The geometric representation of similarity and dissimilarity. Extension to measures of dispersion and comments on the distance model of subjective dissimilarity

In psychometrics and mathematical taxonomy dissimilarity is commonly geometrically represented by the Euclidean (Minkowski) distance function. This is only a part of a general geometric representation of similarity and dissimilarity (Junge, 1978). Gini's index (or ratio) and Pearson's coefficient of variation, used in economics as measures of income distribution inequality, are known in mathematical statistics as standardized measures of dispersion. It is shown that both measures can be identified with the general geometric representation of similarity and dissimilarity.     

A nonspatial methodology for the analysis of two-way proximity data incorporating the distance-density hypothesis

This paper presents a nonspatial operationalization of the Krumhansl (1978, 1982) distancedensity model of similarity. This model assumes that the similarity between two objectsi andj is a function of both the interpoint distance betweeni andj and the density of other stimulus points in the regions surroundingi andj. We review this conceptual model and associated empirical evidence for such a specification. A nonspatial, tree-fitting methodology is described which is sufficiently flexible to fit a number of competing hypotheses of similarity formation. A sequential, unconstrained minimization algorithm is technically presented together with various program options. Three applications are provided which demonstrate the flexibility of the methodology. Finally, extensions to spatial models, three-way analyses, and hybrid models are discussed.     

Netscal: A network scaling algorithm for nonsymmetric proximity data

A simple property of networks is used as the basis for a scaling algorithm that represents nonsymmetric proximities as network distances. The algorithm determines which vertices are directly connected by an arc and estimates the length of each arc. Network distance, defined as the minimum pathlength between vertices, is assumed to be a generalized power function of the data. The derived network structure, however, is invariant across monotonic transformations of the data. A Monte Carlo simulation and applications to eight sets of proximity data support the practical utility of the algorithm.I am grateful to Roger Shepard and Amos Tversky for their helpful comments and guidance throughout this project. The work was supported by National Science Foundation Grant BNS-75-02806 to Roger Shepard and a National Science Foundation Graduate Fellowship to the author. Parts of this paper were drawn from a doctoral dissertation submitted to Stanford University (Hutchinson, 1981).     

“Minimax length links” of a dissimilarity matrix and minimum spanning trees

A theorem is proved stating that the set of all minimax links, defined as links minimizing, over paths, the maximum length of links in any path connecting a pair of objects comprising nodes in an undirected weighted graph, comprise the union of all minimum spanning trees of that graph. This theorem is related to methods of fitting network models to dissimilarity data, particularly a method called Pathfinder due to Schvaneveldt and his colleagues, as well as to single linkage clustering, and results concerning the relationship between minimum spanning trees and single linkage hierarchical trees.Acknowledgments: The author thanks Phipps Arabie, Lawrence J. Hubert, and K. Christoph Klauer for a number of helpful suggestions and comments on various aspects of this paper.     

Judging a book by its cover and its contents: the representation of polysemous and homophonous meanings in four-year-old children

Unlike homophonous meanings, which are semantically unrelated (e.g., the use of bat to refer to a baseball bat and a flying rodent), polysemous meanings are systematically related to one another (e.g., the use of book, CD, and video to refer to physical objects, as in ‘the leather book’, or to the intellectual content they contain, as in ‘the profound book’). But do perceived relations among polysemous meanings reflect the presence of generative lexical or conceptual structures that permit the meanings of these words to shift? If so, these structures may also support children’s early representations of polysemous meanings. In four studies, we demonstrate (1) that four-year-old children can understand both the concrete and abstract meanings of words like book, (2) that when taught a novel label for one of these meanings, children can readily understand an extension of that label to the other meaning, and (3) that extension does not occur between two homophonous meanings, which share a common phonological form but are otherwise unrelated. We conclude that the polysemous meanings of words like book rely on a common representational base early in development, and suggest that this may be the result of foundational, generative properties of the lexicon or conceptual system.