Mappings of Data in Distances

I. Borg and J. Lingoes
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I. Borg: Justus-Liebig-Universität, Department of Psychology
J. Lingoes: University of Michigan, Computing Center

Chapter 18 in Multidimensional Similarity Structure Analysis, 1987, pp 292-308 from Springer

Abstract: Abstract The proposition that given data are ‘distances except for some transformation f’ can be empirically tested under various specifications of f. In ratio scaling, a multiplicative constant (k ≠ 0) can be chosen freely, in interval scaling, any additive constant and a k ≠ 0, and in ordinal scaling, an arbitrary order-preserving function. For typical data, the interval and the ordinal model allow us always to map the data not only into distances, but even into Euclidean distances in at most n — 2 dimensions. A statistical version of this transformation problem is discussed.

Keywords: distances except for some transformation f; distances and Euclidean distances; falsifiability of the proposition f(pij) = dij; tautology; additive constant problem of interval SSA; positive semi-definite matrix; true dimensionality; eigenvalue distribution; error component of proximities; true additive constant; estimation and optimization of the additive constant; double centering; initial configuration of ordinal SSA (search for similar items in EconPapers)
Date: 1987
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DOI: 10.1007/978-1-4612-4768-5_18

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