of empirical orthogonal functions, EOFs. EOFs are a transform of the data; the original set of numbers is transformed into a different set with some desirable properties. In this sense the EOF transform is similar to other transforms, such as the Fourier or Laplace transforms. In all these cases, we project the original data onto a set of functions, thus replacing the original data with the set of projection coefficients on the chosen new set of basis vectors. However, the choice of the specific basis set...

• TWELVE The SVD Analysis of Two Fields
(pp. 261-275)

In the preceding chapter, we discussed one of the many methods available for simultaneously analyzing more than one data set. While powerful and useful (especially for unveiling favored state evolution pathways), the EEOF procedure has some important limitations. Notably, because the state dimensions rapidly expand as state vectors are appended end-to-end, EEOF analysis may not always be numerically tractable. For analyzing two data sets, taking note of their cross-covariance but not explicitly of individual sets’ covariance, the SVD method is the most natural.

SVD analysis (which, unfortunately, shares its name with the related but separate mathematical operation introduced in chapter...

• THIRTEEN Suggested Homework
(pp. 276-312)

For$\text{A}=\left( \begin{array}{*{35}{r}} 3 & -3 & -1 & 7 & 7 & -2 \\ 13 & 14 & 12 & 10 & 7 & 0 \\ 2 & 14 & 10 & 11 & 11 & 6 \\ 9 & -2 & 2 & 4 & 14 & -4 \\ \end{array} \right),$$\text{B}=\left( \begin{array}{*{35}{r}} 2 & 4 & -2 & -7 \\ -3 & 3 & 1 & -5 \\ 6 & 10 & 1 & -8 \\ 3 & 4 & 4 & 6 \\ 9 & 9 & 5 & 12 \\ \end{array} \right),$and$\text{C}=\left( \begin{array}{*{35}{r}} 2 & 0 & 3 & 2 \\ 1 & 5 & -1 & 6 \\ 2 & 1 & 4 & 6 \\ \end{array} \right),$please provide

(a)M,N

(b) rankq; is the matrix full rank? rank deficient?

(c) U and all necessary Eis is

(d) basis$[{\cal R}(\cdot )]\subseteq {{\mathbb{R}}^{M}}$

(e) basis$[{\cal N}({{\cdot }^{T}})]\subseteq {{\mathbb{R}}^{M}}$

(f) basis$[{\cal R}({{\cdot }^{T}})]\subseteq {{\mathbb{R}}^{N}}$

(g) basis$[({\cal N}(\cdot )]\subseteq {{\mathbb{R}}^{N}}$

and show that

(h)${\cal R}(\cdot )\bot {\cal N}({{\cdot }^{T}})$

(i)${\cal R}({{\cdot }^{T}})\bot {\cal N}(\cdot )$

(j)${\cal R}(\cdot )+{\cal N}({{\cdot }^{T}})={{\mathbb{R}}^{M}}$

(k)${\cal R}({{\cdot }^{T}})+{\cal N}(\cdot )={{\mathbb{R}}^{N}}$

In many of the tests below, we need to distinguish values that are practically but not identically zero. For this purpose, format(ʹlongʹ) is useful, as it provides a far better numerical resolution, e.g.,

>> format(ʹlongʹ); disp(2/3)

0.666666666666667

>> format(ʹshortʹ); disp(2/3)

0.6667

(a) To getMandN, use [Ma, Na=size(A), [Mb, Nb=size...

7. Index
(pp. 313-317)