A Guide to Empirical Orthogonal Functions for Climate Data Analysis

by Antonio Navarra, Valeria Simoncini

Overview

Climatology and meteorology have basically been a descriptive science until it became possible to use numerical models, but it is crucial to the success of the strategy that the model must be a good representation of the real climate system of the Earth. Models are required to reproduce not only the mean properties of climate, but also its variability and the strong spatial relations between climate variability in geographically diverse regions. Quantitative techniques were developed to explore the climate variability and its relations between different geographical locations. Methods were borrowed from descriptive statistics, where they were developed to analyze variance of related observations-variable pairs, or to identify unknown relations between variables.

A Guide to Empirical Orthogonal Functions for Climate Data Analysis uses a different approach, trying to introduce the reader to a practical application of the methods, including data sets from climate simulations and MATLAB codes for the algorithms. All pictures and examples used in the book may be reproduced by using the data sets and the routines available in the book .

Though the main thrust of the book is for climatological examples, the treatment is sufficiently general that the discussion is also useful for students and practitioners in other fields.

Supplementary datasets are available via http://extra.springer.com

Product Details

ISBN-13:	9789048137015
Publisher:	Springer Netherlands
Publication date:	02/12/2010
Edition description:	2010
Pages:	151
Product dimensions:	6.40(w) x 9.30(h) x 0.70(d)

Table of Contents

1 Introduction 1

2 Elements of Linear Algebra 5

2.1 Introduction 5

2.2 Elementary Vectors 5

2.3 Scalar Product 6

2.4 Linear Independence and Basis 10

2.5 Matrices 12

2.6 Rank, Singularity and Inverses 16

2.7 Decomposition of Matrices: Eigenvalues and Eigenvectors 17

2.8 The Singular Value Decomposition 19

2.9 Functions of Matrices 21

3 Basic Statistical Concepts 25

3.1 Introduction 25

3.2 Climate Datasets 25

3.3 The Sample and the Population 26

3.4 Estimating the Mean State and Variance 27

3.5 Associations Between Time Series 29

3.6 Hypothesis Testing 32

3.7 Missing Data 36

4 Empirical Orthogonal Functions 39

4.1 Introduction 39

4.2 Empirical Orthogonal Functions 42

4.3 Computing the EOFs 43

4.3.1 EOF and Variance Explained 44

4.4 Sensitivity of EOF Calculation 49

4.4.1 Normalizing the Data 50

4.4.2 Domain of Definition of the EOF 51

4.4.3 Statistical Reliability 55

4.5 Reconstruction of the Data 58

4.5.1 The Singular Value Distribution and Noise 59

4.5.2 Stopping Criterion 62

4.6 A Note on the Interpretation of EOF 64

5 Generalizations: Rotated, Complex, Extended and Combined EOF 69

5.1 Introduction 69

5.2 Rotated EOF 70

5.3 Complex EOF 79

5.4 Extended EOF 87

5.5 Many Field Problems: Combined EOF 90

6 Cross-Covariance and the Singular Value Decomposition 97

6.1 The Cross-Covariance 97

6.2 Cross-Covariance Analysis Using the SVD 99

7 The Canonical Correlation Analysis 107

7.1 The Classical Canonical Correlation Analysis 107

7.2 The Modes 109

7.3 The Barnett-Preisendorfer Canonical Correlation Analysis 114

8 Multiple Linear Regression Methods 123

8.1 Introduction 123

8.1.1 A Slight Digression 125

8.2 A Practical PRO Method 126

8.2.1 A Different Scaling 127

8.2.2 The Relation Between the PRO Method and Other Methods 128

8.3 The Forced Manifold 129

8.3.1 Significance Analysis 136

8.4 The Coupled Manifold 141

References 147

Index 149