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Unfolding a symmetric matrix


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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.

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Paper provided by Department of Economics and Business, Universitat Pompeu Fabra in its series Economics Working Papers with number 154.

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Date of creation: Jan 1996
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Handle: RePEc:upf:upfgen:154

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Keywords: Dimensionality reduction; distances; graphics; multidimensional scaling; symmetric matrices; unfolding;

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  1. Michael Greenacre & Michael Browne, 1986. "An efficient alternating least-squares algorithm to perform multidimensional unfolding," Psychometrika, Springer, vol. 51(2), pages 241-250, June.
  2. M. Browne, 1987. "The Young-Householder algorithm and the least squares multidimensional scaling of squared distances," Journal of Classification, Springer, vol. 4(2), pages 175-190, September.
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