Asymmetric multidimensional scaling of two-mode three-way proximities. (English) Zbl 0905.92035
Summary: An asymmetric multidimensional scaling model and an associated nonmetric algorithm to analyze two-mode three-way proximities (object \(\times\) object \(\times \) source) are introduced. The model consists of a common object configuration and two kinds of weights, i.e., for both symmetry and asymmetry. In the common object configuration, each object is represented by a point and a circle (sphere, hypersphere) in a Euclidean space. The common object configuration represents pairwise proximity relationships between pairs of objects for the ‘group’ of all sources. Each source has its own symmetry weight and a set of asymmetry weights. Symmetry weights represent individual differences among sources of data in symmetric proximity relationships, and asymmetry weights represent individual differences among sources in asymmetric proximity relationships.
The associated nonmetric algorithm, based on J. B. Kruskal’s [Psychometrika 29, 115-129 (1964; Zbl 0123.36804)] nonmetric multidimensional scaling algorithm, is an extension of the algorithm for the asymmetric multidimensional scaling of one mode two-way proximities developed by A. Okada and T. Imaizumi [Behaviormetrika 21, 81-96 (1987)]. As an illustrative example, we analyze intergenerational occupational mobility from 1955 to 1985 in Japan among eight occupational categories.
The associated nonmetric algorithm, based on J. B. Kruskal’s [Psychometrika 29, 115-129 (1964; Zbl 0123.36804)] nonmetric multidimensional scaling algorithm, is an extension of the algorithm for the asymmetric multidimensional scaling of one mode two-way proximities developed by A. Okada and T. Imaizumi [Behaviormetrika 21, 81-96 (1987)]. As an illustrative example, we analyze intergenerational occupational mobility from 1955 to 1985 in Japan among eight occupational categories.
MSC:
| 91C15 | One- and multidimensional scaling in the social and behavioral sciences |
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
Keywords:
asymmetric proximity; two-mode three-way data; asymmetric multidimensional scaling; nonmetric algorithmCitations:
Zbl 0123.36804References:
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