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 Much of the material is hitherto unpublished, thus greatly extending the realm of applicability of PCA, and many suggestions are made for its future application.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. The book is primarily intended for meteorologists and oceanographers at both student and professional levels, however, researchers in other fields, e.g. geophysics and psychometrics who often have data analysis problems similar to those in meteorology and oceanography, should find the book useful.
Author(s): Rudolph W Preisendorfer
Edition: 1
Publisher: Elsevier Science
Year: 1988
Language: English
Pages: 440
Tags: Математика;Теория вероятностей и математическая статистика;Математическая статистика;Прикладная математическая статистика;
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-Loève 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 A4. Rule N. Rule M. Comments on dominant-variance rules. Dynamical origins of the time-history selection rules. Rule KS2. Rules AMP&lgr;. 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-fram
