The analysis of proximities: Multidimensional scaling with an unknown distance function. I.

doi:10.1007/BF02289630

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The analysis of proximities: Multidimensional scaling with an unknown distance function. I.

Journal Article•DOI•

The analysis of proximities: Multidimensional scaling with an unknown distance function. I.

Roger N. Shepard¹•Institutions (1)

Bell Labs¹

01 Jun 1962-Psychometrika (Springer-Verlag)-Vol. 27, Iss: 2, pp 125-140

TL;DR: The results of two kinds of test applications of a computer program for multidimensional scaling on the basis of essentially nonmetric data are reported to measures of interstimulus similarity and confusability obtained from some actual psychological experiments.

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Abstract: A computer program is described that is designed to reconstruct the metric configuration of a set of points in Euclidean space on the basis of essentially nonmetric information about that configuration. A minimum set of Cartesian coordinates for the points is determined when the only available information specifies for each pair of those points—not the distance between them—but some unknown, fixed monotonic function of that distance. The program is proposed as a tool for reductively analyzing several types of psychological data, particularly measures of interstimulus similarity or confusability, by making explicit the multidimensional structure underlying such data.

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Citations

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Journal Article•DOI•

A circumplex model of affect

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James A. Russell¹•Institutions (1)

University of British Columbia¹

01 Dec 1980-Journal of Personality and Social Psychology

12,519 citations

Journal Article•DOI•

Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis

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Joseph B. Kruskal¹•Institutions (1)

Bell Labs¹

01 Mar 1964-Psychometrika

TL;DR: The fundamental hypothesis is that dissimilarities and distances are monotonically related, and a quantitative, intuitively satisfying measure of goodness of fit is defined to this hypothesis.

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Abstract: Multidimensional scaling is the problem of representingn objects geometrically byn points, so that the interpoint distances correspond in some sense to experimental dissimilarities between objects. In just what sense distances and dissimilarities should correspond has been left rather vague in most approaches, thus leaving these approaches logically incomplete. Our fundamental hypothesis is that dissimilarities and distances are monotonically related. We define a quantitative, intuitively satisfying measure of goodness of fit to this hypothesis. Our technique of multidimensional scaling is to compute that configuration of points which optimizes the goodness of fit. A practical computer program for doing the calculations is described in a companion paper.

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6,875 citations

Journal Article•DOI•

Family Resemblances: Studies in the Internal Structure of Categories

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Eleanor Rosch¹, Carolyn B. Mervis²•Institutions (2)

University of California, Berkeley¹, Cornell University²

01 Oct 1975-Cognitive Psychology

TL;DR: In this paper, the authors explored the hypothesis that the members of categories which are considered most prototypical are those with most attributes in common with other members of the category and least attributes with other categories and found that family resemblance offers an alternative to criterial features in defining categories.

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5,002 citations

Cites methods from "The analysis of proximities: Multid..."

...The similarity ratings were scaled by M-D scale (Shepard, 1962; Shepard, Romney, & Nerlove, Vol. 1, 1972)....
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Journal Article•DOI•

Toward a consensual structure of mood.

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David Watson¹, Auke Tellegen•Institutions (1)

Southern Methodist University¹

01 Sep 1985-Psychological Bulletin

TL;DR: Reanalyses of a number of studies of self-reported mood indicate that Positive and Negative Affect consistently emerge as the first two Varimax rotated dimensions in orthogonal factor analyses or as thefirst two second-order factors derived from oblique solutions.

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Abstract: Reanalyses of a number of studies of self-reported mood indicate that Positive and Negative Affect consistently emerge as the first two Varimax rotated dimensions in orthogonal factor analyses or as the first two second-order factors derived from oblique solutions. The two factors emerged with varyi

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4,741 citations

Journal Article•DOI•

Nonmetric multidimensional scaling: A numerical method

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Joseph B. Kruskal¹•Institutions (1)

Bell Labs¹

01 Jun 1964-Psychometrika

TL;DR: The numerical methods required in the approach to multi-dimensional scaling are described and the rationale of this approach has appeared previously.

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Abstract: We describe the numerical methods required in our approach to multi-dimensional scaling. The rationale of this approach has appeared previously.

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4,561 citations

Cites methods from "The analysis of proximities: Multid..."

...In a companion paper [7] we describe the rationale for our approach to scaling, which is related to that of Shepard [ 9 ]....
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References

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Book•

Modern factor analysis

[...]

Harry Horace Harman

01 Jan 1960

TL;DR: The third edition of HARMAN's authoritative text as mentioned in this paper incorporates the many new advances made in computer science and technology over the last ten years The author gives full coverage to both theoretical and applied aspects of factor analysis from its foundations through the most advanced techniques This highly readable text will be welcomed by researchers and students working in psychology, statistics, economics and related disciplines

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Abstract: This thoroughly revised third edition of Harry H Harman's authoritative text incorporates the many new advances made in computer science and technology over the last ten years The author gives full coverage to both theoretical and applied aspects of factor analysis from its foundations through the most advanced techniques This highly readable text will be welcomed by researchers and students working in psychology, statistics, economics, and related disciplines

...read moreread less

5,419 citations

Book•

Theory and Methods of Scaling

[...]

Warren S. Torgerson

15 Jan 1958

3,060 citations

Book•

Multiple-factor analysis

[...]

Louis Leon Thurstone

01 Jan 1947

2,920 citations

"The analysis of proximities: Multid..." refers background in this paper

...The fact that correlations or covariances can be interpreted as scalar products of vectors ([ 31 ], pp. 89-91) turns out to provide a much stronger starting position than is possible in the analysis of proximities with an unknown distance function....
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Journal Article•DOI•

Multidimensional scaling : I. theory and method

[...]

Warren S. Torgerson¹•Institutions (1)

Social Science Research Council¹

01 Dec 1952-Psychometrika

TL;DR: In this paper, a scale of comparative distances between all pairs of stimuli is obtained, which is analogous to the scale of stimuli obtained in the traditional paired comparisons methods, but instead of locating each stimulus-object on a given continuum, the distances between each pair of stimuli are located on a distance continuum.

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Abstract: Multidimensional scaling can be considered as involving three basic steps. In the first step, a scale of comparative distances between all pairs of stimuli is obtained. This scale is analogous to the scale of stimuli obtained in the traditional paired comparisons methods. In this scale, however, instead of locating each stimulus-object on a given continuum, the distances between each pair of stimuli are located on a distance continuum. As in paired comparisons, the procedures for obtaining a scale of comparative distances leave the true zero point undetermined. Hence, a comparative distance is not a distance in the usual sense of the term, but is a distance minus an unknown constant. The second step involves estimating this unknown constant. When the unknown constant is obtained, the comparative distances can be converted into absolute distances. In the third step, the dimensionality of the psychological space necessary to account for these absolute distances is determined, and the projections of stimuli on axes of this space are obtained. A set of analytical procedures was developed for each of the three steps given above, including a least-squares solution for obtaining comparative distances by the complete method of triads, two practical methods for estimating the additive constant, and an extension of Young and Householder's Euclidean model to include procedures for obtaining the projections of stimuli on axes from fallible absolute distances.

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1,984 citations

"The analysis of proximities: Multid..." refers methods in this paper

...There are many examples in this class: Richardson's method of triadic combinations [24], Klingberg's multidimensional method of rank orders [19], Torgerson's complete method of triads ([ 32 ]; [33], pp. 263-268), and Messick and Abelson's method of successive intervals [1, 22]....
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Mathematics, measurement, and psychophysics.

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S. S. Stevens

01 Jan 1951

1,005 citations

"The analysis of proximities: Multid..." refers background or methods in this paper

...In fact the computer program described here can be regarded, in the terminology of Stevens ([ 30 ], p. 25), as an automatic method for essentially transforming the given "ordinal scale" of similarities into a "ratio scale" of distances....
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...a unit of measurement [5], it has been widely recognized that much can be accomplished even when the measuring operations are not sufficiently quantitative to yield what Stevens [ 30 ] has termed an "interval scale." Following Coombs, scales obtained by these less quantitative operations have been called "ordered metric scales" because the operations usuutly determined partial ordering on the distances between stimuli....
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...In such cases an ordered metric scale is stronger than an ordinal but weaker than an interval scale (el., [ 30 ], p. 25)....
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...In this case the analysis begins, essentially, with a rank ordering of 105 proximity measures and yields, as output, 30 coordinates on axes with the properties of an interval scale (see [ 30 ], p. 25)....
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