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Graphical Representation of Asymmetric Matrices

Author

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  • A. G. Constantine
  • J. C. Gower

Abstract

When sample units are plotted against orthogonal principal axes in a components analysis, the concept of (Euclidean) distance plays a central point in interpretation. Variants of such diagrams are common throughout multivariate analysis. Because distance between a pair of points is independent of the order in which they are taken, displays of this kind are especially associated with symmetric matrices, but methods are also required for displaying asymmetric matrices. In this paper two methods for displaying asymmetric square matrices are presented and illustrated by examples. In the first method (multidimensional unfolding) the square matrix is regarded as part of an otherwise unknown symmetric matrix and the resulting diagram is interpreted using distances, much as with classical methods. In the second method the matrix is partitioned into its symmetric and skew‐symmetric components. While the symmetric part is represented by some established distance‐based method, the skew‐symmetric part is represented by points whose relationships are interpreted in terms of areas of triangles and co‐linearities.

Suggested Citation

  • A. G. Constantine & J. C. Gower, 1978. "Graphical Representation of Asymmetric Matrices," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 27(3), pages 297-304, November.
    • Handle: RePEc:bla:jorssc:v:27:y:1978:i:3:p:297-304
    DOI: 10.2307/2347165
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    Citations

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    Cited by:

    1. Berrie Zielman & Willem Heiser, 1993. "Analysis of asymmetry by a slide-vector," Psychometrika, Springer;The Psychometric Society, vol. 58(1), pages 101-114, March.
    2. Opeoluwa FO & Sugnet L, 2017. "Biplots in Covariance Analysis," Biostatistics and Biometrics Open Access Journal, Juniper Publishers Inc., vol. 3(5), pages 147-154, November.
    3. Mark de Rooij, 2008. "The analysis of change, Newton's law of gravity and association models," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 171(1), pages 137-157, January.
    4. Wayne DeSarbo & Michael Johnson & Ajay Manrai & Lalita Manrai & Elizabeth Edwards, 1992. "Tscale: A new multidimensional scaling procedure based on tversky's contrast model," Psychometrika, Springer;The Psychometric Society, vol. 57(1), pages 43-69, March.
    5. Giuseppe Bove & Akinori Okada, 2018. "Methods for the analysis of asymmetric pairwise relationships," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 12(1), pages 5-31, March.
    6. Saburi, S. & Chino, N., 2008. "A maximum likelihood method for an asymmetric MDS model," Computational Statistics & Data Analysis, Elsevier, vol. 52(10), pages 4673-4684, June.
    7. ten Berge, Jos M. F., 1997. "Reduction of asymmetry by rank-one matrices," Computational Statistics & Data Analysis, Elsevier, vol. 24(3), pages 357-366, May.
    8. Nora Lado & Anna Torres & Oscar Licandro, 2006. "Changes in the Importance of Bank Attributes Provoked by a Financial Crisis: A Dynamic Analysis of the Uruguayan Case," Economics Working Papers ECO2006/4, European University Institute.
    9. Michael Brusco & Stephanie Stahl, 2001. "An interactive multiobjective programming approach to combinatorial data analysis," Psychometrika, Springer;The Psychometric Society, vol. 66(1), pages 5-24, March.
    10. Frank Critchley & Lawrence Jones & Hubert Feger & Tapas Sen & David Swofford & Arthur Kendall & Frank Critchley & William Day & Gilbert Saporta & George Estabrook & John Sonquist & Charles Jones, 1986. "Book reviews," Journal of Classification, Springer;The Classification Society, vol. 3(1), pages 135-173, March.
    11. PETER G. M. van der HEIJDEN & AB MOOIJAART, 1995. "Some New Log-Bilinear Models for the Analysis of Asymmetry in a Square Contingency Table," Sociological Methods & Research, , vol. 24(1), pages 7-29, August.

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