Advertisement

On multi-way metricity, minimality and diagonal planes

  • Matthijs J. Warrens
Open Access
Regular Article

Abstract

Validity of the triangle inequality and minimality, both axioms for two-way dissimilarities, ensures that a two-way dissimilarity is nonnegative and symmetric. Three-way generalizations of the triangle inequality and minimality from the literature are reviewed and it is investigated what forms of symmetry and nonnegativity are implied by the three-way axioms. A special form of three-way symmetry that can be deduced is equality of the diagonal planes of the three-dimensional cube. Furthermore, it is studied what diagonal plane equalities hold for the four-dimensional tesseract.

Keywords

Diagonal plane equality Tetrahedron inequality Multi-way symmetry Three-way block Tesseract Multi-way dissimilarity 

Mathematics Subject Classification (2000)

51K05 

Notes

Acknowledgments

The author thanks Hans-Hermann Bock and three anonymous reviewers for their helpful comments and valuable suggestions on earlier versions of this article.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution,and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. Bennani-Dosse M (1993) Analyses Métriques á Trois Voies. Unpublished PhD thesis, Université de Haute Bretagne Rennes II, FranceGoogle Scholar
  2. Cox TF, Cox MAA, Branco JA (1991) Multidimensional scaling for n-tuples. Br J Math Stat Psychol 44: 195–206MATHGoogle Scholar
  3. Daws JT (1996) The analysis of free-sorting data: Beyond pairwise cooccurrences. J Classif 13:57–80MATHCrossRefGoogle Scholar
  4. Deza M-M, Rosenberg IG (2000) n-Semimetrics. Eur J Comb Spec Issue Discrete Metric Spaces 21: 797–806MATHMathSciNetGoogle Scholar
  5. Deza M-M, Rosenberg IG (2005) Small cones of m-hemimetrics. Discrete Math 291: 81–97MATHCrossRefMathSciNetGoogle Scholar
  6. Diatta J (2004) A relation between the theory of formal concepts and multiway clustering. Pattern Recogn Lett 25: 1183–1189CrossRefGoogle Scholar
  7. Diatta J (2006) Description-meet compatible multiway dissimilarities. Discrete Appl Math 154: 493–507MATHCrossRefMathSciNetGoogle Scholar
  8. Diatta J (2007) Galois closed entity sets and k-balls of quasi-ultrametric multi-way dissimilarities. Adv Data Anal Classif 1: 53–65MATHCrossRefMathSciNetGoogle Scholar
  9. Gower JC, De Rooij M (2003) A comparison of the multidimensional scaling of triadic and dyadic distances. J Classif 20: 115–136MATHCrossRefMathSciNetGoogle Scholar
  10. Heiser WJ, Bennani M (1997) Triadic distance models: axiomatization and least squares representation. J Math Psychol 41: 189–206MATHCrossRefMathSciNetGoogle Scholar
  11. Joly S, Le Calvé G (1995) Three-way distances. J Classif 12: 191–205MATHCrossRefGoogle Scholar
  12. Nakayama A (2005) A multidimensional scaling model for three-way data analysis. Behaviormetrika 32: 95–110MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2008

Authors and Affiliations

  1. 1.Psychometrics and Research Methodology Group, Leiden University Institute for Psychological ResearchLeiden UniversityLeidenThe Netherlands

Personalised recommendations