Anmerkung: Intro -- Preface -- Pattern Identification and Data Mining in Weather and Climate -- -- Acknowledgements -- Contents -- 1 Introduction -- 1.1 Complexity of the Climate System -- 1.2 Data Exploration, Data Mining and Feature Extraction -- 1.3 Major Concern in Climate Data Analysis -- 1.3.1 Characteristics of High-Dimensional SpaceGeometry -- Volume Paradox of Hyperspheres -- Two Further Paradoxical Examples -- 1.3.2 Curse of Dimensionality and Empty SpacePhenomena -- 1.3.3 Dimension Reduction and Latent Variable Models -- 1.3.4 Some Problems and Remedies in Dimension Reduction -- Known Difficulties -- Some Remedies -- 1.4 Examples of the Most Familiar Techniques -- 2 General Setting and Basic Terminology -- 2.1 Introduction -- 2.2 Simple Visualisation Techniques -- 2.3 Data Processing and Smoothing -- 2.3.1 Preliminary Checking -- 2.3.2 Smoothing -- Moving Average -- Exponential Smoothing -- Spline Smoothing -- Kernel Smoothing -- 2.3.3 Simple Descriptive Statistics -- 2.4 Data Set-Up -- 2.5 Basic Notation/Terminology -- 2.5.1 Centring -- 2.5.2 Covariance Matrix -- 2.5.3 Scaling -- 2.5.4 Sphering -- 2.5.5 Singular Value Decomposition -- 2.6 Stationary Time Series, Filtering and Spectra -- 2.6.1 Univariate Case -- 2.6.2 Multivariate Case -- 3 Empirical Orthogonal Functions -- 3.1 Introduction -- 3.2 Eigenvalue Problems in Meteorology: Historical Perspective -- 3.2.1 The Quest for Climate Patterns: Teleconnections -- 3.2.2 Eigenvalue Problems in Meteorology -- 3.3 Computing Principal Components -- 3.3.1 Basis of Principal Component Analysis -- 3.3.2 Karhunen-Loéve Expansion -- 3.3.3 Derivation of PCs/EOFs -- 3.3.4 Computing EOFs and PCs -- Singular Value Decomposition and Similar Algorithms -- Basic Iterative Approaches -- 3.4 Sampling, Properties and Interpretation of EOFs -- 3.4.1 Sampling Variability and Uncertainty. , Uncertainty Based on Asymptotic Approximation -- Probabilistic PCA -- Monte-Carlo Resampling Methods -- Bootstrap Application to EOFs of Atmospheric Fields -- 3.4.2 Independent and Effective Sample Sizes -- Serial Correlation -- Time Varying Fields -- 3.4.3 Dimension Reduction -- 3.4.4 Properties and Interpretation -- 3.5 Covariance Versus Correlation -- 3.6 Scaling Problems in EOFs -- 3.7 EOFs for Multivariate Normal Data -- 3.8 Other Procedures for Obtaining EOFs -- 3.9 Other Related Methods -- 3.9.1 Teleconnectivity -- 3.9.2 Regression Matrix -- 3.9.3 Empirical Orthogonal Teleconnection -- 3.9.4 Climate Network-Based Methods -- 4 Rotated and Simplified EOFs -- 4.1 Introduction -- 4.2 Rotation of EOFs -- 4.2.1 Background on Rotation -- 4.2.2 Derivation of REOFs -- 4.2.3 Computing REOFs -- Rotation or Simplicity Criteria -- Computation of REOFs -- 4.3 Simplified EOFs: SCoTLASS -- 4.3.1 Background -- 4.3.2 LASSO-Based Simplified EOFs -- 4.3.3 Computing the Simplified EOFs -- 5 Complex/Hilbert EOFs -- 5.1 Background -- 5.2 Conventional Complex EOFs -- 5.2.1 Pairs of Scalar Fields -- 5.2.2 Single Field -- 5.3 Frequency Domain EOFs -- 5.3.1 Background -- 5.3.2 Derivation of FDEOFs -- 5.4 Complex Hilbert EOFs -- 5.4.1 Hilbert Transform: Continuous Signals -- 5.4.2 Hilbert Transform: Discrete Signals -- 5.4.3 Application to Time Series -- 5.4.4 Complex Hilbert EOFs -- Complexified Field -- Computational Aspects -- 5.5 Rotation of HEOFs -- 6 Principal Oscillation Patterns and Their Extension -- 6.1 Introduction -- 6.2 POP Derivation and Estimation -- 6.2.1 Spatial Patterns -- 6.2.2 Time Coefficients -- 6.2.3 Example -- 6.3 Relation to Continuous POPs -- 6.3.1 Basic Relationships -- 6.3.2 Finite Time POPs -- 6.4 Cyclo-Stationary POPs -- 6.5 Other Extensions/Interpretations of POPs -- 6.5.1 POPs and Normal Modes -- 6.5.2 Complex POPs. , 6.5.3 Hilbert Oscillation Patterns -- 6.5.4 Dynamic Mode Decomposition -- 6.6 High-Order POPs -- 6.7 Principal Interaction Patterns -- 7 Extended EOFs and SSA -- 7.1 Introduction -- 7.2 Dynamical Reconstruction and SSA -- 7.2.1 Background -- 7.2.2 Dynamical Reconstruction and SSA -- 7.3 Examples -- 7.3.1 White Noise -- 7.3.2 Red Noise -- 7.4 SSA and Periodic Signals -- 7.5 Extended EOFs or Multivariate SSA -- 7.5.1 Background -- 7.5.2 Definition and Computation of EEOFs -- 7.5.3 Data Filtering and Oscillation Reconstruction -- 7.6 Potential Interpretation Pitfalls -- 7.7 Alternatives to SSA and EEOFs -- 7.7.1 Recurrence Networks -- 7.7.2 Data-Adaptive Harmonic Decomposition -- 8 Persistent, Predictive and Interpolated Patterns -- 8.1 Introduction -- 8.2 Background on Persistence and Prediction of Stationary Time Series -- 8.2.1 Decorrelation Time -- 8.2.2 The Prediction Problem and Kolmogorov Formula -- 8.3 Optimal Persistence and Average Predictability -- 8.3.1 Derivation of Optimally Persistent Patterns -- 8.3.2 Estimation from Finite Samples -- 8.3.3 Average Predictability Patterns -- 8.4 Predictive Patterns -- 8.4.1 Introduction -- 8.4.2 Optimally Predictable Patterns -- 8.4.3 Computational Aspects -- 8.5 Optimally Interpolated Patterns -- 8.5.1 Background -- 8.5.2 Interpolation and Pattern Derivation -- 8.5.3 Numerical Aspects -- 8.5.4 Application -- 8.6 Forecastable Component Analysis -- 9 Principal Coordinates or Multidimensional Scaling -- 9.1 Introduction -- 9.2 Dissimilarity Measures -- 9.3 Metric Multidimensional Scaling -- 9.3.1 The Problem of Classical Scaling -- 9.3.2 Principal Coordinate Analysis -- Classical Scaling in Presence of Errors -- 9.3.3 Case of Non-Euclidean Dissimilarity Matrix -- 9.4 Non-metric Scaling -- 9.5 Further Extensions -- 9.5.1 Replicated and Weighted MDS -- 9.5.2 Nonlinear Structure. , 9.5.3 Application to the Asian Monsoon -- 9.5.4 Scaling and the Matrix Nearness Problem -- 10 Factor Analysis -- 10.1 Introduction -- 10.2 The Factor Model -- 10.2.1 Background -- 10.2.2 Model Definition and Terminology -- 10.2.3 Model Identification -- Case of Autocorrelated Factors -- 10.2.4 Non-unicity of Loadings -- 10.3 Parameter Estimation -- 10.3.1 Maximum Likelihood Estimates -- 10.3.2 Expectation Maximisation Algorithm -- Necessary Optimality Condition -- Expectation Maximisation Algorithm -- Model Assessment -- 10.4 Factor Rotation -- 10.4.1 Oblique and Orthogonal Rotations -- 10.4.2 Examples of Rotation Criteria -- Quartimax -- Quartimin -- Oblimin -- Oblimax -- Entropy -- 10.5 Exploratory FA and Application to SLP Anomalies -- 10.5.1 Factor Analysis as a Matrix Decomposition Problem -- 10.5.2 A Factor Rotation -- 10.6 Basic Difference Between EOF and Factor Analyses -- 10.6.1 Comparison Based on the Standard Factor Model -- 10.6.2 Comparison Based on the Exploratory Factor Analysis Model -- 11 Projection Pursuit -- 11.1 Introduction -- 11.2 Definition and Purpose of Projection Pursuit -- 11.2.1 What Is Projection Pursuit? -- 11.2.2 Why Projection Pursuit? -- 11.3 Entropy and Structure of Random Variables -- 11.3.1 Shannon Entropy -- 11.3.2 Differential Entropy -- 11.4 Types of Projection Indexes -- 11.4.1 Quality of a Projection Index -- 11.4.2 Various PP Indexes -- Friedman and Tukey's Index -- Jones and Sibson's Index -- Entropy/Information Index -- Moments-Based Indexes -- Friedman and Related L2 Norm-Based Indices -- Chi-Square Index -- Clustering Index -- 11.4.3 Practical Implementation -- 11.5 PP Regression and Density Estimation -- 11.5.1 PP Regression -- 11.5.2 PP Density Estimation -- 11.6 Skewness Modes and Climate Application of PP -- 12 Independent Component Analysis -- 12.1 Introduction. , 12.2 Background and Definition -- 12.2.1 Blind Deconvolution -- 12.2.2 Blind Source Separation -- 12.2.3 Definition of ICA -- 12.3 Independence and Non-normality -- 12.3.1 Statistical Independence -- 12.3.2 Non-normality -- 12.4 Information-Theoretic Measures -- 12.4.1 Entropy -- 12.4.2 Kullback-Leibler Divergence -- Properties of the K-L Divergence -- 12.4.3 Mutual Information -- 12.4.4 Negentropy -- 12.4.5 Useful Approximations -- 12.5 Independent Component Estimation -- 12.5.1 Choice of Objective Function for ICA -- Negentropy -- Non-normality -- Information-Theoretic Approach -- Likelihood Maximisation Approach -- Information Maximisation Approach -- A Non-parametric Approach -- Other Methods -- 12.5.2 Numerical Implementation -- Sphering/Whitening -- Optimisation Algorithms -- 12.6 ICA via EOF Rotation and Weather and Climate Application -- 12.6.1 The Standard Two-Way Problem -- 12.6.2 Extension to the Three-Way Data -- 12.7 ICA Generalisation: Independent Subspace Analysis -- 13 Kernel EOFs -- 13.1 Background -- 13.2 Kernel EOFs -- 13.2.1 Formulation of Kernel EOFs -- 13.2.2 Practical Details of Kernel EOF Computation -- 13.2.3 Illustration with Concentric Clusters -- 13.3 Relation to Other Approaches -- 13.3.1 Spectral Clustering -- 13.3.2 Modularity Clustering -- 13.4 Pre-images in Kernel PCA -- 13.5 Application to An Atmospheric Model and Reanalyses -- 13.5.1 Application to a Simplified Atmospheric Model -- 13.5.2 Application to Reanalyses -- 13.6 Other Extensions of Kernel EOFs -- 13.6.1 Extended Kernel EOFs -- Direct Formulation -- Alternative Formulations -- 13.6.2 Kernel POPs -- 14 Functional and Regularised EOFs -- 14.1 Functional EOFs -- 14.2 Functional PCs and Discrete Sampling -- 14.3 An Example of Functional PCs from Oceanography -- 14.4 Regularised EOFs -- 14.4.1 General Setting -- 14.4.2 Case of Spatial Fields. , The Example of the RBF Solution.