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Psychometrika

, Volume 33, Issue 4, pp 469–506 | Cite as

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

  • Louis Guttman
Article

Abstract

LetA1,A2, ...,An be anyn 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 (Ai,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 calculatingm coordinates {x ia } for eachA i for a semi-metricd of preassigned formd ij =d(|xi1 -x j1 |, |xi2 -xj2|, ..., |x im -x jm |). The dimensionalitym is sought to be as small as possible, yet satisfy the monotonicity condition thatd ij <d kl whenever the observed data indicate thatA i is “closer” toA j thanA k is toA l . Minkowski and Euclidean spaces are special metric examples ofd. A general coefficient of monotonicityμ 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Psychometric Society 1968

Authors and Affiliations

  • Louis Guttman
    • 1
    • 2
  1. 1.The Hebrew UniversityIsrael
  2. 2.The Israel Institute of Applied Social ResearchIsrael

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