Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis

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.

[1]  H. Eggleston,et al.  Elements of the theory of functions , 1952 .

[2]  G. Ekman Dimensions of Color Vision , 1954 .

[3]  R. Shepard Stimulus and response generalization: A stochastic model relating generalization to distance in psychological space , 1957 .

[4]  E. Rothkopf A measure of stimulus similarity and errors in some paired-associate learning tasks. , 1957, Journal of experimental psychology.

[5]  C. H. Coombs,et al.  An Application of a Nonmetric Model for Multidimensional Analysis of Similarities , 1958 .

[6]  Joseph B. Kruskal,et al.  The coefficients in an allocation problem , 1958 .

[7]  Joseph B. Kruskal,et al.  Assigning quantitative values to qualitative factors in the naval electronics problem , 1959 .

[8]  D. J. Bartholomew,et al.  A TEST OF HOMOGENEITY FOR ORDERED ALTERNATIVES. II , 1959 .

[9]  T. Indow,et al.  Multidimensional mapping of Munsell colors varying in hue and chroma. , 1960, Journal of experimental psychology.

[10]  T. Indow,et al.  Multidimensional mapping of Munsell colors varying in hue, chroma, and value. , 1960, Journal of experimental psychology.

[11]  Richard C. W. Kao,et al.  On a connection between factor analysis and multidimensional unfolding , 1960 .

[12]  S. Fomin,et al.  Elements of the Theory of Functions and Functional Analysis , 1961 .

[13]  Joseph L. Zinnes,et al.  Theory and Methods of Scaling. , 1958 .

[14]  R. Shepard The analysis of proximities: Multidimensional scaling with an unknown distance function. II , 1962 .

[15]  R. Shepard The analysis of proximities: Multidimensional scaling with an unknown distance function. I. , 1962 .

[16]  G. Sperling A Model for Visual Memory Tasks1 , 1963, Human factors.

[17]  R N SHEPARD,et al.  Analysis of Proximities as a Technique for the Study of Information Processing in Man1 , 1963, Human factors.

[18]  J. Kruskal Nonmetric multidimensional scaling: A numerical method , 1964 .