Principal Component Analysis in Meteorology and Oceanography: (Developments in Atmospheric Science)

This book, the only one of its kind available, presents PCA from its simplest form through its abstract formalism, including applications. Furthermore, it extends the use of PCA far beyond its well-known applications to scalar (e.g. temperature) or vector (e.g. wind) fields. Much of the material is hitherto unpublished, thus greatly extending the realm of applicability of PCA, and many suggestions are made for its future application. The first half of the book provides a comprehensive discussion of PCA, including solved numerical examples, beginning with a simple bivariate data set and progressing to the PCA of multivariate fields. The use of selection rules to establish statistical significance is emphasized. The second half of the book compares PCA with other analysis techniques such as Factor Analysis, Linear Regression Analysis, and Canonical Factor Analysis. The book also discusses the use of PCA in construction of statistical-dynamical models, in the detection of moving patterns in data sets, and in studies of stationary random processes.

Table of Contents:
List of figures. List of tables. 1. Introduction. An overview of principal component analysis (PCA). Outline of the book. A brief history of PCA. Acknowledgements. 2. Algebraic foundations of PCA. Introductory example: Bivariate data sets. Principal component analysis: Real-valued scalar fields. Principal component analysis: Complex-valued scalar fields, and beyond. Bibliographic notes and miscellaneous topics. 3. Dynamical origins of PCA. One-dimensional harmonic motion. Two-dimensional wave motion. Dynamical origins of linear regression (LR). Random processes and Karhunen-Loeve analysis. Stationary processes and PCA. Bibliographic notes. 4. Extensions of PCA to multivariate fields. Categories of data and modes of analysis. Local PCA of a general vector field. Global PCA of a general vector field: Time-modulation form. Global PCA of a general vector field: Space-modulation form. PCA of spectral components of a general vector field. Bibliographic notes and miscellaneous topics. 5. Selection rules for PCA. Random reference data sets. Dynamical origins of the dominant-variance selection rules. Rule A 4 . Rule N. Rule M. Comments on dominant-variance rules. Dynamical origins of the time-history selection rules. Rule KS2. Rules AMPl. Rule Q. Selection rules for vector-valued fields. A space-map selection rule. Bibliographic notes and miscellaneous topics. 6. Factor analysis (FA) and PCA. Comparison of PCA, LRA, and FA. The central problems of FA. Bibliographic notes. 7. Diagnostic procedures via PCA and FA. Dual interpretations of a data set: state space and sample space. Interpreting E-frames in PCA state space. Informative and uninformative E-frames in PCA state space. Rotating E-frames in PCA state space (varimax). Projections onto E-frames in PCA state space (procrustes). Interpreting A-frames in PCA sample space. Rotating A-frames in PCA sample space (varimax). Projections onto A-frames in PCA sample space (procrustes). Detecting clusters of points in PCA state or sample spaces. The analogous PCA interpretations and transformations in FA. Bibliographic notes. 8. Canonical correlation analysis (CCA) and PCA. The singular value decomposition (SVD) of two data sets. The correlation probe. Maximizing the correlation function g(r,s). Canonical correlations. Canonical component representations of data sets. Selection rules for CCA. Bibliographic notes. 9. Linear regression analysis (LRA) and PCA. Basic regression equations. Regression using PCA frames. Regression using CCA frames. Regression hindcast skill. Regression signal-to-noise ratio. Significant hindcast skill. Bibliographic notes. 10. Statistical-dynamical models and PCA. Example 1: A linear two-dimensional damped-wave model. Example 2: Linearized primitive equations for the atmosphere and oceans. Bibliographic notes. 11. The eigenvector-partition problem. PCA on partitioned domains. Iterative improvement of the optimal combinations.