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Modern multidimensional scaling: theory and applications. (English) Zbl 0862.62052

Springer Series in Statistics. New York, NY: Springer. xvii, 471 p. (1997).
This book provides a comprehensive presentation of multidimensional scaling (MDS), a statistical technique for the analysis of similarity or dissimilarity data on a set of objects in multidimensional scale. MDS attempts to model data as distances among points in a geometric space of low dimensionality, which makes the data structure easier to understand than an array of numbers. The contents of the book are divided into five parts.
Part I is sufficient for readers with applied interests only. It explains the basic notions of ordinary MDS and deals with some special models that are important in particular applications of MDS. Part II discusses technical aspects of MDS (matrix algebra for MDS, majorization algorithm for solving MDS problems, ordinal transformations of similiarity data, fit measures, classical scaling). Part III introduces the idea of unfolding and some variants of this model. Part IV treats the geometry of MDS as a substantive model (MDS as a psychological model, scalar products, mapping proximities into Euclidean distances). The last part, Part V, discusses some methods and models closely related to MDS (principal components analysis, models for asymmetric data, correspondence analysis).
The first six chapters of Part I make up a complete introductory course to MDS that assumes only elementary knowledge of descriptive statistics. The book is also suited for various advanced courses on MDS oriented either to data analysis or to the psychology of similarity.

MSC:

62H25 Factor analysis and principal components; correspondence analysis
91C15 One- and multidimensional scaling in the social and behavioral sciences
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62P15 Applications of statistics to psychology
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