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Cluster Differences Unfolding for Two-Way Two-Mode Preference Rating Data

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Abstract

Classification and spatial methods can be used in conjunction to represent the individual information of similar preferences by means of groups. In the context of latent class models and using Simulated Annealing, the cluster-unfolding model for two-way two-mode preference rating data has been shown to be superior to a two-step approach of first deriving the clusters and then unfolding the classes. However, the high computational cost makes the procedure only suitable for small or medium-sized data sets, and the hypothesis of independent and normally distributed preference data may also be too restrictive in many practical situations. Therefore, an alternating least squares procedure is proposed, in which the individuals and the objects are partitioned into clusters, while at the same time the cluster centers are represented by unfolding. An enhanced Simulated Annealing algorithm in the least squares framework is also proposed in order to address the local optimum problem. Real and artificial data sets are analyzed to illustrate the performance of the model.

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References

  • BÖCKENHOLT, U., and BÖCKENHOLT, I., (1991), “Constrained Latent Class Analysis: Simultaneous Classification and Scaling of Discrete Choice Data”, Psychometrika 56, 699–716.

    Article  Google Scholar 

  • BORG I., and GROENEN P.J.F., (2005), Modern Multidimensional Scaling: Theory and Applications (2nd ed.), New York: Springer.

    Google Scholar 

  • BUSING, F.M.T.A., (2006), “Avoiding Degeneracy in Metric Unfolding by Penalizing the Intercept”, British Journal of Mathematical and Statistical Psychology 59, 419–427.

    Article  MathSciNet  Google Scholar 

  • BUSING, F.M.T.A., HEISER, W., and GROENEN, P.J.F., (2005), “Avoiding Degeneracy in Multidimensional Unfolding by Penalizing on the Coefficient of Variation”, Psychometrika 70(1), 71–98.

    Article  MathSciNet  Google Scholar 

  • CALINSKI, R.B., and HARABASZ, J. (1974), “A Dendrite Method for Cluster Analysis”, Communications in Statistics 3, 1–27.

    MathSciNet  MATH  Google Scholar 

  • CHINTAGUNTA, P.K., (1994), “Heterogeneous Logit Implications for Brand Positioning”, Journal of Marketing Research 31, 304–311.

    Article  Google Scholar 

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

    Google Scholar 

  • DESARBO, W.S., HOWARD, D.J., and JEDIDI, K., (1990), “MULTICLUS: A New Method for Simultaneously Performing Multidimensional Scaling and Cluster Analysis”, Psychometrika 56, 121–136.

    Article  Google Scholar 

  • DESARBO, W. S., JEDIDI, K.J., COOL, K., and SCHENDEL, O., (1991), “Simultaneous Multidimensional Unfolding and Cluster Analysis: An Investigation of Strategic Groups”, Marketing Letters 2, 129–146.

    Article  Google Scholar 

  • DESARBO, W.S., MANRAI, A.K., and MANRAI, L.A., (1994), “Latent Class Multidimensional Scaling: A Review of Recent Developments in the Marketing and Psychometric Literature”, in Advanced Methods of Marketing Research, ed. R.P. Bagozzi, Cambridge: Blackwell, pp. 190–222.

  • DESARBO, W. S., RAMASWAMY, V., and CHATTERJEE, R., (1995), “Analyzing Constant-Sum Multiple Criterion Data: A Segment-Level Approach”, Journal of Marketing Research 32, 222–232.

    Article  Google Scholar 

  • DESARBO, W.S., YOUNG, M.R., and RANGASWAMY, A., (1997), “A Parametric Multidimensional Unfolding Procedure for Incomplete Nonmetric Preference/Choice Set Data in Marketing Research”, Technical Report, The Pennsylvania State University.

  • DE SOETE, G., and CARROLL, J.D., (1994). “K-means Clustering in a Low Dimensional Euclidean Space”, in New Approaches in Classification and Data Analyisis, eds. E. Diday et al., Heidelberg: Springer Verlag, pp. 212–219.

  • DE SOETE, G., and HEISER, W.J., (1993), “A Latent Class Unfolding Model for Analyzing Single Stimulus Preference Ratings”, Psychometrika, 58, 545–565.

    Article  MathSciNet  MATH  Google Scholar 

  • DE SOETE, G., and WINSBERG, S., (1993), “A Latent Class Vector Model for Preference Ratings”, Journal of Classification 10, 195–208.

    Article  MATH  Google Scholar 

  • EVERIT, B. S., LANDAU, S., and LEESE, M., (2001). Cluster Analysis (4th ed.), London: Arnold.

  • HEISER, W.J., (1981), ”Unfolding Analysis of Proximity Data”, unpublished doctoral dissertation, University of Leiden, The Netherlands.

  • HEISER, W.J., (1987), “The Unfolding Technique”, in Developments in Numerical Ecology, eds. P. Legendre & L. Legendre, Berlin: Springer-Verlag, pp. 189–221.

  • HEISER, W.J., (1991), “A Generalized Majorization Method for Least Squares Multidimensional Scaling of Pseudodistances that May Be Negative”, Psychometrika, 56, 7-27.

    Article  MathSciNet  MATH  Google Scholar 

  • HEISER W.J., (1993), “Clustering in Low-Dimensional Space”, in Information and Classification: Concepts, Methods and Applications, eds. B. Lausen, R. Klar, and O. Opitz, Heidelberg: Springer Verlag, pp. 162–173.

  • HEISER W.J, and GROENEN P.J.F. (1997), “Cluster Differences Scaling with a Within-Clusters Loss Component and a Fuzzy Succesive Approximation Strategy to Avoid Local Minima”, Psychometrika, 62, 63–83.

    Article  MathSciNet  MATH  Google Scholar 

  • KIERS, H.A.L., VICARI, D., and VICHI, M., (2005), “Simultaneous Classification and Multidimensional Scaling with External Information”, Psychometrika 70, 433–460.

    Article  MathSciNet  Google Scholar 

  • KRANTZ, D.H., (1967), “Rational Distance Functions for Multidimensional Scaling”, Journal of Mathematical Psychology 4, 226–245

    Article  MATH  Google Scholar 

  • LUCE, R.D., (1961), “A Choice Theory Analysis of Similarity Judgments”, Psychometrika, 26, 151–163.

    Article  MATH  Google Scholar 

  • METROPOLIS, N.A., ROSENBLUTH, M., ROSENBLUTH, A., TELLER, A., and TELLER, E. (1953), “Equation of State Calculations by Fast Computing Machines”, Journal of Chemical Physics, 21, 1087–1092.

    Article  Google Scholar 

  • MILLIGAN, G.W., and COOPER, M.C., (1985), “An Examination of Procedures for Determining the Number of Clusters in a Data Set”, Psychometrika, 50, 159–179.

    Article  Google Scholar 

  • MURILLO, A., VERA, J.F., and HEISER, W.J., (2005), “A Permutation-Translation Simulated Annealing Algorithms for l1 and l2 Unidimensional Scaling”, Journal of Classification 22, 119–138.

    Article  MathSciNet  MATH  Google Scholar 

  • ROCCI, R., and VICHI, M., (2008), “Two-Mode Multi-Partitioning”, Computational Statistics and Data Analysis, 52(8),1984-2003.

    Article  MathSciNet  MATH  Google Scholar 

  • SUGAR, C.A., and JAMES, G.M., (2003), “Finding the Number of Clusters in a Dataset: An Information-Theoretic Approach”, Journal of the American Statistical Asssociation, 98, 750–762.

    Article  MathSciNet  MATH  Google Scholar 

  • VAN BUUREN, S., and HEISER, W.J., (1989), “Clustering Objects into Groups under Optimal Scaling of Variables”, Psychometrika 54, 699–706.

    Article  MathSciNet  Google Scholar 

  • VAN DEUN, K., MARCHAL, K., HEISER, W.J., ENGELEN, K., and VAN MECHELEN, I., (2007), “Joint Mapping of Genes and Conditions via Multidimensional Unfolding Analysis”, BMC Bioinformatics, 8:181.

    Article  Google Scholar 

  • VAN MECHELEN, I., BOCK, H.-H., and DE BOECK, P. (2004), ”Two-Mode Clustering Methods: A Structured Overview”, Statistical Methods in Medical Research, 13, 363-394.

    Article  MathSciNet  MATH  Google Scholar 

  • VAN ROSMALEN, J., GROENEN, P.J.F., TREJOS, J., and CASTILLO, W., (2009), “Optimization Strategies for Two-Mode Partitioning”, Journal of Classification 26, 155-181.

    Article  MathSciNet  Google Scholar 

  • VERA, J.F., and DÍAZ-GARCÍA, J.A., (2008), “A Global Simulated Annealing Heuristic for the Three-Parameter Lognormal Maximum Likelihood Estimation”, Computational Statistics and Data Analysis, 52(12), 5055–5065.

    Article  MathSciNet  MATH  Google Scholar 

  • VERA, J.F., HEISER, W.J., and MURILLO, A., (2007), “Global Optimization in any Minkowski Metric: A Permutation-Translation Simulated Annealing Algorithm for Multidimensional Scaling”, Journal of Classification, 24, 277–301.

    Article  MathSciNet  MATH  Google Scholar 

  • VERA, J.F., MACÍAS, R., and ANGULO, J.M., (2008), “Non-Stationary Spatial Covariance Structure Estimation in Oversampled Domains by Cluster Differences Scaling with Spatial Constraints”, Stochastic Environmental Research and Risk Assessment, 22, 95–106.

    Article  MathSciNet  MATH  Google Scholar 

  • VERA, J.F., MACÍAS, R., and ANGULO, J.M., (2009), “A Latent Class MDS Model withSpatial Constraints for Non-Stationary Spatial Covariance Estimation”, Stochastic Environmental Research and Risk Assessment. 23(6), 769–779.

    Article  MathSciNet  Google Scholar 

  • VERA, J.F., MACÍAS, R., and HEISER, W.J., (2009a), “A Latent Class MultidimensionalScaling Model for Two-Way One-Mode Continuous Rating Dissimilarity Data”, Psychometrika, 74(2), 297–315.

    Article  MathSciNet  MATH  Google Scholar 

  • VERA, J.F., MACÍAS, R., and HEISER, W.J., (2009b), “A Dual Latent Class Unfolding Model for Two-Way Two-Mode Preference Rating Data”, Computational Statistics and Data Analysis, 53(8), 3231–3244.

    Article  MathSciNet  MATH  Google Scholar 

  • VICHI, M., and KIERS, H.A.L., (2001), “Factorial k-Means Analysis for Two-Way Data”, Computational Statistics and Data Analysis 37, 49–64.

    Article  MathSciNet  MATH  Google Scholar 

  • WANG, M.M, SCHÖNEMANN, P.H., and Rusk, J.B., (1975), “A Conjugate Gradient Algorithm for the Multidimensional Analysis of Preference Data”, Multivariate Behavioral Research, 10, 45–79.

    Article  Google Scholar 

  • WEDEL, M., and DESARBO, W.S., (1996), “An Exponential-Family Multidimensional Scaling Mixture Methodology”, Journal of Business and Economic Statistics 14, 447-459.

    Google Scholar 

  • WINKLER, G., (1995), Image Analysis, Random Fields and Dynamic Monte Carlo Methods, New York: Springer-Verlag.

    Book  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Statistics and O.R., Faculty of Sciences, University of Granada, 18071, Granada, Spain

    J. Fernando Vera

  2. Centro de Investigación en Matemáticas, Guanajuato, México

    Rodrigo Macías

  3. Leiden University, Leiden, The Netherlands

    Willem J. Heiser

Authors
  1. J. Fernando Vera
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  2. Rodrigo Macías
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  3. Willem J. Heiser
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Corresponding author

Correspondence to J. Fernando Vera.

Additional information

This work has been partially supported by the project CTM2009-13411 of the MICINN, Spain, and CONACYT, CB 105657, Mexico.

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Vera, J.F., Macías, R. & Heiser, W.J. Cluster Differences Unfolding for Two-Way Two-Mode Preference Rating Data. J Classif 30, 370–396 (2013). https://doi.org/10.1007/s00357-013-9144-5

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