Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-02-11T01:49:02.267Z Has data issue: false hasContentIssue false
  • English
  • Français

Was Euclid an Unnecessarily Sophisticated Psychologist?

Published online by Cambridge University Press:  01 January 2025

Phipps Arabie
Phipps Arabie*
Affiliation:
Graduate School of Management, Rutgers University
*
Requests for reprints should be sent to Phipps Arabic, Graduate School of Management, Rutgers University, 92 New Street, Newark, NJ 07102-1895.
Get access Rights & Permissions [Opens in a new window]

Abstract

A survey of the current state of multidimensional scaling using the city-block metric is presented. Topics include substantive and theoretical issues, recent algorithmic developments and their implications for seemingly straightforward analyses, isometries with other metrics, links to graph-theoretic models, and future prospects.

Keywords

multidimensional scalingfacility planingcity-block distancesMinkowski metricsrectangular metricrectilinear metricManhattan metriccombination rules
Type
Original Paper
Information
Psychometrika , Volume 56 , Issue 4 , December 1991 , pp. 567 - 587
Copyright
Copyright © 1991 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Presented as the 1991 Psychometric Society Presidential Address. I am indebted to Doug Carroll, John Daws, Jan de Leeuw, Geert De Soete, Wayne DeSarbo, Eric Holman, Larry Hubert, Chingis Izmailov, Joe Kruskal, Rob Nosofsky, Akinori Okada, Roger Shepard, Auke Tellegen, and Wijbrandt van Schuur for many helpful comments on this research and to Yuko Minowa for bibliographic assistance. Doug Carroll's comments on an early draft were especially useful. Parts of this research were supported by a grant from AT&T to the University of Illinois. Some of the computational results reported here were obtained during the early 1970's at the Institute for Mathematical Studies in the Social Sciences at Stanford. Grateful acknowledgment is made to the Institute's staff and its Co-Directors, Richard C. Atkinson and Patrick Suppes.

References

References

Arabie, P. (1973). Concerning Monte Carlo evaluations of nonmetric scaling algorithms. Psychometrika, 38, 607–608.CrossRefGoogle Scholar
Arabie, P., Boorman, S. A. (1973). Multidimensional scaling of measures of distance between partitions. Journal of Mathematical Psychology, 10, 148–203.CrossRefGoogle Scholar
Arabie, P., Carroll, J. D., DeSarbo, W. S. (1987). Three-way scaling and clustering, Newbury Park, CA: Sage.CrossRefGoogle Scholar
Arabie, P., Hubert, L. (1992). Combinatorial data analysis. Annual Review of Psychology, 43, 169–203.CrossRefGoogle Scholar
Arnold, J. B. (1971). A multidimensional scaling study of semantic distance. Journal of Experimental Psychology Monograph, 90(2), 349–372.CrossRefGoogle Scholar
Attneave, F. (1950). Dimensions of similarity. American Journal of Psychology, 63, 516–556.CrossRefGoogle ScholarPubMed
Backhaus, W., Menzel, R., Kreissl, S. (1987). Multidimensional scaling of color similarity in bees. Biological Cybernetics, 56, 293–304.CrossRefGoogle Scholar
Beckman, P. (1971). A history of π (pi) 3rd ed.,, New York: St. Martin's Press.Google Scholar
Blaschke, W. (1916). Räumliche Variationsprobleme mit symmetrischer Transversalitätsbedingung [Spatial variations problems with a symmetric transversal condition]. Berichten Sächsich Akademie Wissenschaft Leipzig, 68, 50–55.Google Scholar
Blough, D. S. (1972). Recognition by the pigeon of stimuli varying in two dimensions. Journal of the Experimental Analysis of Behavior, 18, 345–367.CrossRefGoogle ScholarPubMed
Boorman, S. A., Arabie, P. (1972). Structural measures and the method of sorting. In Shepard, R. N., Romney, A. K., Nerlove, S. B. (Eds.), Multidimensional scaling: Theory and applications in the behavioral sciences. Vol. 1: Theory (pp. 225–249). New York: Seminar Press.Google Scholar
Boorman, S. A., Olivier, D. C. (1973). Metrics on spaces of finite trees. Journal of Mathematical Psychology, 10, 26–59.CrossRefGoogle Scholar
Borg, I. (1983). Scaling: A review of the German scaling literature of the past 15 years. German Journal of Psychology, 7, 63–79.Google Scholar
Borg, I., Leutner, D. (1983). Dimensional models for the perception of rectangles. Perception & Psychophysics, 34, 257–267.CrossRefGoogle ScholarPubMed
Borg, I., Lingoes, J. (1987). Multidimensional similarity structure analysis, New York: Springer-Verlag.CrossRefGoogle Scholar
Bortz, J. (1974). Kritische Bemerkungen ueber den Einsatz nicht-euklidischer Metriken im Rahmen der multidimensionalen Skalierung [Critical remarks on the use of non-Euclidean metrics in the context of multidimensional scaling]. Archiv fuer Psychologie, 126, 196–212.Google Scholar
Boyd, J. P. (1972). Information distance for discrete structures. In Shepard, R. N., Romney, A. K., Nerlove, S. B. (Eds.), Multidimensional scaling: Theory and applications in the behavioral sciences (pp. 213–223). New York: Seminar Press.Google Scholar
Brandeau, M. L., Chiu, S. S. (1988). Parametric facility location on a tree network with an L p norm cost function. Transportation Science, 22, 59–69.CrossRefGoogle Scholar
Buja, A., Logan, B., Reeds, J., & Shepp, L. (1991, June). Garbage in, structure out—the performance of MDS on null data. Paper presented at joint meeting of the Classification Society of North America and the Psychometric Society, New Brunswick, NJ.Google Scholar
Busemann, H. (1950). The foundations of Minkowskian geometry. Commentarii Mathematici Helvetici, 24, 156–186.CrossRefGoogle Scholar
Busemann, H. (1955). The geometry of geodesics, New York: Academic Press.Google Scholar
Busemeyer, J. R., Jones, L. E. (1983). Analysis of multiplicative combination rules when the causal variables are measured with error. Psychological Bulletin, 93, 549–562.CrossRefGoogle Scholar
Carroll, J. D. (1976). Spatial, non-spatial and hybrid models for scaling. Psychometrika, 41, 439–463.CrossRefGoogle Scholar
Carroll, J. D., Arabie, P. (1980). Multidimensional scaling. In Rosenzweig, M. R., Porter, L. W. (Eds.), Annual review of psychology (pp. 607–649). Palo Alto, CA: Annual Reviews.Google Scholar
Carroll, J. D., Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition. Psychometrika, 35, 283–319.CrossRefGoogle Scholar
Carroll, J. D., Chang, J. J. (1973). A method for fitting a class of hierarchical tree structure models to dissimilarities data and its application to some “body parts” data of Miller's. Proceedings of the 81st Annual Convention of the American Psychological Association, 8, 1097–1098.Google Scholar
Carroll, J. D., Wish, M. (1974). Multidimensional perceptual models and measurement methods. In Carterette, E. C., Friedman, M. P. (Eds.), Handbook of perception (pp. 391–447). New York: Academic Press.Google Scholar
Celeux, G., & Govaert, G. (1991). Clustering criteria for discrete data and latent class models. Journal of Classification, 8.CrossRefGoogle Scholar
Coombs, C. H. (1964). A theory of data, New York: Wiley.Google Scholar
Cross, D. V. (1965). Multidimensional stimulus control of the discriminative response in experimental conditioning and psychophysics, Ann Arbor: University of Michigan.Google Scholar
Cross, D. V. (1965). Metric properties of multidimensional stimulus generalization. In Mostofsky, D. I. (Eds.), Stimulus generalization (pp. 72–93). Stanford: Stanford University Press.Google Scholar
Daws, J. T. (1991, June). The analysis of free-sort data: Beyond pairwise cooccurrences. Paper presented at joint meeting of the Classification Society of North America and the Psychometric Society, New Brunswick, NJ.Google Scholar
Daws, J. T., Arabie, P., & Hubert, L. J. (1990, June). A note on expected stress values in nonmetric multidimensional scaling. Paper presented at meeting of the Classification Society of North America, Logan, UT.Google Scholar
de Leeuw, J., Heiser, W. (1977). Convergence of correction-matrix algorithms for multidimensional scaling. In Lingoes, J. C. (Eds.), Geometric representations of relational data: Readings in multidimensional scaling (pp. 735–752). Ann Arbor, MI: Mathesis.Google Scholar
de Leeuw, J., Heiser, W. (1982). Theory of multidimensional scaling. In Krishnaiah, P. R., Kanal, L. N. (Eds.), Handbook of statistics Vol.2: Classification, pattern recognition and reduction of dimensionality (pp. 285–316). Amsterdam: North-Holland.CrossRefGoogle Scholar
De Soete, G., Carroll, J. D., DeSarbo, W. S. (1987). Least squares algorithms for constructing constrained ultrametric and additive tree representations of symmetric proximity data. Journal of Classification, 4, 155–173.CrossRefGoogle Scholar
Dress, A., Krüger, M. (1987). Parsimonious phylogenetic trees in metric spaces and simulated annealing. Advances in Applied Mathematics, 8, 8–37.CrossRefGoogle Scholar
Dunn, T. R., Harshman, R. A. (1982). A multidimensional scaling model for the size-weight illusion. Psychometrika, 47, 25–45.CrossRefGoogle Scholar
Eaton, B. C., Lipsey, R. G. (1980). The block metric and the law of markets. Journal of Urban Economics, 7, 337–347.CrossRefGoogle Scholar
Eisler, H. (1973). The algebraic and statistical tractability of the city block metric. British Journal of Mathematical and Statistical Psychology, 26, 212–218.CrossRefGoogle Scholar
Eisler, H., Lindman, R. (1990). Representations of dimensional models of similarity. In Geissler, H.-O. (Eds.), Psychophysical explorations of mental structures (pp. 165–171). Toronto: Hogrefe & Huber.Google Scholar
Ekman, G. (1954). Dimensions of color vision. Journal of Psychology, 38, 467–474.CrossRefGoogle Scholar
Evans, M. G. (1991). The problem of analyzing multiplicative composites. American Psychologist, 46, 6–15.CrossRefGoogle Scholar
Felfoldy, G. L. (1974). Repetition effects in choice reaction time to multidimensional stimuli. Perception & Psychophysics, 15, 453–459.CrossRefGoogle Scholar
Fichet, B. (1987). The role played by L 1 in data analysis. In Dodge, Y. (Eds.), Statistical data analysis based on the L1-norm and related methods (pp. 185–193). New York: North-Holland.Google Scholar
Fichet, B. (1988). L p spaces in data analysis. In Bock, H.-H. (Eds.), Classification and related methods of data analysis (pp. 439–444). Amsterdam: North-Holland.Google Scholar
Fischer, W., Micko, H. C. (1972). More about metrics of subjective spaces and attention distributions. Journal of Mathematical Psychology, 9, 36–54.CrossRefGoogle Scholar
Francis, R. L., White, J. A. (1974). Facility layout and location: An analytical approach, Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
Garner, W. R. (1974). The processing of information and structure, Hillsdale, NJ: Erlbaum.Google Scholar
Garner, W. R. (1976). Interaction of stimulus dimensions in concept and choice processes. Cognitive Psychology, 8, 98–123.CrossRefGoogle Scholar
Govaert, G. (1984). Classification simultanée de tableaux binaires [Simultaneous classification of binary tables]. In Diday, E., Jambu, M., Lebart, L., Pagès, J., Tomassone, R. (Eds.), Data analysis and informatics, Vol. III (pp. 223–236). Amsterdam: North-Holland.Google Scholar
Gower, J. C. (1966). Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika, 53, 325–338.CrossRefGoogle Scholar
Gower, J. C. (1971). A general coefficient of similarity and some of its properties. Biometrics, 27, 857–871.CrossRefGoogle Scholar
Gower, J. C., Legendre, P. (1986). Metric and Euclidean properties of dissimilarity coefficients. Journal of Classification, 3, 5–48.CrossRefGoogle Scholar
Green, P. E., Rao, V. R. (1969). A note on proximity measures and cluster analysis. Journal of Marketing Research, 6, 359–364.CrossRefGoogle Scholar
Green, P. E., Srinivasan, V. (1990). Conjoint analysis in marketing: New developments with implications for research and practice. Journal of Marketing, 54, 3–19.CrossRefGoogle Scholar
Guénoche, A. (1987). Cinq algorithmes d'approximation d'une dissimilarite par des arbres a distances additives [Five algorithms of dissimilarity approximation by additive trees]. Mathématiques et Sciences humaines, 25(98), 21–40.Google Scholar
Hancock, H. (1964). Development of the Minkowski geometry of numbers, New York: Dover.Google Scholar
Hansen, P., Peeters, D., Thisse, J.-F. (1981). Constrained location and the Weber-Rawls problem. Annals of Discrete Mathematics, 11, 147–166.Google Scholar
Hansen, P., Peeters, D., Thisse, J.-F. (1982). An algorithm for a constrained Weber problem. Management Science, 11, 1285–1295.CrossRefGoogle Scholar
Hansen, P., Perreur, J., Thisse, J.-F. (1980). Location theory, dominance, and convexity: Some further results. Operations Research, 28, 1241–1250.CrossRefGoogle Scholar
Harary, F., Melter, R. A., Tomescu, I. (1984). Digital metrics: A graph-theoretical approach. Pattern Recognition Letters, 2, 159–163.CrossRefGoogle Scholar
Hardzinski, M., Pachella, R. G. (1977). A psychophysical analysis of complex integrated displays, Ann Arbor: University of Michigan, Human Performance Center.Google Scholar
Heath, T. L. (1956). The thirteen books of Euclid's Elements 2nd ed.,, New York: Dover.Google Scholar
Heiser, W. J. (1989). The city-block model for three-way multidimensional scaling. In Coppi, R., Bolasco, S. (Eds.), Multiway data analysis (pp. 395–404). Amsterdam: North-Holland.Google Scholar
Hewes, A. (1900). Social institutions and the Riemann surface. American Journal of Sociology, 5, 392–403.CrossRefGoogle Scholar
Householder, A. S., Landahl, H. D. (1945). Mathematical biophysics of the central nervous system, Bloomington, IN: Principia Press.CrossRefGoogle Scholar
Hubert, L. J. (1984). Statistical applications of linear assignment. Psychometrika, 49, 449–473.CrossRefGoogle Scholar
Hubert, L., Arabie, P. (1985). Comparing partitions. Journal of Classification, 2, 193–218.CrossRefGoogle Scholar
Hubert, L., Arabie, P. (1986). Unidimensional scaling and combinatorial optimization. In de Leeuw, J., Heiser, W., Meulman, J., Critchley, F. (Eds.), Multidimensional data analysis (pp. 181–196). Leiden, The Netherlands: DSWO-Press.Google Scholar
Hubert, L. J., Arabie, P. (1988). Relying on necessary conditions for optimization: Unidimensional scaling and some extensions. In Bock, H.-H. (Eds.), Classification and related methods of data analysis (pp. 463–472). Amsterdam: North-Holland.Google Scholar
Hubert, L. J., & Arabie, P., & Hesson-Mcinnis, M. (1991). Multidimensional scaling in the city-block metric: A combinatorial approach. Manuscript submitted for publication.Google Scholar
Hubert, L. J., Busk, P. (1976). Normative location theory: Placement in continuous space. Journal of Mathematical Psychology, 14, 187–210.CrossRefGoogle Scholar
Joly, S., & Le Calvé, G. (1991). Realisable 0-1 matrices and city block distance. Manuscript submitted for publication.Google Scholar
Jones, L. E. (1982). Construal of social environments: Multidimensional models of interpersonal perception and attraction. In Hirschberg, N., Humphreys, L. G. (Eds.), Multivariate applications in the social sciences (pp. 61–85). Hillsdale, NJ: Erlbaum.Google Scholar
Koopman, R. F., & Cooper, M. (1974, March). Some problems with Minkowski distance models in multidimensional scaling. Paper presented at the meeting of the Psychometric Society, Stanford, CA.Google Scholar
Krause, E. F. (1975). Taxicab geometry, Menlo Park, CA: Addison-Wesley.Google Scholar
Kruskal, J. B. (1964). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1–27.CrossRefGoogle Scholar
Kruskal, J. B. (1964). Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115–129.CrossRefGoogle Scholar
Kruskal, J. B., Carmone, F. (1970). How to use M-D-SCAL (Version 5M) and other useful information, Cambridge: Marketing Science Institute.Google Scholar
Kruskal, J. B., Carroll, J. D. (1969). Geometrical models and badness-of-fit functions. In Krishnaiah, P. R. (Eds.), Multivariate analysis II (pp. 639–671). New York: Academic Press.Google Scholar
Kruskal, J. B., Young, F. W., Seery, J. B. (1977). How to use KYST2, a very flexible program to do multidimensional scaling and unfolding, Murray Hill, NJ: AT&T Bell Laboratories.Google Scholar
Landahl, H. D. (1945). Neural mechanisms for the concepts of difference and similarity. Bulletin of Mathematical Biophysics, 7, 83–88.CrossRefGoogle Scholar
Larson, R. C., Sadiq, G. (1983). Facility locations with the Manhattan metric in the presence of barriers to travel. Operations Research, 31, 652–669.CrossRefGoogle Scholar
Lashley, K. S. (1942). An examination of the “continuity theory” as applied to discriminative learning. Journal of General Psychology, 26, 241–265.CrossRefGoogle Scholar
Le Calvé, G. (1987). L 1-embeddings of a data structure (I, D). In Dodge, Y. (Eds.), Statistical data analysis based on the L1-norm and related methods (pp. 195–202). New York: North-Holland.Google Scholar
Lew, J. S. (1978). Some counterexamples in multidimensional scaling. Journal of Mathematical Psychology, 17, 247–254.CrossRefGoogle Scholar
Lingoes, J. C., Borg, I. (1983). A quasi-statistical model for choosing between alternative configurations derived from ordinally constrained data. British Journal of Mathematical and Statistical Psychology, 36, 36–53.CrossRefGoogle Scholar
Lockhead, G. R., & Pomerantz, J. R. (Eds.)] (in press). The perception of structure. Arlington VA: American Psychological Association.Google Scholar
Lorentz, H. A., Einstein, A., Minkowski, H., Weyl, H. (1952). In Perrett, W., Jeffery, G. B. (Eds.), The principle of relativity, New York: Dover.Google Scholar
Melara, R. D. (1989). Similarity relations among synesthetic stimuli and their attributes. Journal of Experimental Psychology: Human Perception and Performance, 15, 212–231.Google ScholarPubMed
Mencken, H. L. (1967). The American language, New York: Knopf.Google Scholar
Micko, H. C., Fischer, W. (1970). The metric of multidimensional psychological spaces as a function of the differential attention to subjective attributes. Journal of Mathematical Psychology, 7, 118–143.CrossRefGoogle Scholar
Miller, G. A. (1969). A psychological method to investigate verbal concepts. Journal of Mathematical Psychology, 6, 169–191.CrossRefGoogle Scholar
Minkowski, H. (1891). Über die positiven quadratischen Formen und über kettenbruchahnliche Algorithmen [On positive quadratic forms and on algorithms suggesting continued fractions]. Journal für die reine und angewandte Mathematik (Crelle's Journal), 107, 278–297.Google Scholar
Minkowski, H. (1908). Space and time. In Lorentz, H. A., Einstein, A., Minkowski, H., Weyl, H. (Eds.), The principle of relativity (pp. 73–91). New York: Dover.Google Scholar
Minkowski, H. (1957). Diophantische Approximationen [Diophantine approximations], New York: Chelsea.Google Scholar
Mirkin, B. G., Chernyi, L. B. (1970). Measurement of the distance between distinct partitions of a finite set of objects [Avtomatika i Telemekhanika]. Automation and Remote Control, 1, 786–792.Google Scholar
Muller, J.-C. (1982). Non-Euclidean geographic spaces: Mapping functional distances. Geographical Analysis, 14, 189–203.Google Scholar
Nosofsky, R. M. (1985). Overall similarity and the identification of separable-dimension stimuli: A choice model analysis. Perception & Psychophysics, 38, 414–432.CrossRefGoogle ScholarPubMed
Nosofsky, R. M. (1986). Attention, similarity, and the identification-categorization relationship. Journal of Experimental Psychology: General, 115, 39–57.CrossRefGoogle ScholarPubMed
Nosofsky, R. M. (1989). Further tests of an exemplar-similarity approach to relation identification and categorization. Perception & Psychophysics, 45, 279–290.CrossRefGoogle ScholarPubMed
Nosofsky, R. M. (1992). Similarity scaling and cognitive process models. Annual Review of Psychology, 43, 25–53.CrossRefGoogle Scholar
Okada, A., Imaizumi, T. (1980). Nonmetric method for extended INDSCAL model. Behaviormetrika, 7, 13–22.CrossRefGoogle Scholar
Pachella, R. G., Somers, P., Hardzinski, M. (1981). A psychophysical approach to dimensional integrality. In Getty, D. J., Howard, J. H. Jr. (Eds.), Auditory and visual pattern recognition (pp. 107–126). Hillsdale, NJ: Erlbaum.Google Scholar
Pliner, V. M. (1986). The problem of multivariate metric scaling. Automation and Remote Control, 4, 140–148.Google Scholar
Poole, K. T. (1990). Least squares metric, unidimensional scaling of multivariate linear models. Psychometrika, 55, 123–149.CrossRefGoogle Scholar
Ramsay, J. O. (1977). Maximum likelihood estimation in multidimensional scaling. Psychometrika, 42, 241–266.CrossRefGoogle Scholar
Rand, W. M. (1971). Objective criteria for the evaluation of clustering methods. Journal of the American Statistical Association, 66, 846–850.CrossRefGoogle Scholar
Rao, V. R., Sabavala, D. J. (1981). Inference of hierarchical choice processes from panel data. Journal of Consumer Research, 8, 85–96.CrossRefGoogle Scholar
Richards, J. L. (1988). Mathematical visions: The pursuit of geometry in Victorian England, San Diego: Academic Press.Google Scholar
Richardson, G. D. (1981). The appropriateness of using various Minkowskian metrics for representing cognitive configurations. Environment and Planning A, 13, 475–485.CrossRefGoogle Scholar
Richardson, H. W., Anjomani, A. (1981). The diamond city: The case for rectangular grid models. Socio-Economic Planning Sciences, 15, 295–303.CrossRefGoogle Scholar
Robinson, W. S. (1951). A method for chronologically ordering archaeological deposits. American Antiquity, 16, 293–301.CrossRefGoogle Scholar
Rosenberg, S. (1982). The method of sorting in multivariate research with applications selected from cognitive psychology and person perception. In Hirschberg, N., Humphreys, L. G. (Eds.), Multivariate applications in the social sciences (pp. 117–142). Hillsdale, NJ: Erlbaum.Google Scholar
Rushton, G., Thill, J. C. (1989). The effect of distance metric on the degree of spatial competition between firms. Environment and Planning A, 21, 499–508.CrossRefGoogle Scholar
Russell, B. (1956). An essay on the foundations of geometry, New York: Dover.Google Scholar
Russell, J. A. (1980). A circumplex model of affect. Journal of Personality and Social Psychology, 39, 1161–1178.CrossRefGoogle Scholar
Schmalensee, R., Thisse, J. F. (1986). Perceptual maps and the optimal location of new products (Report No. 86-103), Cambridge, MA: Marketing Science Institute.Google Scholar
Schoenberg, I. J. (1937). On certain metric spaces arising from Euclidean spaces by a change of metric and their imbedding in Hilbert space. Annals of Mathematics, 38, 787–793.CrossRefGoogle Scholar
Schoenberg, I. J. (1938). Metric spaces and positive definite functions. Transactions of the American Mathematical Society, 44, 522–536.CrossRefGoogle Scholar
Schönemann, P. H. (1990). Psychophysical maps for rectangles. In Geissler, H.-O. (Eds.), Psychophysical explorations of mental structures (pp. 149–164). Toronto: Hogrefe & Huber.Google Scholar
Schönemann, P. H., Dorcey, T., Kienapple, K. (1985). Subadditive concatenation in dissimilarity judgments. Perception & Psychophysics, 38, 1–17.CrossRefGoogle ScholarPubMed
Shepard, R. N. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance function. I. Psychometrika, 27, 125–140.CrossRefGoogle Scholar
Shepard, R. N. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance function. II. Psychometrika, 27, 219–246.CrossRefGoogle Scholar
Shepard, R. N. (1964). Attention and the metric structure of the stimulus space. Journal of Mathematical Psychology, 1, 54–87.CrossRefGoogle Scholar
Shepard, R. N. (1974). Representation of structure in similarity data: Problems and prospects. Psychometrika, 39, 373–421.CrossRefGoogle Scholar
Shepard, R. N. (1978). The circumplex and related topological manifolds in the study of perception. In Shye, S. (Eds.), Theory construction and data analysis in the behavioral sciences (pp. 29–80). San Francisco: Jossey-Bass.Google Scholar
Shepard, R. N. (1980a). Internal representation. Unpublished manuscript.Google Scholar
Shepard, R. N. (1980). Multidimensional scaling, tree-fitting, and clustering. Science, 210, 390–398.CrossRefGoogle ScholarPubMed
Shepard, R. N. (1986). Discrimination and generalization in identification and classification: Comment on Nosofsky. Journal of Experimental Psychology: General, 115, 58–61.CrossRefGoogle Scholar
Shepard, R. N. (1987). Toward a universal law of generalization. Science, 237, 1317–1323.CrossRefGoogle Scholar
Shepard, R. N. (in press). Integrality versus separability of stimulus dimensions: From an early convergence of evidence to a proposed theoretical basis. In Lockhead, G. R. & Pomerantz, J. R. (Eds.), The perception of structure. Arlington VA: American Psychological Association.Google Scholar
Shepard, R. N., Arabie, P. (1979). Additive clustering: Representation of similarities as combinations of discrete overlapping properties. Psychological Review, 86, 87–123.CrossRefGoogle Scholar
Shepard, R. N., Cermak, G. W. (1973). Perceptual-cognitive explorations of a toroidal set of free-form stimuli. Cognitive Psychology, 4, 351–377.CrossRefGoogle Scholar
Sherman, C. R. (1972). Nonmetric multidimensional scaling: A Monte Carlo study of the basic parameters. Psychometrika, 37, 323–355.CrossRefGoogle Scholar
Shreider, Y. A. (1974). What is distance?, Chicago: University of Chicago Press.Google Scholar
Suppes, P., Krantz, D. M., Luce, R. D., Tversky, A. (1989). Foundations of measurement: Vol. II. Geometrical, threshold, and probabilistic representations, New York: Academic Press.Google Scholar
Thisse, J.-F., Ward, J. E., Wendell, R. E. (1984). Some properties of location problems with block and round norms. Operations Research, 32, 1309–1327.CrossRefGoogle Scholar
Tong, Y. C., Dowell, R. C., Blamey, P. J., Clark, G. M. (1983). Two component hearing sensations produced by two-electrode stimulation in the cochlea of deaf patient. Science, 219, 993–994.CrossRefGoogle ScholarPubMed
Torgerson, W. S. (1958). Theory and methods of scalling, New York: Wiley.Google Scholar
Townes, B. D., Abbott, R. D. (1974). The psychological interpretation of metrics in ordinal multidimensional scaling (JSAS #604), Washington, DC: American Psychological Association.Google Scholar
Trudeau, R. J. (1987). The Non-Euclidean revolution, Boston: Birkhäuser.Google Scholar
Tversky, A., Gati, I. (1982). Similarity, separability, and the triangle inequality. Psychological Review, 89, 123–154.CrossRefGoogle ScholarPubMed
Viegas, J., Hansen, P. (1985). Finding shortest paths in the plane in the presence of barriers to travel (for any l p-norm). European Journal of Operational Research, 20, 373–381.CrossRefGoogle Scholar
Von Hohenbalken, B., West, D. S. (1984). Manhattan versus Euclid: Market areas computed and compared. Regional Science and Urban Economics, 14, 19–35.CrossRefGoogle Scholar
Ward, J. E., Wendell, R. E. (1985). Using block norms for location modeling. Operations Research, 33, 1074–1090.CrossRefGoogle Scholar
Wiener-Ehrlich, W. K. (1978). Dimensional and metric structures in multidimensional stimuli. Perception & Psychophysics, 24, 399–414.CrossRefGoogle ScholarPubMed
Wiggins, J. S. (1982). Circumplex models of interpersonal behavior in clinical psychology. In Kendall, P. C., Butcher, J. N. (Eds.), Handbook of research methods in clinical psychology (pp. 183–221). New York: Wiley.Google Scholar
Wilkie, D. M. (1989). Evidence that pigeons represent Euclidean properties of space. Journal of Experimental Psychology: Animal Behavior Processes, 15, 114–123.Google Scholar
Wise, J. A., Mockovak, W. P. (1973). Descriptive modeling of subjective probabilities. Organizational Behavior and Human Performance, 9, 292–306.CrossRefGoogle Scholar
Wolfrum, C. (1976). Zum Auftreten quasiaequivalenter Lösungen bei einer Verallgemeinerung des Skalierungsverfahrens von Kruskal auf metrische Räume mit einer Minkowski-Metrik [On the existence of quasi-equivalent solutions with a generalization of Kruskal's scaling procedure in metric spaces with a Minkowski metric]. Archiv für Psychologie, 128, 96–111.Google Scholar
Wolfrum, C. (1976). Zur Bestimmung eines optimalen Metrikoeffizienten r mit dem Skailerungsverfahren von Kruskal [On the determination of optimal metric coefficients r with Kruskal's scaling procedure]. Zeitschrift für experimentelle und angewandte Psychologie, 23, 339–350.Google Scholar

Reference

Glazer, R., Nakamoto, K. (1991). Cognitive geometry: An analysis of structure underlying representations of similarity. Marketing Science, 10, 205–228.CrossRefGoogle Scholar