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Unfolding a symmetric matrix

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Abstract

Graphical displays which show inter--sample distances are important for the interpretation and presentation of multivariate data. Except when the displays are two--dimensional, however, they are often difficult to visualize as a whole. A device, based on multidimensional unfolding, is described for presenting some intrinsically high--dimensional displays in fewer, usually two, dimensions. This goal is achieved by representing each sample by a pair of points, say $R_i$ and $r_i$, so that a theoretical distance between the $i$-th and $j$-th samples is represented twice, once by the distance between $R_i$ and $r_j$ and once by the distance between $R_j$ and $r_i$. Self--distances between $R_i$ and $r_i$ need not be zero. The mathematical conditions for unfolding to exhibit symmetry are established. Algorithms for finding approximate fits, not constrained to be symmetric, are discussed and some examples are given.

Suggested Citation

  • John C. Gower & Michael Greenacre, 1996. "Unfolding a symmetric matrix," Economics Working Papers 154, Department of Economics and Business, Universitat Pompeu Fabra.
    • Handle: RePEc:upf:upfgen:154
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    1. J. Kruskal, 1964. "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis," Psychometrika, Springer;The Psychometric Society, vol. 29(1), pages 1-27, March.
    2. E. Gold, 1973. "Metric unfolding: Data requirement for unique solution & clarification of Schönemann's algorithm," Psychometrika, Springer;The Psychometric Society, vol. 38(4), pages 555-569, December.
    3. Michael Greenacre & Michael Browne, 1986. "An efficient alternating least-squares algorithm to perform multidimensional unfolding," Psychometrika, Springer;The Psychometric Society, vol. 51(2), pages 241-250, June.
    4. M. Browne, 1987. "The Young-Householder algorithm and the least squares multidimensional scaling of squared distances," Journal of Classification, Springer;The Classification Society, vol. 4(2), pages 175-190, September.
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    More about this item

    Keywords

    Dimensionality reduction; distances; graphics; multidimensional scaling; symmetric matrices; unfolding;
    All these keywords.

    JEL classification:

    • C19 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Other
    • C88 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Other Computer Software

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