Skip to main content
SpringerLink
Log in

Analysis of Symmetry and Skew-Symmetry

  • Chapter
  • First Online:
Methods for the Analysis of Asymmetric Proximity Data

Part of the book series: Behaviormetrics: Quantitative Approaches to Human Behavior ((BQAHB,volume 7))

  • 311 Accesses

Abstract

The properties of the decomposition of a square data matrix in its symmetric and skew-symmetric components are firstly presented and discussed. The orthogonal breakdown of the sum of squares of the proximities characterizing the decomposition is emphasized. Afterwards, an overview of models and methods proposed to represent, separately or jointly, symmetry and skew-symmetry is presented. The relationships among the models are outlined and suggestions for strategies of data analysis are provided. Applications of the models to a real data set and a software section are provided at the end of the chapter.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Borg, I., & Groenen, P. J. F. (2005). Modern multidimensional scaling. Theory and applications. (2nd ed.). Springer.

    Google Scholar 

  2. Heiser, W. J., & Busing, F. M. T. A. (2004). Multidimensional scaling and unfolding of symmetric and asymmetric proximity relations. In D. Kaplan (Ed.), The Sage handbook of quantitative methodology for the social sciences (pp. 25–48). Sage Publications.

    Google Scholar 

  3. Constantine, A. G., & Gower, J. C. (1978). Graphical representation of asymmetric matrices. Applied Statistics, 3, 297–304.

    Article  Google Scholar 

  4. Bowker, A. H. (1948). A test for symmetry in contingency tables. Journal of the American Statistical Association, 43, 572–574.

    Article  Google Scholar 

  5. Hubert, L. J., & Baker, F. B. (1979). Evaluating the symmetry of a proximity matrix. Quality and Quantity, 43, 81–91.

    Google Scholar 

  6. Saburi, S., & Chino, N. (2008). A maximum likelihood method for an asymmetric MDS model. Computational Statistics and Data Analysis, 52, 4673–4684.

    Article  MathSciNet  Google Scholar 

  7. Zielman, B., & Heiser, W. J. (1996). Models for asymmetric proximities. British Journal of Mathematical and Statistical Psychology, 49, 127–146.

    Article  Google Scholar 

  8. Kuriki, S. (2010). Distributions of the largest singular values of skew-symmetric random matrices and their applications to paired comparisons. Communications in Statistics–Theory and Methods, 39, 1522–1535

    Google Scholar 

  9. Critchley, F. (1988). On characterization of the inner products of the space of square matrices of given order which render orthogonal the symmetric and the skew-symmetric subspaces. Warwick Statistics Research Report 171. University of Warwick. 

    Google Scholar 

  10. Carroll, J. D., & Arabie, P. (1998). Multidimensional scaling. In M. H. Birnbaum (Ed.), Measurement, Judgment and Decision Making (pp. 179–250). Academic Press.

    Chapter  Google Scholar 

  11. Gower, J. C. (1977). The analysis of asymmetry and orthogonality. In J. R. Barra et al. (Eds.), Recent developments in statistics (pp. 109–123). North Holland.

    Google Scholar 

  12. Gower, J. C. (2018). Skew symmetry in retrospect. Advances in Data Analysis and Classification, 12, 33–41.

    Article  MathSciNet  Google Scholar 

  13. Weeks, D. G., & Bentler, P. M. (1982). Restricted multidimensional scaling models for asymmetric proximities. Psychometrika, 47, 201–207.

    Article  Google Scholar 

  14. Caussinus, H., & de Falguerolles, A. (1987). Tableaux carrés: Modélisation et methods factorielles. Revue De Statistique Appliquée, 35, 35–52.

    Google Scholar 

  15. De Rooij, M., & Heiser, W. J. (2003). A distance representation of the quasi-symmetry model and related distance models. In H. Yanai & A. Okada (Eds.), New Developments in Psychometrics (pp. 487–494). Springer-Verlag.

    Chapter  Google Scholar 

  16. De Rooij, M., & Heiser, W. J. (2005). Graphical representations and odds ratios in a distance association for the analysis of cross-classified data. Psychometrika, 70, 99–122.

    Article  MathSciNet  Google Scholar 

  17. De Rooij, M. (2020). Asymmetric scaling models for square contingency tables: points, circles, arrows and odds ratios. In T. Imaizumi, A. Nakayama & S. Yokoyama (Eds.), Advanced studies in behaviormetrics and data science. Behaviormetrics: Quantitative Approaches to Human Behavior (pp. 43–61). Springer Nature.

    Google Scholar 

  18. Holman, E. W. (1979). Monotonic models for asymmetric proximities. Journal of Mathematical Psychology, 20, 1–15.

    Article  Google Scholar 

  19. Nosofsky, R. M. (1991). Stimulus bias, asymmetric similarity, and classification. Cognitive Psychology, 23, 94–140.

    Article  Google Scholar 

  20. Brossier, G. (1982). Classification hiérarchique à partir de matrices carrées non symétriques. Statistique et Analyse des Données, 7, 22–40.

    MathSciNet  MATH  Google Scholar 

  21. Bove, G. (2012). Exploratory approaches to seriation by asymmetric multidimensional scaling. Behaviormetrika, 39, 63–73.

    Article  Google Scholar 

  22. Okada, A., & Imaizumi, T. (1987). Non-metric multidimensional scaling of asymmetric similarities. Behaviormetrika, 21, 81–96.

    Article  Google Scholar 

  23. Okada, A. (1990). A generalization of asymmetric multidimensional scaling. In M. Schader & W. Gaul (Eds.), Knowledge, data and computer-assisted decisions (pp. 127–138). Springer.

    Chapter  Google Scholar 

  24. Mair, P., Groenen, P. J. F., & De Leeuw, J. (2020). More on multidimensional scaling and unfolding in R: smacof Version 2. https://cran.r-project.org/web/packages/smacof/vignettes/smacof.pdf

  25. Chino, N. (1978). A graphical technique for representing the asymmetric relationships between N objects. Behaviormetrika, 5, 23–40.

    Article  Google Scholar 

  26. Escoufier, Y., & Grorud, A. (1980). Analyse factorielle des matrices carrées non symmétriques. In E. Diday et al. (Eds.), Data analysis and informatics (pp. 263–276). North Holland.

    Google Scholar 

  27. Chino, N. (1990). A generalized inner product model for the analysis of asymmetry. Behaviormetrika, 17, 25–46.

    Article  Google Scholar 

  28. Rocci, R., & Bove, G. (2002). Rotation techniques in asymmetric multidimensional scaling. Journal of Computational and Graphical Statistics, 11, 405–419.

    Article  MathSciNet  Google Scholar 

  29. Harshman, R. A. (1978). Models for analysis of asymmetrical relationships among N objects or stimuli. Paper presented at the First Joint Meeting of the Psychometric Society and the Society for Mathematical Psychology. McMaster University.

    Google Scholar 

  30. Kiers, H. A. L., & Takane, Y. (1994). A generalization of GIPSCAL for the analysis of asymmetric data. Journal of Classification, 11, 79–99.

    Article  Google Scholar 

  31. Saito, T., & Yadohisa, H. (2005). Data analysis of asymmetric structures. Advanced approaches in computational statistics. Marcel Dekker.

    Google Scholar 

  32. Chino, N. (2012). A brief survey of asymmetric MDS and some open problems. Behaviormetrika, 39, 127–165.

    Article  Google Scholar 

  33. Rocci, R. (2004). A general algorithm to fit constrained DEDICOM models. Statistical Methods & Applications, 13, 139–150.

    MathSciNet  MATH  Google Scholar 

  34. Bove, G., & Critchley, F. (1993). Metric multidimensional scaling for asymmetric proximities when the asymmetry is one-dimensional. In R. Steyer (Ed.), Psychometric Methodology (pp. 55–60). Gustav Fischer Verlag.

    Google Scholar 

  35. IBM-SPSS. (2019). IBM-SPSS Categories 26. IBM Corp.

    Google Scholar 

  36. Borg, I., Groenen, P. J. F., & Mair, P. (2018). Applied multidimensional scaling and unfolding (2nd ed.). Springer.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Education, Roma Tre University, Rome, Italy

    Giuseppe Bove

  2. Rikkyo University, Tokyo, Japan

    Akinori Okada

  3. Department of Statistical Sciences, Sapienza University of Rome, Rome, Italy

    Donatella Vicari

Authors
  1. Giuseppe Bove
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akinori Okada
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Donatella Vicari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giuseppe Bove .

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bove, G., Okada, A., Vicari, D. (2021). Analysis of Symmetry and Skew-Symmetry. In: Methods for the Analysis of Asymmetric Proximity Data. Behaviormetrics: Quantitative Approaches to Human Behavior, vol 7. Springer, Singapore. https://doi.org/10.1007/978-981-16-3172-6_3

Download citation

Publish with us

Policies and ethics

Search

Navigation