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Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation

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Abstract

Multidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances. Most degeneracies are easily recognized: the solutions are perfect but trivial, characterized by approximately equal distances between points from different sets. A definition of an absolutely degenerate solution is proposed, which makes clear that these solutions only occur when an intercept is present in the transformation function. Many solutions for the degeneracy problem have been proposed and tested, but with little success so far. In this paper, we offer a substantial modification of an approach initiated bythat introduced a normalization factor based on thevariance in the usual least squares loss function. Heiser unpublishedthesis, (1981) and showed that the normalization factor proposed by Kruskal and Carroll was not strong enough to avoid degeneracies. The factor proposed in the present paper, based on the coefficient of variation, discourages or penalizes nonmetric transformations of the proximities with small variation, so that the procedure steers away from solutions with small variation in the interpoint distances. An algorithm is described for minimizing the re-adjusted loss function, based on iterative majorization. The results of a simulation study are discussed, in which the optimal range of the penalty parameters is determined. Two empirical data sets are analyzed by our method, clearly showing the benefits of the proposed loss function.

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References

  • Bennett J.F., Hayes W.L. (1960) Multidimensional unfolding: Determining the dimensionality of ranked preference data. Psychometrika, 25:36–48

    Google Scholar 

  • Borg I., Bergermaier R. (1982) Degenerationsprobleme im Unfolding und Ihre ösung. Zeitschrift ür Sozialpsychologie, 13:287–299

    Google Scholar 

  • Borg I., Groenen P.J.F. (1997) Modern Multidimensional Scaling: Theory and Applications. New York: Springer

    Google Scholar 

  • Borg I., Lingoes J. (1987) Multidimensional Similarity Structure Analysis. Berlin: Springer

    Google Scholar 

  • Busing F.M.T.A., Heiser W.J. (2003) PREFSCAL Progress Report: two-way models (Tech. Rep.) Leiden, The Netherlands: Leiden University. (working paper)

    Google Scholar 

  • Carroll, J.D. (1972). In R.N. Shepard, A.K. Romney S.B. Nerlove (Eds), Multidimensional Scaling: Theory and Applications in the Behavioral Sciences (Vol. 1, pp 105–155) New York: Seminar Press

  • Coombs C.H. (1950) Psychological scaling without a unit of measurement. Psychological Review, 57:148–158

    Google Scholar 

  • Coombs C.H. (1964) A Theory of Data. New York: Wiley

    Google Scholar 

  • Coombs C.H., Kao R.C. (1960) On a connection between factor analysis and multidimensional unfolding. Psychometrika, 25:219–231

    Google Scholar 

  • Dagpunar A. (1988) Principles of Random Variate Generation. Oxford: Clarendon Press

    Google Scholar 

  • De Leeuw J. (1977) Applications of convex analysis to multidimensional scaling. In J.R. Barra, F. Brodeau, G. Romier B.van Cutsem (Eds), Recent Developments in Statistics (pp. 133–145)Amsterdam, The Netherlands: North-Holland

    Google Scholar 

  • De Leeuw, J. (1983). On Degenerate Nonmetric Unfolding Solutions (Tech. Rep.). Department of Data Theory, FSW/RUL.

  • De Leeuw J., Heiser W.J. (1980) Multidimensional scaling with restrictions on the configuration. In P.R. Krishnaiah (Ed.), Multivariate Analysis (VOL 5, pp 501–522)Amsterdam, The Netherlands: North-Holland

    Google Scholar 

  • Dennis J.E., Schnabel R.B. (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs, New Jersey: Prentice-Hall. (Republished by SIAM, Philadelphia, in 1996 as Vol. 16 of Classics in Applied Mathematics)

  • DeSarbo W.S., Carroll J.D. (1985) Three-way metric unfolding via alternating weighted least squares. Psychometrika, 50(3):275–300

    Google Scholar 

  • DeSarbo W.S., Rao V.R. (1984) GENFOLD2: A set of models and algorithms for the GENeral unFOLDing analysis of preference/dominance data. Journal of Classification, 1:147–186

    Google Scholar 

  • DeSarbo W.S., Young M.R., Rangaswamy A. (1997). A parametric Multidimensional Unfolding Procedure for Incomplete Nonmetric Preference/Choice Set Data in Marketing Research (Tech. Rep.). The Pennsylvania State University.(Working paper)

  • Dinkelbach W. (1967) On nonlinear fractional programming. Management Science, 13:492–498

    Google Scholar 

  • Gower J.C. (1966) Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika, 53:325–338

    Google Scholar 

  • Green P.E., Rao V. (1972) Applied multidimensional scaling. Hinsdale, IL: Dryden Press

    Google Scholar 

  • Groenen P.J.F. (1993). The majorization Approach to Multidimensional Scaling: Some Problems and Extensions.Leiden, The Netherlands: DSWO Press

    Google Scholar 

  • Groenen P.J.F., Heiser W.J. (1996) The tunneling method for global optimization in multidimensional scaling. Psychometrika, 61:529–550

    Google Scholar 

  • Hardy G.H., Littlewood J.E., Polya G. (1952) Inequalities. Cambridge: Cambridge University Press

    Google Scholar 

  • Hayes W.L., Bennett J.F. (1961) Multidimensional unfolding: Determining configuration from complete rank order preference data. Psychometrika, 26:221–238

    Google Scholar 

  • Heiser, W.J. (1981). Unfolding analysis of proximity data. Unpublished doctoral dissertation, Leiden University.

  • Heiser W.J. (1987) Joint ordination of species and sites: The unfolding technique. In P.Legendre L.Legendre (Eds.), Developments in Numerical Ecology (pp. 189–221)Berlin, Heidelberg: Springer

    Google Scholar 

  • Heiser W.J. (1989) Order invariant unfolding analysis under smoothness restrictions. In G. De Soete, H. Feger, K.C. Klauer (Eds.) New Developments in Psychological Choice Modeling (pp. 3–31)Amsterdam: North-Holland

    Google Scholar 

  • Heiser W.J. (1995) Convergent computation by iterative majorization: Theory and applications in multidimensional data analysis. In W.J. Krzanowski(Ed), Recent Advances in Descriptive Multivariate Analysis (pp. 157–189)Oxford: Oxford University Press

    Google Scholar 

  • Heiser, W.J., & De Leeuw J. (1979). How to Use SMACOF-III: A Program for Metric Multidimensional Unfolding (Tech. Rep.). Leiden University, Department of Data Theory.

  • Katsnelson J., Kotz S. (1957) On the upper limits of some measures of variability. Archivfür Meterologie, Geophysik and Bioklimatologie (B), 8:103

    Google Scholar 

  • Kim C., Rangaswamy A., DeSarbo W.S. (1999) A quasi-metric approach to multidimensional unfolding for reducing the occurence of degenerate solutions. Multivariate Behavioral Research, 34(2):143–180

    Article  Google Scholar 

  • Kruskal J.B. (1964a) Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis. Psychometrika, 29:1–27

    Google Scholar 

  • Kruskal J.B. (1964b) Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29:115–129

    Google Scholar 

  • Kruskal J.B. (1977) Multidimensional scaling and other methods for discovering structure. In K. Enslein, A. Ralston H.S. Wilf (Eds.), Mathematical Methods for Digital Computers (VOL. 2, pp 296-339)New York: Wiley

    Google Scholar 

  • Kruskal J.B., Carroll J.D. (1969) Geometrical models and badness-of-fit functions.In P.R. Krishnaiah (Ed.), Multivariate Analysis(vol. 2, pp 639–671)New York: Academic Press

    Google Scholar 

  • Kruskal J.B., Young F.W., Seery J.B. (1978) How to Use KYST, A Very Flexible Program to do Multidimensional Scaling and Unfolding(Tech. Rep.)Murray Hill, NJ: Bell Laboratories

    Google Scholar 

  • L’Ecuyer P. (1999) Tables of maximally equidistributed combined LFSR generators. Mathematics of Computing, 68:261–269

    Article  Google Scholar 

  • Marden J. I. (1995) Analyzing and Modeling Rank Data. London: Chapman and Hall

    Google Scholar 

  • McClelland G.H., Coombs C.H. (1975) ORDMET: A general algorithm for constructing all numerical solutions to ordered metric solutions. Psychometrika, 40:269–290

    Google Scholar 

  • Pearson K. (1896) Regression, heridity, and panmixia. Philosophical Transactions of the Royal Society of London, Series A, 187:253–318

    Google Scholar 

  • Roskam E.E. Ch.I. (1968) Metric Analysis of Ordinal Data. Voorschoten: VAM. Shepard, R.N. (1974) Representation of structure in similarity data: Problems and prospects. Psychometrika, 39(4):373–421

    Google Scholar 

  • Takane Y., Young F.W., Leeuw J. (1977) Nonmetric individual differences MDS: An alternating least squares method with optimal scaling features. Psychometrika, 42:7–67

    Google Scholar 

  • Takane Y., Young F.W., Leeuw J. (1977) Nonmetric individual differences MDS: An alternating least squares method with optimal scaling features. Psychometrika, 42:7–67

    Google Scholar 

  • Torgerson W.S. (1958) Theory and Methods of Scaling. New York: Wiley

    Google Scholar 

  • Trosset M.W. (1998) A new formulation of the nonmetric STRAIN problem in multidimensional scaling. Journal of Classification, 15:15–35

    Article  Google Scholar 

  • Van Blokland-Vogelesang A.W. (1989) Unfolding and consensus ranking: A prestige ladder for technical occupations. In G. De Soete, H. Feger K. C. Klauer (Eds.), New Developments in Psychological Choice Modeling (pp. 237–258)Amsterdam, The Netherlands: North-Holland

    Google Scholar 

  • Van Blokland-Vogelesang A.W. (1993) A nonparametric distance model for unidimensional unfolding. In M. A. Fligner J. S. Verducci (Eds.), Probability Models and Statistical Analyses for Ranking Data (pp. 241–276)New York: Springer-Verlag

    Google Scholar 

  • Wagenaar W.A., Padmos P. (1971) Quantitative interpretation of Stress in Kruskal’s method multidimensional scaling technique. British Journal of Mathematical and Statistical Psychology, 24:101–110

    Google Scholar 

  • Winsberg, S., Carroll, J. D. (1989) A quasi-nonmetric method for multidimensional scaling via an extended Euclidean model. Psychometrika, 54:217–229

    MathSciNet  Google Scholar 

  • Young F. W. (1972) A model for polynomial conjoint analysis algorithms. In R.N. Shepard, A.K. Romney S.B. Nerlove (Eds.), Multidimensional Scaling, Theory(VOL I, pp 69–104)New York

    Google Scholar 

  • Young F.W., Torgerson W.S. (1967) T0RSCA, a Fortran IV program for Shepard-Kruskal multidimensional scaling analysis. Behavioral Science, 12:498

    Google Scholar 

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Authors and Affiliations

  1. Leiden University, Leiden

    Frank M. T. A. Busing

  2. Erasmus University Rotterdam, Leiden

    Patrick J. K. Groenen

  3. Leiden University, Leiden

    Willem J. Heiser

  4. Department of Psychology, Leiden University, 9555, 2300, RB, Leiden

    Frank M. T. A. Busing

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  1. Frank M. T. A. Busing
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  2. Patrick J. K. Groenen
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  3. Willem J. Heiser
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Correspondence to Frank M. T. A. Busing.

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The authors would like to thank the editor, an associate editor, and three reviewers for their valuable comments and suggestions to improve the quality of this work.

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Busing, F.M.T.A., Groenen, P.J.K. & Heiser, W.J. Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation. Psychometrika 70, 71–98 (2005). https://doi.org/10.1007/s11336-001-0908-1

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