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Convex Optimization as a Tool for Correcting Dissimilarity Matrices for Regular Minimality

  • Matthias Trendtel
  • Ali Ünlü
Conference paper
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)

Abstract

Fechnerian scaling as developed by Dzhafarov and Colonius (e.g., Dzhafarov and Colonius, J Math Psychol 51:290–304, 2007) aims at imposing a metric on a set of objects based on their pairwise dissimilarities. A necessary condition for this theory is the law of Regular Minimality (e.g., Dzhafarov EN, Colonius H (2006) Regular minimality: a fundamental law of discrimination. In: Colonius H, Dzhafarov EN (eds) Measurement and representation of sensations. Erlbaum, Mahwah, pp. 1–46 ). In this paper, we solve the problem of correcting a dissimilarity matrix for Regular Minimality by phrasing it as a convex optimization problem in Euclidean metric space. In simulations, we demonstrate the usefulness of this correction procedure.

Notes

Acknowledgements

We are deeply indebted to Professor Ehtibar N. Dzhafarov for introducing us to this topic.

References

  1. Boyd, S., & Vandenberghe, L. (2009). Convex optimization. New York: Cambridge University Press.Google Scholar
  2. Dattorro, J. (2009). Convex optimization & euclidean distance geometry. Palo Alto: Meboo.Google Scholar
  3. Dzhafarov, E. N. (2002). Multidimensional Fechnerian scaling: pairwise comparisons, regular minimality, and nonconstant self-similarity. Journal of Mathematical Psychology, 46, 583–608.MathSciNetMATHCrossRefGoogle Scholar
  4. Dzhafarov, E. N., & Colonius, H. (2006a). Reconstructing distances among objects from their discriminability. Psychometrika, 71, 365–386.MathSciNetCrossRefGoogle Scholar
  5. Dzhafarov, E. N., & Colonius, H. (2006b). Regular minimality: a fundamental law of discrimination. In: H. Colonius & E. N. Dzhafarov (Eds.), Measurement and representation of sensations (pp. 1–46). Mahwah: Erlbaum.Google Scholar
  6. Dzhafarov, E. N., & Colonius, H. (2007). Dissimilarity cumulation theory and subjective metrics. Journal of Mathematical Psychology, 51, 290–304.MathSciNetMATHCrossRefGoogle Scholar
  7. Dzhafarov, E. N., Ünlü, A., Trendtel, M., & Colonius, H. (2011). Matrices with a given number of violations of regular minimality. Journal of Mathematical Psychology, 55, 240–250.MathSciNetMATHCrossRefGoogle Scholar
  8. Ekeland, I., & Temam, R. (1999). Convex analysis and variational problems. Philadelphia: SIAM.MATHCrossRefGoogle Scholar
  9. Goldfarb, D., & Idnani, A. (1983). A numerically stable dual method for solving strictly convex quadratic programs. Mathematical Programming, 27, 1–33MathSciNetMATHCrossRefGoogle Scholar
  10. Hiriart-Urruty, J., & Lemaréchal, C. (2001). Fundamentals of convex analysis. Berlin: Springer.MATHCrossRefGoogle Scholar
  11. Kruskal, J. B., & Wish, M. (1978). Multidimensional scaling. Beverly Hills: Sage.Google Scholar
  12. Roberts, A. W., & Varberg, D. E. (1973). Convex functions. New York: Academic.MATHGoogle Scholar
  13. Trendtel, M., Ünlü, A., & Dzhafarov, E. N. (2010). Matrices satisfying Regular Minimality. Frontiers in Quantitative Psychology and Measurement, 1, 1–6.Google Scholar
  14. Ünlü, A., & Trendtel, M. (2010). Testing for regular minimality. In A. Bastianelli & G. Vidotto (Eds.), Fechner Day 2010 (pp. 51–56). Padua: The International Society for Psychophysics.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Division for Methodology and Statistics, Federal Institute for Educational ResearchInnovation & Development of the Austrian School System (BIFIE)SalzburgAustria
  2. 2.Chair for Methods in Empirical Educational Research, TUM School of EducationTechnische Universität MünchenMünchenGermany

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