ALBERT

Note: Contents: List of Figures. - List of Tables. - 1. Introduction. - a. An Overview of Principal Component Analysis (PCA). - b. Outline of the Book. - c. A Brief History of PCA. - d. Acknowledgments. - 2. Algebraic Foundations of PCA. - a. Introductory Example: Bivariate Data Sets. - Monterey, California air temperatures. - Centering and rotating the data set. - Variances in the rotated frame. - Principal angles. - Principal variances. - Principal covariance. - Principal directions. - Principal components; principal directions as basis vectors. - Matrix representation. - The PCA property. - Invariance of the total variance under rotation. - Principal variances for standardized data sets. - PCA and estimates of the statistical parameters of normal populations. - PCA and the construction of Monte Carlo experiments. - Eigenvalues and eigenvectors of the covariance and scatter matrices. - b. Principal Component Analysis: Real-valued Scalar Fields. - t-centering the data set. - The scatter probe and the scatter matrix. - The eigenstructures of PCA. - The basic data set representations; analysis and synthesis formulas. - The PCA property. - Second-order properties of PCA; the total scatter . - The singular value decomposition (SVD) of a data set. - Second-order properties of PCA; correlations. - PCA characterized by the PCA property. - The asymptotic PCA property and dynamical systems. - PCA of spatial composites of data sets. - PCA of temporal composites of data sets. - c. Principal Component Analysis: Complex-valued Scalar Fields, and Beyond. - PCA of complex-valued data sets (C-PCA). - Complex algebra conventions. - The scatter probe and scatter matrix for C-PCA. - Derivation of the eigenstructures of C-PCA. - The fundamental formulas of C-PCA. - Generalization of PCA to quaternion-valued data sets (Q-PCA). - Matrix representations of complex and quaternion numbers. - PCA of matrix-valued data sets (M-PCA). - Reduction of M-PCA to C-PCA form. - d. Bibliographic Notes and Miscellaneous Topics. - Alternate interpretation of the scatter probe. - Numerical calculations of eigenstructures of a scatter matrix. - Some elementary properties of eigenstructures of a scatter matrix. - Sample space vs. state space: choosing the dual computation. - PCA for continuous domains. - PCA for continuous domains: the viewpoint of empirical orthogonal functions. - The sixteen possible domain pairs for PCA: abstract PCA. - 3. Dynamical Origins of PCA. - a. One-dimensional Hannonic Motion. - A spring-linked-mass model; general form. - A spring-linked-mass model; special form. - A numerical example of the asymptotic PCA property. - Further investigations of the asymptotic PCA property and of EOF's. - b. Two-dimensional Wave Motion. - Solution of a two-dimensional damped-wave model. - Demonstration of the asymptotic PCA property (forcing and friction absent). - Demonstration of the asymptotic PCA property (forcing and friction present). - Physical basis for eigenframe rotations. - c. Dynamical Origins of Linear Regression (LR). - From continuous to discrete solutions to the regression model. - The linear regression procedure. - Comparison of LRA and PCA. - d. Random Processes and Karhunen-Loeve Analysis. - Origins of random processes in linear settings. - Karhunen-Loeve representation of random data sets and comparison with PCA. - e. Stationary Processes and PCA. - Derivation of the PCA representation of a one-dimensional stationary process via a simple wave model. - Connections between PCA and stationary processes: the case of one dimension. - Connections between PGA and stationary processes: extension to two dimensions. - f. Bibliographic Notes. - 4. Extensions of PCA to Multivariate Fields. - a. Categories of Data and Modes of Analysis. - Examples. - Generalized notation: the concepts of "individual" and "variable" in PCA. - b. Local PCA of a General Vector Field. - The PCA formalism. - Squared correlations. - Variational origin of the scatter matrix. - Examples. - c. Global PCA of a General Vector Field: Time-Modulation Form. - The PGA formalism. - Squared correlations. - Degeneracy of global PGA to local PGA. - Variational origin of the scatter matrix. - d. Global PCA of a General Vector Field: Space-Modulation Form. - The PCA formalism. - Squared correlations. - Variational origin of the scatter matrix. - e. PCA of Spectral Components of a General Vector Field. - Fourier analysis of the vector field components. - The scatter matrix in the spectral setting. - Example of spectral PCA of a windfield. - f. Bibliographic Notes and Miscellaneous Topics. - The eight modes of analysis and Cattell's classifications. - Time-modulation PGA as a special case of matrix-valued PGA. - Applications to the PGA of wind fields. - Distinction between time-modulation PGA and complex PGA. - Applications to the PGA of storm tracks. - 5. Selection Rules for PCA. - a. Random Reference Data Sets. - b. Dynamical Origins of the Dominant-Variance Selection Rules. - A dynamical model. - Rationale for selection rules. - c. Rule A4. - Statistical basis and discussion. - Choice of λ0. - d. Rule N . - Statistical basis and discussion. - Adjustments for correlated data: effective sample size. - Asymptotic eigenvalues for large data sets. - e. Rule M. - f. Comments on Dominant-Variance Rules . - g. Dynamical Origins of the Time-History Selection Rules. - h. Rule KS2. - The white spectrum and the cumulative periodogram. - Statement of Rule KS2. - i. Rules AMPλ. - Fisher's test. - Siegel's test. - Statement of Rules AMPλ. - j. Rule Q. - k. Selection Rules for Vector-Valued Fields. - Local PCA rules. - Global PCA (time-modulated) rules. - Global PCA (space-modulated) rules. - I. A Space-map Selection Rule. - Canonic direction angles. - Differential relations between unit vectors and canonic direction angles. - An r-tile metric for comparing canonic direction angles. - Statistical aspects: critical values for class errors. - Statement of the selection rule. - m. Bibliographic Notes and Miscellaneous Topics. - Puzzles and problems underlying Rule N; the logarithmic eigenvalue curve. - Numerical intractability of the classical formulas for the eigenvalues of a random matrix. - Monte Carlo approaches to the eigenvalue distribution problem. - Comparison of Monte Carlo methods and asymptotic formulas for eigenvalue distributions. - The problem of closely spaced eigenvalues; tests for equal eigenvalues. - The generalized basis for dominant variance selection rules. - Parallel work in atomic physics. - 6. Factor Analysis (FA) and PCA. - a. Comparison of PCA, LRA, and FA. - Similarities between PCA, LRA, and FA. - Dissimilarities between PCA, LRA, and FA. - The usual algebraic form of FA; its PC and LR interpretations. - b. The Central Problems of FA. - The matrix formulation of FA. - The detailed sub-problems of FA. - c. Bibliographic Notes. - The selection rule problem in FA. - The parameter estimation problem in FA. - 7. Diagnostic Procedures via PCA and FA. - a. Dual Interpretations of a Data Set: State Space and Sample Space. - b. Interpreting E-frames in PCA State Space. - Example: graphical display of eigenvectors. - Rationales for interpreting eigenmaps and time series. - PCA as a means, rather than an end. - c. Informative and Uninformative E-frames in PCA State Space. - d. Rotating E-frames in PCA State Space (varimax). - A two-dimensional example of the varimax procedure. - The general varimax procedure. - The loss of the PCA property for rotated E-frames. - e. Projections onto E-frames in PCA State Space (procrustes). - Derivation of the procrustes technique. - Some observations on the generality of the procrustes technique. - f. Interpreting A-frames in PCA Sample Space. - g. Rotating A-frames in PCA Sample Space (varimax). - h. Projections onto A-frames in PCA Sample Space (procrustes). - i. Detecting Clusters of Points in PCA State or Sample Spaces. - Minimal spanning trees. - Defining cluster pairs, and te