SSA Models, Measures of Fit, and Their Optimization

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 4 in Multidimensional Similarity Structure Analysis, 1987, pp 41-57 from Springer

Abstract: Abstract Some SSA models are defined by specifying their representation functions, i.e., how data are to be mapped into distances. For practical applications, these representation functions are too strict. They are modified into loss functions which define data-distance correspondences that have to be optimized rather than satisfied. Loss functions lead to general badness-of-fit measures such as stress and alienation, and to procedures for iteratively constructing SSA representations by computation. A 2-phase algorithm for that purpose is described. In one phase, target distances for a given SSA configuration are computed. In the second phase, the points are moved such that the resulting distances approximate the given target distances as closely as possible. The latter problem is solved here by the gradient method. Some problems arising from this method are discussed.

Keywords: Cartesian coordinate system; SSA model; absolute SSA; interval SSA; nominal SSA; metric and nonmetric SSA models; error theory; total error of an SSA solution; stress; alienation; monotonicity coefficient; exact error measure; minimization problem; minimization parameter; 2-phase algorithm; ping-pong process; stopping criterion; re-norming of the configuration; configuration point; graph of a function; local and global extremum; slope of a tangent; limiting value; limiting value operator; differentiation; slope function; saddle point; partial differentiation; centering of a configuration; partial minimum; step size; gradient; infinitesimal neighborhood; method of steepest descent; rational starting configuration (search for similar items in EconPapers)
Date: 1987
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DOI: 10.1007/978-1-4612-4768-5_4

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