Summary
In this paper a new algorithmic approach via an eigenvector iteration to fit l p -distances to given dissimilarities with respect to the weighted least squares loss function (Stress) is presented. The optimization problem is cast in the general setting of maximizing a nondifferentiable convex function over the level set of another convex function. Necessary conditions are shown to exhibit a nonlinear eigenproblem, which is approached by a new eigenvector iteration process. Global convergence properties of this algorithm for 1 ≤ p ≤ 2 are established and a psychometric application is given.
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References
ARABIE, P. (1991): Was Euclid an unnecessarily sophisticated psychologist? Psychometrika56, 567–587.
ATTNEAVE, F. (1950): Dimensions of similarity. Amer. J. Psychol.63, 516–556.
FICHET, B. (1988): L p spaces in data analysis. In: H.H. Bock (ed.): Classification and related methods of data analysis, North-Holland, Amsterdam, 439–444.
GREGSON, R.A.M. (1965): Representation of taste mixture cross-modal-matching in a Minkowski-r-metric. Australian Journal of Psychology17, 3, 195–204.
GROENEN, P.J.F., MATHAR, R., HEISER, W.J. (1993): The majorization approach to multidimensional scaling for Minkowski distances. Journal of Classification, submitted.
HEISER, W.J. (1989): The city-block model for three-way multidimensional scaling. In: R. Coppi, S. Bolasco (eds.): Multivariate data analysis, North-Holland, Amsterdam, 395–404.
HUBERT, L., ARABIE, P., HESSON-MCINNIS, M. (1992): Multidimensional scaling in the city-block metric: a combinatorial approach. Journal of Classification 9, 211–236.
KRUSKAL, J.B. (1964): Nonmetric multidimensional scaling: a numerical method. Psy-chometrika 29, 115–130.
KRUSKAL, J.B., YOUNG, F.W., SEERY, J.B. (1977): How to use KYST-2, a flexible program to do MDS and unfolding. Murray Hill, N.J., AT&T Bell Labs.
MATHAR, R., MEYER, R. (1993): Algorithms in convex analysis to fit l p-distance matrices. Journal of Multivariate Analysis, submitted.
MEULMAN, J.J. (1986): A distance approach to nonlinear multivariate analysis. Leiden, DSWO Press.
MEYER, R. (1991a): Multidimensional scaling as a framework for correspondence analysis and its extensions. In: M. Schader (ed.): Analyzing and modeling data and knowledge, Springer, Berlin, 99–108.
MEYER, R. (1991b): Canonical correlation analysis as a starting point for extensions of correspondence analysis. Statistique et Analyse des Données16, 1, 55–77.
OSBORNE, M.R. (1975): Some special nonlinear least squares problems. SIAM J. Numer. Anal.12, 571–592.
PARLETT, B.N. (1980): The symmetric eigenvalue problem. Prentice Hall, E.C.
RESTLE, F. (1959): A metric and an ordering on sets. Psychometrika 24, 207–220.
ROBERT, F. (1967): Calcul du rapport maximal de deux normes sur IRn . R.I.R.O.5, 97–118.
ROCKAFELLAR, R.T. (1970): Convex analysis. Princeton University Press, Princeton.
ROSKAM, E.E. (1981): Data theory and scaling: a methodological essay on the role and use of data theory and scaling in psychology. In: I. BORG (ed.): Multidimensional data representations: when and why, Mathesis Press, Ann Arbor, Michigan.
SHEPARD, R.N. (1964): Attention and the metric structure of the stimulus space. Journal of Math. Psychology 1, 54–87.
TVERSKY, A., KRANTZ, D.H. (1970): The dimensional representation and the metric structure of similarity data. J. Math. Psychology 7, 572–596.
WATSON, G.A. (1985): On a class of algorithms for total approximation. Journal of Approximation Theory 45, 219–231.
WILKINSON, L. (1988): SYSTAT: The system for statistics. Evanston, IL, SYSTAT, Inc.
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Meyer, R. (1994). An Eigenvector Algorithm to Fit l p -Distance Matrices. In: Diday, E., Lechevallier, Y., Schader, M., Bertrand, P., Burtschy, B. (eds) New Approaches in Classification and Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-51175-2_58
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DOI: https://doi.org/10.1007/978-3-642-51175-2_58
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58425-4
Online ISBN: 978-3-642-51175-2
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