# A general nonmetric technique for finding the smallest coordinate space for a configuration of points

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## Abstract

Let*A*_{1},*A*_{2}, ...,*A*_{n} be any*n* objects, such as variables, categories, people, social groups, ideas, physical objects, or any other. The empirical data to be analyzed are coefficients of similarity or distance within pairs (*A*_{i},*A*_{ i }), such as correlation coefficients, conditional probabilities or likelihoods, psychological choice or confusion, etc. It is desired to represent these data parsimoniously in a coordinate space, by calculating*m* coordinates {*x*_{ ia }} for each*A*_{ i } for a semi-metric*d* of preassigned form*d*_{ ij } =*d*(|*x*_{i1} -*x*_{ j1 }|, |*x*_{i2} -*x*_{j2}|, ..., |*x*_{ im } -*x*_{ jm }|). The dimensionality*m* is sought to be as small as possible, yet satisfy the monotonicity condition that*d*_{ ij } <*d*_{ kl } whenever the observed data indicate that*A*_{ i } is “closer” to*A*_{ j } than*A*_{ k } is to*A*_{ l }. Minkowski and Euclidean spaces are special metric examples of*d*. A general coefficient of monotonicity*μ* is defined, whose maximization is equivalent to optimal satisfaction of the monotonicity condition, and which allows various options both for treatment of ties and for weighting error-of-fit. A general rationale for algorithm construction is derived for maximizing μ by gradient-guided iterations; this provides a unified mathematical solution to the basic operational problems of norming the gradient to assure proper convergence, of trading between speed and robustness against undesired stationary values, and of a rational first approximation. Distinction is made between single-phase (quadratic) and two-phase (bilinear) strategies for algorithm construction, and between “hard-squeeze” and “soft-squeeze” tactics within these strategies. Special reference is made to the rank-image and related transformational principles, as executed by current Guttman-Lingoes families of computer programs.

## Keywords

Public Policy Conditional Probability Euclidean Space Empirical Data Social Group## Preview

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