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Download the Paper (577 K, PDF file) - January 30, 2007 Tell a colleague about it. Printing Tips: Select 'print as image' in the Acrobat print dialog if you have trouble printing.

ABSTRACT:
Horseshoes (quadratic curves) routinely show up in multidimensional scaling and correspondence analysis solutions. We review some of the empirical situations in which they are known to occur, and we discuss some of the mathematical models that produce them. One particular model, discussed by Diaconis et al. in a recent paper, is the Kac-Murdock-Szegö matrix A with elements a_ij = ρ ^{| i-j |}. In this paper we analyze this example in some detail. We point out that A is both totally positive and Toeplitz, and that the horseshoes also occur for other matrices with these properties. It is shown that double centering of a Toeplitz matrix leads to a centro-symmetric matrix, which again will produce horseshoes.

SUGGESTED CITATION:
Jan de Leeuw, "A Horseshoe for Multidimensional Scaling" (January 30, 2007). Department of Statistics, UCLA. Department of Statistics Papers. Paper 2007010122.
http://repositories.cdlib.org/uclastat/papers/2007010122