Representation of structure in similarity data: Problems and prospects
- Roger N. Shepard
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
After struggling with the problem of representing structure in similarity data for over 20 years, I find that a number of challenging problems still remain to be overcome—even in the simplest case of the analysis of a single symmetric matrix of similarity estimates. At the same time, I am more optimistic than ever that efforts directed toward surmounting the remaining difficulties will reap both methodological and substantive benefits. The methodological benefits that I forsee include both an improved efficiency and a deeper understanding of “discovery” methods of data analysis. And the substantive benefits should follow, through the greater leverage that such methods will provide for the study of complex empirical phenomena—perhaps particularly those characteristic of the human mind.
Presidential address delivered before the annual meeting of the Psychometric Society at Stanford University on March 28, 1974 (in a somewhat revised form necessarily, for purposes of publication, with more words and fewer pictures). Much of the work surveyed in this paper was supported by the Bell Telephone Laboratories, during my eight years as a member of the technical staff there, and by grants (principally, GS-1302, GS-2283, and GB-31971X) from the National Science Foundation, during my subsequent eight years at Harvard and at Stanford. I wish to acknowledge, too, the important contributions to this work made by my many students and associates during these 16 years including, particularly, Doug Carroll, Jih-Jie Chang, Steve Johnson, and Joe Kruskal (at the Bell Laboratories) and Phipps Arabie, Glen Crawford, and Jim Cunningham (at Stanford). Finally, I am indebted to Sherry Huntsberger for suggestions and extensive help in connection with the preparation of the paper itself, and to Bert Green for useful editorial comments.
Abelson, R. P., and Tukey, J. W. Efficient conversion of nonmetric information into metric information.Proceedings of the American Statistical Association Meetings, Social Statistics Section, 1959, 226–230.
Anderson, J. R., and Bower, G. H. Human associative memory. Washington, D. C.: V. H. Winston & Sons, 1973.
Arabie, P. Concerning Monte Carlo evaluations of nonmetric multidimensional scaling algorithms.Psychometrika, 1973,38, 607–608.
Arabie, P. and Boorman, S. A. Multidimensional scaling of measures of distance between partitions.Journal of Mathematical Psychology, 1973,10, 148–203.
Arabie, P. and Shepard, R. N. Representation of similarities as additive combinations of discrete overlapping properties. Presented at the Mathematical Psychology Meeting in Montreal, August, 1973.
Arnold, J. B. A multidimensional scaling study of semantic distance.Journal of Experimental Psychology Monograph, 1971,90, 349–372.
Attneave, F. Dimensions of similarity.American Journal of Psychology, 1950,63, 516–556.
Beals, R., Krantz, D. H., and Tversky, A. Foundations of multidimensional scaling.Psychological Review, 1968,75, 127–142.
Bennett, R. S. The intrinsic dimensionality of signal collections. Doctoral dissertation, The Johns Hopkins University, 1965. University Microfilms, Inc., Ann Arbor, Michigan.
Bennett, R. S. The intrinsic dimensionality of signal collections.IEEE Transactions on Information Theory, 1969, Vol. IT-15, No. 5, 517–525.
Blank, A. A. Axiomatics of binocular vision: The foundations of metric geometry in relation to space perception.Journal of the Optical Society of America, 1958,48, 328–334.
Blank, A. A. The Luneburg theory of binocular space perception. In S. Koch (Ed.),Psychology: A study of a science. New York: McGraw-Hill, 1959. Pp. 395–426.
Blumenthal, L. M.Theory and applications of distance geometry. Oxford: Clarendon Press, 1953.
Boyd, J. P. Information distance for discrete structures. In R. N. Shepard, A. K. Romney, & S. B. Nerlove (Eds.),Multidimensional scaling: Theory and applications in the behavioral sciences, Vol. I. New York: Seminar Press, 1972. Pp. 213–223.
Busemann, H.The geometry of geodesics. New York: Academic Press, 1955.
Carroll, J. D. Individual differences and multidimensional scaling. In R. N. Shepard, A. K. Romney, & S. B. Nerlove (Eds.),Multidimensional scaling: Theory and applications in the behavioral sciences, Vol. I. New York: Seminar Press, 1972. (a) Pp. 105–155.
Carroll, J. D. Algorithms for rotation and interpretation of dimensions and for configuration matching. Bell Telephone Laboratories, 1972. (b) (Multilithed handout prepared for the workshop on multidimensional scaling, held at the University of Pennsylvania in June, 1972.)
Carroll, J. D., and Chang, J.-J. Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition.Psychometrika, 1970,35, 283–319.
Chang, J.-J., and Shepard, R. N. Exponential fitting in the proximity analysis of confusion matrices. Presented at the annual meeting of the Eastern Psychological Association, New York, April 14, 1966.
Clark, H. H. Linguistic processes in deductive reasoning.Psychological Review, 1969,76, 387–404.
Clark, H. H. Word associations and linguistic theory. In J. Lyons (Ed.),New horizons in linguistics. Baltimore, Maryland: Penguin Books, 1970. Pp. 271–286.
Coombs, C.A theory of data. New York: Wiley, 1964.
Cross, D. V. Metric properties of multidimensional stimulus generalization. In D. I. Mostofsky (Ed.),Stimulus generalization. Stanford, Calif.: Stanford University Press, 1965. Pp. 72–93.
Cunningham, J. P. On finding an optimal tree realization of a proximity matrix. Presented at the mathematical psychology meeting, Ann Arbor, Michigan, August 29, 1974.
Cunningham, J. P., and Shepard, R. N. Monotone mapping of similarities into a general metric space.Journal of Mathematical Psychology, 1974,11, (in press).
Degerman, R. L. Multidimensional analysis of complex structure: Mixtures of class and quantitative variation.Psychometrika, 1970,35, 475–491.
Ekman, G. Dimensions of color vision.Journal of Psychology, 1954,38, 467–474.
Fillenbaum, S., and Rapoport, A.Structures in the subjective lexicon, New York: Academic Press, 1971.
Foley, J. M. The size-distance relation and the intrinsic geometry of visual space.Vision Research, 1972,12, 323–332.
Greenberg, J. H.Language universals. The Hague: Mouton, 1966.
Guttman, L. A new approach to factor analysis: The radex. In P. F. Lazarsfeld (Ed.),Mathematical thinking in the social sciences. Glencoe, Illinois: Free Press, 1954. Pp. 258–348.
Guttman, L. A generalized simplex for factor analysis.Psychometrika, 1955,20, 173–192.
Guttman, N., and Kalish, H. I. Discriminability and stimulus generalization.Journal of Experimental Psychology, 1956,51, 79–88.
Halle, M. On the bases of phonology. In J. A. Fodor and J. J. Katz (Eds.),The structure of language. Englewood Cliffs, N. J.: Prentice-Hall, 1964. Pp. 324–333.
Harshman, R. A. Foundations of the parafac procedure: Models and conditions for an explanatory multi-modal factor analysis. Unpublished doctoral dissertation, University of California at Los Angeles, 1970.
Haviland, S. E., and Clark, E. V. ‘This man's father is my father's son’: A study of the acquisition of English kin terms.Journal of Child Language, 1974,1, 23–47.
Helm, C. E. Multidimensional ratio scaling analysis of perceived color relations.Journal of the Optical Society of America, 1964,54, 256–262.
Hyman, R., and Well, A. Judgments of similarity and spatial models.Perception and Psychophysics, 1967,2, 233–248.
Hyman, R., and Well, A. Perceptual separability and spatial models.Perception and Psychophysics, 1968,3, 161–165.
Indow, T. Applications of multidimensional scaling in perception. In E. C. Carterette and M. P. Friedman (Eds.),Handbook of perception, Vol. 2. New York: Academic Press, in press.
Jardine, N., and Sibson, R.Mathematical taxonomy. New York: Wiley, 1971.
Johnson, S. C. Hierarchical clustering schemes.Psychometrika, 1967,32, 241–254.
Klahr, D. A Monte Carlo investigation of the statistical significance of Kruskal's nonmetric scaling procedure.Psychometrika, 1969,34, 319–330.
Koopman, R. F., and Cooper, M. Some problems with Minkowski distance models in multidimensional scaling. Presented at the annual meeting of the Psychometric Society, Stanford University, March 28, 1974.
Kruskal, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis.Psychometrika, 1964,29, 1–27. (a)
Kruskal, J. B. Nonmetric multidimensional scaling: A numerical method.Psychometrika, 1964,29, 28–42. (b)
Kruskal, J. B. Linear transformation of multivariate data to reveal clustering. In R. N. Shepard, A. K. Romney, & S. B. Nerlove (Eds.),Multidimensional scaling: Theory and applications in the behavioral sciences. Vol I. New York: Seminar Press, 1972. Pp. 179–191.
Kruskal, J. B. Multidimensional scaling and other methods for discovering structure. In A. J. Ralston, H. S. Wilf, and K. Enslein, (Eds.),Statistical methods for digital computers. New York: Wiley, in press.
Levelt, W. J. M., Van de Geer, J. P., and Plomp, R. Triadic comparisons of musical intervals.British Journal of Mathematical and Statistical Psychology, 1966,19, 163–179.
Lévi-Strauss, C.The savage mind. Chicago: University of Chicago Press, 1967.
Liberman, A. M., Cooper, F. S., Shankweiler, D. P., and Studdert-Kennedy, M. Perception of the speech code.Psychological Review, 1967,74, 431–461.
Lingoes, J. C. A general survey of the Guttman-Lingoes nonmetric program series. In R. N. Shepard, A. K. Romney, & S. B. Nerlove (Eds.),Multidimensional scaling: Theory and applications in the behavioral sciences. Vol. I. New York: Seminar Press, 1972.
Luneburg, R. K.Mathematical analysis of binocular vision. Princeton, N. J.: Princeton University Press, 1947.
Luneburg, R. K. The metric of binocular visual space.Journal of Optical Society of America, 1950,40, 637–642.
Miller, G., and Nicely, P. E. An analysis of perceptual confusions among some English consonants.Journal of the Acoustical Society of America, 1955,27, 338–352.
Osgood, C. E., Suci, G. J., and Tannenbaum, P. H.The measurement of meaning. Urbana, Ill.: University of Illinois Press, 1957.
Peters, R. W. Dimensions of perception of consonants.Journal of the Acoustical Society of America, 1963,35, 1985–1989.
Quillian, M. R. Semantic memory. In M. Minsky (Ed.),Semantic information processing. Cambridge, Mass.: M.I.T. Press, 1968.
Rips, L. J., Shoben, E. J., and Smith, E. E. Semantic distance and the verification of semantic relations.Journal of Verbal Learning and Verbal Behavior, 1973,12, 1–20.
Romney, A. K., and D'Andrade, R. G. Cognitive aspects of English kinship terms.American Anthropologist, 1964,66, 146–170.
Rumelhart, D. E., and Abrahamson, A. A. A model for analogical reasoning.Cognitive Psychology, 1973,5, 1–28.
Rumelhart, D. E., Lindsay, P. H., and Norman, D. A. A process model for long-term memory. In E. Tulving and W. Donaldson (Eds.),Organization of memory. New York: Academic Press, 1972.
Rund, H.The differential geometry of Finsler space. Berlin: Springer-Verlag, 1959.
Shepard, R. N. Multidimensional scaling of concepts based upon sequences of restricted associative responses.American Psychologist, 1957,12, 440–441. (abstract) (a)
Shepard, R. N. Stimulus and response generalization: A stochastic model relating generalization to distance in psychological space.Psychometrika, 1957,22, 325–345. (b)
Shepard, R. N. Stimulus and response generalization: Deduction of the generalization gradient from a trace model.Psychological Review, 1958,65, 242–256. (a)
Shepard, R. N. Stimulus and response generalization: Tests of a model relating generalization to distance in psychological space.Journal of Experimental Psychology, 1958,55, 509–523. (b)
Shepard, R. N. Similarity of stimuli and metric properties of behavioral data. In H. Gulliksen and S. Messick (Eds.),Psychological scaling: Theory and applications. New York: Wiley, 1960. Pp. 33–43.
Shepard, R. N. The analysis of proximities: Multidimensional scaling with an unknown distance function. I.Psychometrika, 1962,27, 125–140. (a)
Shepard, R. N. The analysis of proximities: Multidimensional scaling with an unknown distance function. II.Psychometrika, 1962,27, 219–246. (b)
Shepard, R. N. Analysis of proximities as a technique for the study of information processing in man.Human Factors, 1963,5, 33–48. (a)
Shepard, R. N. Comments on Professor Underwood's paper “Stimulus selection in verbal learning.” In C. N. Cofer and B. S. Musgrave (Eds.),Verbal behavior and learning: Problems and processes. New York: McGraw-Hill, 1963. Pp. 48–70. (b)
Shepard, R. N. Attention and the metric structure of the stimulus space.Journal of Mathematical Psychology, 1964,1, 54–87. (a)
Shepard, R. N. Polynomial fitting in the analysis of proximities.Proceedings of the XVIIth international congress of psychology. Amsterdam: North-Holland Publisher, 1964, 345–346. (abstract) (b)
Shepard, R. N. Approximation to uniform gradients of generalization by monotone transformations of scale. In D. I. Mostofsky (Ed.),Stimulus Generalization. Stanford, Calif.: Stanford University Press, 1965. Pp. 94–110.
Shepard, R. N. Computer explorations of psychological space. Invited research address presented at the annual meeting of the American Psychological Association, New York, September 3, 1966. (a)
Shepard, R. N. Metric structures in ordinal data.Journal of Mathematical Psychology, 1966,3, 287–315. (b)
Shepard, R. N. Continuity versus the triangle inequality as a central principle for the spatial analysis of similarity data. Presented in a symposium on alternative models for the geometric representation of psychological data, at the mathematical psychology meetings, Stanford University, August 28, 1968.
Shepard, R. N. Introduction to Volume I. In R. N. Shepard, A. K. Romney, & S. B. Nerlove (Eds.),Multidimensional scaling: Theory and applications in the behavioral sciences, Vol. I. New York: Seminar Press, 1972. Pp. 1–20. (a)
Shepard, R. N. A taxonomy of some principal types of data and of multidimensional methods for their analysis. In R. N. Shepard, A. K. Romney, & S. B. Nerlove (Eds.),Multidimensional scaling: Theory and applications in the behavioral sciences, Vol. I. New York: Seminar Press, 1972. Pp. 21–47. (b)
Shepard, R. N. Psychological representation of speech sounds. In E. E. David & P. B. Denes (Eds.),Human communication: A unified view. New York: McGraw-Hill, 1972. Pp. 67–113. (c)
Shepard, R. N., and Carroll, J. D. Parametric representation of nonlinear data structures. In P. R. Krishnaiah (Ed.),Multivariate analysis: Proceedings of an international symposium. New York: Academic Press, 1966. Pp. 561–592.
Shepard, R. N., and Cermak, G. W. Perceptual-cognitive explorations of a toroidal set of free-form stimuli.Cognitive Psychology, 1973,4, 351–377.
Shepard, R. N., Hovland, C. I. and Jenkins, H. M. Learning and memorization of classifications.Psychological Monographs, 1961,75, (13, whole no. 517).
Shepard, R. N., Kilpatric, D. W., and Cunningham, J. P. The internal representation of numbers.Cognitive Psychology, in press.
Silberstein, L. Investigations of the intrinsic properties of the color domain.Journal of the Optical Society of America, 1938,28, 63–85.
Silberstein, L., and MacAdam, D. L. The distribution of color matchings around a color center.Journal of the Optical Society of America, 1945,35, 32–39.
Sokal, R. R., and Sneath, P. H. A.Principles of numerical taxonomy. San Francisco: W. H. Freeman, 1963.
Stenson, H. H., and Knoll, R. L. Goodness of fit for random rankings in Kruskal's nonmetric scaling procedure.Psychological Bulletin, 1969,71, 122–126.
Thomas, H. Spatial models and multidimensional scaling of random shapes.American Journal of Psychology, 1968,81, 551–558.
Torgerson, W. S. Multidimensional scaling: I. theory and method.Psychometrika, 1952,17, 401–419.
Torgerson, W. S.Theory and methods of scaling. New York: Wiley, 1958.
Torgerson, W. S. Multidimensional scaling of similarity.Psychometrika, 1965,30, 379–393.
Tucker, L. R. Relations between multidimensional scaling and three-mode factor analysis.Psychometrika, 1972,37, 3–27.
Wish, M., and Carroll, J. D. Applications of “INDSCAL” to studies of human perception and judgment. In E. C. Carterette and M. P. Friedman (Eds.),Handbook of perception, Vol. 2. New York: Academic Press, in press.
Young, F. W. Nonmetric multidimensional scaling: Recovery of metric information.Psychometrika, 1970,35, 455–473.
Young, F. W., and Torgerson, W. S. TORSCA, A FORTRAN IV program for Shepard-Kruskal multidimensional scaling analysis.Behavioral Science, 1967,12, 498.
- Representation of structure in similarity data: Problems and prospects
Volume 39, Issue 4 , pp 373-421
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Industry Sectors
- Roger N. Shepard (1)
- Author Affiliations
- 1. Stanford University, USA