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
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