Hostname: page-component-669899f699-swprf Total loading time: 0 Render date: 2025-04-26T10:19:57.509Z Has data issue: false hasContentIssue false
  • English
  • Français

The Integration of Multidimensional Scaling and Multivariate Analysis with Optimal Transformations

Published online by Cambridge University Press:  01 January 2025

Jacqueline J. Meulman
Jacqueline J. Meulman*
Affiliation:
Department of Data Theory, University of Leiden
*
Requests for reprints should be sent to Jacqueline J. Meulman, Department of Data Theory, Faculty of Social Sciences, University of Leiden, PO Box 9555, 2300 RB Leiden, THE NETHERLANDS.
Get access Rights & Permissions [Opens in a new window]

Abstract

The recent history of multidimensional data analysis suggests two distinct traditions that have developed along quite different lines. In multidimensional scaling (MDS), the available data typically describe the relationships among a set of objects in terms of similarity/dissimilarity (or (pseudo-)distances). In multivariate analysis (MVA), data usually result from observation on a collection of variables over a common set of objects. This paper starts from a very general multidimensional scaling task, defined on distances between objects derived from one or more sets of multivariate data. Particular special cases of the general problem, following familiar notions from MVA, will be discussed that encompass a variety of analysis techniques, including the possible use of optimal variable transformation. Throughout, it will be noted how certain data analysis approaches are equivalent to familiar MVA solutions when particular problem specifications are combined with particular distance approximations.

Keywords

multivariate analysismultidimensional scalingnonlinear transformationsdistance approximationmajorization
Type
Original Paper
Information
Psychometrika , Volume 57 , Issue 4 , December 1992 , pp. 539 - 565
Copyright
Copyright © 1992 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.)

Article purchase

Temporarily unavailable

Footnotes

This research was supported by the Royal Netherlands Academy of Arts and Sciences (KNAW). An earlier version of this paper was written during a stay at McGill University in Montréal; this visit was supported by a travel grant from the Netherlands Organization for Scientific Research (NWO). I am grateful to Jim Ramsay and Willem Heiser for their encouragement and helpful suggestions, and to the Editor and referees for their constructive comments.

References

Benzécri, J.-P. et al. (1973). L'analyse des données. 1. La Taxinomie. 2. L'analyse des Correspondances [Data analysis. 1. Taxonomy. 2. Correspondence analysis], Paris: Dunod.Google Scholar
Burt, C. (1950). The factorial analysis of qualitative data. British Journal of Psychology, Statistical Section, 3, 166–185.CrossRefGoogle Scholar
Carroll, J. D. (1968). Generalization of canonical correlation analysis to three or more sets of variables. Proceedings of the 76th annual convention of the American Psychological Association, 3, 227–228.Google Scholar
Carroll, J. D., & Chang, J. J. (1972, September). IDIOSCAL (Individual differences in orientation scaling): A generalization of INDSCAL allowing IDIOsyncratic reference systems as well as an analytic approximation to INDSCAL. Paper presented at the Psychometric Society Meeting, Princeton, NJ.Google Scholar
Carroll, J. D., Kruskal, J. B. (1968). Scaling, multidimensional. In Kruskal, W. H., Tanur, J. M. (Eds.), International encyclopedia of statistics (pp. 892–907). New York: The Free Press.Google Scholar
Chambers, J. M., Klein, B. (1982). Graphical techniques for multivariate data and for clustering. In Krisnaiah, P. R., Kanal, L. N. (Eds.), Handbook of statistics (pp. 209–244). Amsterdam: North-Holland.Google Scholar
Cliff, N. (1966). Orthogonal rotation to congruence. Psychometrika, 31, 33–42.CrossRefGoogle Scholar
Dauxois, J., Pousse, A. (1976). Les analyses factorielles en calcul des probabilités et en statistique: essai d'étude synthétique [Components analysis in probability and statistics: An attempt at a synthesis], Toulouse: Université Paul Sabatier.Google Scholar
de Leeuw, J. (1973). Canonical analysis of categorical data Second edition,, Leiden: DSWO Press.Google Scholar
de Leeuw, J. (1977). A normalized cone regression approach to alternating least squares algorithms, Leiden: University of Leiden, Department of Data Theory.Google Scholar
de Leeuw, J. (1988). Convergence of the majorization algorithm for multidimensional scaling. Journal of Classification, 5, 163–180.CrossRefGoogle Scholar
de Leeuw, J., Heiser, W. J. (1980). Multidimensional scaling with restrictions on the configuration. In Krisnaiah, P. R. (Eds.), Multivariate analysis (pp. 501–522). Amsterdam: North-Holland.Google Scholar
de Leeuw, J., Meulman, J. J. (1986). Principal component analysis and restricted multidimensional scaling. In Gaul, W., Schader, M. (Eds.), Classification as a tool of research (pp. 83–96). Amsterdam: North-Holland.Google Scholar
Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Annals of Statistics, 7, 1–26.CrossRefGoogle Scholar
Efron, B. (1982). The jackknife, the bootstrap, and other resampling plans (CMBS-NSF regional conference series in applied mathematics, Monograph 38), Philadelphia: SIAM.Google Scholar
Everitt, B. S. (1978). Graphical techniques for multivariate data, London: Heinemann.Google Scholar
Fisher, R. A. (1938). Statistical methods for research workers, Edinburgh: Oliver and Boyd.Google Scholar
Gifi, A. (1985). PRINCALS, Leiden: University of Leiden, Department of Data Theory.Google Scholar
Gifi, A. (1990). Nonlinear multivariate analysis First edition,, Chichester: Wiley.Google Scholar
Gower, J. C. (1966). Some distance properties of latent roots and vector methods used in multivariate analysis. Biometrika, 53, 325–338.CrossRefGoogle Scholar
Gower, J. C. (1967). Multivariate analysis and multidimensional geometry. The Statistician, 17, 13–28.CrossRefGoogle Scholar
Gower, J. C. (1968). Adding a point to vector diagrams in multivariate analysis. Biometrika, 55, 582–585.CrossRefGoogle Scholar
Gower, J. C., Harding, S. A. (1988). Nonlinear biplots. Biometrika, 75, 445–455.CrossRefGoogle Scholar
Greenacre, M. J. (1984). Theory and applications of correspondence analysis, London: Academic Press.Google Scholar
Guttman, L. et al. (1941). The quantification of a class of attributes: A theory and method of scale construction. In Horst, P. et al. (Eds.), The prediction of personal adjustment (pp. 319–348). New York: Social Science Research Council.Google Scholar
Hayashi, C. (1952). On the prediction of phenomena from qualitative data and the quantification of qualitative data from the mathematico-statistical point of view. Annals of the Institute of Statistical Mathematics, 2, 93–96.Google Scholar
Heiser, W. J. (1987). Correspondence analysis with least absolute residuals. Computational Statistics and Data Analysis, 5, 337–356.CrossRefGoogle Scholar
Heiser, W. J. (1987). Joint ordination of species and sites: the unfolding technique. In Legendre, P., Legendre, L. (Eds.), Developments in numerical ecology (pp. 189–221). New York: Springer.CrossRefGoogle Scholar
Heiser, W. J., Meulman, J. J. (1983). Analyzing rectangular tables by joint and constrained multidimensional scaling. Journal of Econometrics, 22, 139–167.CrossRefGoogle Scholar
Heiser, W. J., Meulman, J. J. (1983). Constrained multidimensional scaling, including confirmation. Applied Psychological Measurement, 7, 381–404.CrossRefGoogle Scholar
Heiser, W. J., & Meulman, J. J. (1990, June). Nonlinear biplots for nonlinear mappings. Paper presented at the XXIIème Journeés de Statistique, Tours, France.Google Scholar
Hotelling, H. (1936). Relations between two sets of variates. Biometrika, 28, 321–377.CrossRefGoogle Scholar
Jolliffe, I. T. (1986). Principal component analysis, New York: Springer Verlag.CrossRefGoogle Scholar
Kettenring, J. R. (1971). Canonical analysis of several sets of variables. Biometrika, 58, 433–460.CrossRefGoogle Scholar
Kruskal, J. B. (1964). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1–28.CrossRefGoogle Scholar
Kruskal, J. B. (1964). Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115–129.CrossRefGoogle Scholar
Kruskal, J. B. (1978). Factor analysis and principal components analysis: Bilinear methods. In Kruskal, W. H., Tanur, J. M. (Eds.), International encyclopedia of statistics (pp. 307–330). New York: The Free Press.Google Scholar
Kruskal, J. B., Carroll, J. D. (1969). Geometrical models and badness-of-fit functions. In Krisnaiah, P. R. (Eds.), Multivariate analysis (pp. 639–671). New York: Academic Press.Google Scholar
Kruskal, J. B., Shepard, R. N. (1974). A nonmetric variety of linear factor analysis. Psychometrika, 39, 123–157.CrossRefGoogle Scholar
Lebart, L., Morineau, A., Warwick, K. M. (1984). Multivariate descriptive statistical analysis, New York: Wiley.Google Scholar
Max, J. (1960). Quantizing for minimum distortion. Information Theory, 6, 7–12.CrossRefGoogle Scholar
Meulman, J. J. (1986). A distance approach to nonlinear multivariate analysis, Leiden: DSWO Press.Google Scholar
Meulman, J. J. (1988). Nonlinear redundancy analysis via distances. In Bock, H. H. (Eds.), Classification and related methods of data analysis (pp. 515–522). Amsterdam: North-Holland.Google Scholar
Meulman, J. J. (1989). Distance analysis in reduced canonical spaces. In Coppi, R., Bolasco, S. (Eds.), Multiway data analysis (pp. 233–244). Amsterdam: North-Holland.Google Scholar
Meulman, J. J. (1990, December). Multivariate analysis with nonlinear transformations, nonlinear mappings, and nonlinear biplots. Paper presented at the workshop “Linear models and multivariate statistical analysis,” Oberwolfach, Germany.Google Scholar
Meulman, J. J. (1991). Principal components analysis using distances, including weights for variables and dimensions. In Zidak, S. (Eds.), Proceedings of DIANA III, International Meeting on Discriminant Analysis (pp. 178–196). Prague: Czechoslovak Academy of Sciences.Google Scholar
Meulman, J. J., Heiser, W. J. (1988). Second order regression and distance analysis. In Gaul, W., Schader, M. (Eds.), Data, expert knowledge and decisions (pp. 368–380). Berlin: Springer-Verlag.CrossRefGoogle Scholar
Meulman, J. J., Zeppa, P., Boon, M. E., Rietveld, W. J. (1992). Prediction of various grades of cervical preneoplasia and neoplasia on plastic embedded cytobrush samples: Discriminant analysis with qualitative and quantitative predictors. Analytical and Quantitative Cytology and Histology, 14, 60–72.Google ScholarPubMed
Nishisato, S. (1980). Analysis of categorical data: Dual scaling and its applications, Toronto: University of Toronto Press.CrossRefGoogle Scholar
Ramsay, J. O. (1989). Monotone regression splines in action. Statistical Science, 4, 425–441.Google Scholar
Roskam, E. E. C. I. (1968). Metric analysis of ordinal data in psychology, Voorschoten: VAM.Google Scholar
Saporta, G. (1975). Liaisons entre plusieurs ensembles de variables et codage de données qualitatives [Relationships between various sets of variables and scaling of qualitative data], Paris: Université Paris VI.Google Scholar
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. (1966). Metric structures in ordinal data. Journal of Mathematical Psychology, 3, 287–315.CrossRefGoogle Scholar
Takane, Y., Young, F. W., de Leeuw, J. (1977). Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features. Psychometrika, 42, 7–67.CrossRefGoogle Scholar
Tenenhaus, M., Young, F. W. (1985). An anlaysis and synthesis of multiple correspondenece analysis, optimal scaling, dual scaling, homogeneity analysis and other methods for quantifying categorical multivariate data. Psychometrika, 50, 91–119.CrossRefGoogle Scholar
Torgerson, W. S. (1958). Theory and methods of scaling, New York: Wiley.Google Scholar
van de Geer, J. P. (1984). Relations among K sets of variables. Psychometrika, 49, 79–94.CrossRefGoogle Scholar
van de Geer, J. P. (1986). Introduction to linear multivariate data analysis, Leiden: DSWO Press.Google Scholar
van der Burg, E. (1988). Nonlinear canonical correlation and some related techniques, Leiden: DSWO Press.Google Scholar
van der Burg, E., de Leeuw, J. (1983). Nonlinear canonical correlation. British Journal of Mathematical and Statistical Psychology, 36, 54–80.CrossRefGoogle Scholar
van der Burg, E., de Leeuw, J., Verdegaal, R. (1988). Homogeneity analysis with k sets of variables: An alternating least squares method with optimal scaling features. Psychometrika, 53, 177–197.CrossRefGoogle Scholar
Weeks, D. G., Bentler, P. M. (1979). A comparison of linear and monotone multidimensional scaling models. Psychological Bulletin, 86, 349–354.CrossRefGoogle Scholar
Winsberg, S., Ramsay, J. O. (1983). Monotone spline transformations for dimension reduction. Psychometrika, 48, 575–595.CrossRefGoogle Scholar
Wolkowicz, H., Styan, G. P. H. (1980). Bounds for eigenvalues using traces. Linear Algebra and its Applications, 29, 471–506.CrossRefGoogle Scholar
Young, F. W., Takane, Y., de Leeuw, J. (1978). The principal components of mixed measurement level multivariate data: An alternating least squares method with optimal scaling features. Psychometrika, 43, 279–281.CrossRefGoogle Scholar
Young, G., Householder, A. S. (1938). Discussion of a set of points in terms of their mutual distances. Psychometrika, 3, 19–22.CrossRefGoogle Scholar