Multidimensional Similarity Structure Analysis comprises a class of models that represent similarity among entities (for example, variables, items, objects, persons, etc.) in multidimensional space to permit one to grasp more easily the interrelations and patterns present in the data. The book is oriented to both researchers who have little or no previous exposure to data scaling and have no more than a high school background in mathematics and to investigators who would like to extend their analyses in the direction of hypothesis and theory testing or to more intimately understand these analytic procedures. The book is repleted with examples and illustrations of the various techniques drawn largely, but not restrictively, from the social sciences, with a heavy emphasis on the concrete, geometric or spatial aspect of the data representations.

Multidimensional scaling (MDS) is a technique for the analysis of similarity or dissimilarity data on a set of objects. Such data may be intercorrelations of test items, ratings of similarity on political candidates, or trade indices for a set of countries. MDS attempts to model such data as distances among points in a geometric space. The main reason for doing this is that one wants a graphical display of the structure of the data, one that is much easier to understand than an array of numbers and, moreover, one that displays the essential information in the data, smoothing out noise. There are numerous varieties of MDS. Some facets for distinguishing among them are the particular type of geometry into which one wants to map the data, the mapping function, the algorithms used to find an optimal data representation, the treatment of statistical error in the models, or the possibility to represent not just one but several similarity matrices at the same time. Other facets relate to the different purposes for which MDS has been used, to various ways of looking at or "interpreting" an MDS representation, or to differences in the data required for the particular models. In this book, we give a fairly comprehensive presentation of MDS. For the reader with applied interests only, the first six chapters of Part I should be sufficient. They explain the basic notions of ordinary MDS, with an emphasis on how MDS can be helpful in answering substantive questions.

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Outlines a set of techniques that enables a researcher to explore the hidden structure of large databases. These techniques use proximities to find a configuration of points that reflect the structure in the data.

This book introduces multidimensional scaling (MDS) and unfolding as data analysis techniques for applied researchers. MDS is used for the analysis of proximity data on a set of objects, representing the data as distances between points in a geometric space (usually of two dimensions). Unfolding is a related method that maps preference data (typically evaluative ratings of different persons on a set of objects) as distances between two sets of points (representing the persons and the objects, resp.). This second edition has been completely revised to reflect new developments and the coverage of unfolding has also been substantially expanded. Intended for applied researchers whose main interests are in using these methods as tools for building substantive theories, it discusses numerous applications (classical and recent), highlights practical issues (such as evaluating model fit), presents ways to enforce theoretical expectations for the scaling solutions, and addresses the typical mistakes that MDS/unfolding users tend to make. Further, it shows how MDS and unfolding can be used in practical research work, primarily by using the smacof package in the R environment but also Proxscal in SPSS. It is a valuable resource for psychologists, social scientists, and market researchers, with a basic understanding of multivariate statistics (such as multiple regression and factor analysis).

"Geometric Structure of High-Dimensional Data and Dimensionality Reduction" adopts data geometry as a framework to address various methods of dimensionality reduction. In addition to the introduction to well-known linear methods, the book moreover stresses the recently developed nonlinear methods and introduces the applications of dimensionality reduction in many areas, such as face recognition, image segmentation, data classification, data visualization, and hyperspectral imagery data analysis. Numerous tables and graphs are included to illustrate the ideas, effects, and shortcomings of the methods. MATLAB code of all dimensionality reduction algorithms is provided to aid the readers with the implementations on computers. The book will be useful for mathematicians, statisticians, computer scientists, and data analysts. It is also a valuable handbook for other practitioners who have a basic background in mathematics, statistics and/or computer algorithms, like internet search engine designers, physicists, geologists, electronic engineers, and economists. Jianzhong Wang is a Professor of Mathematics at Sam Houston State University, U.S.A.