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Characterizations of continuous adequate objective functions for ordinal or interval scaled data

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  • Gianni Bosi
  • Gabriele Sbaiz
  • Magal`i Zuanon

Abstract

Objective functions (goodness criteria which have to be optimized) that are considered, for instance, in cluster analysis, factor analysis, (linear) structural equation modeling, (linear) regression, multidimensional scaling, choice theory, and utility theory, must be {\em adequate}, i.e. carefully adapted to the structure of the observed data. Adequateness of an objective function means, in our terminology, that (certain) transformations of its arguments, e.g. changes in the unit measures of the quantities involved, do not influence the solutions of the optimization procedure. In this paper we concentrate our attention on affine strictly increasing transformations and we also incorporate the case of continuous adequate objective functions accordingly. The characterization of adequate dissimilarity coefficients for interval scaled data shows the appropriateness of the concept of adequateness that is developed in this paper.

Suggested Citation

  • Gianni Bosi & Gabriele Sbaiz & Magal`i Zuanon, 2026. "Characterizations of continuous adequate objective functions for ordinal or interval scaled data," Papers 2608.20074, arXiv.org.
    • Handle: RePEc:arx:papers:2608.20074
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    References listed on IDEAS

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    3. Herden, Gerhard & Pallack, Andreas, 2002. "Consistency in ordinal data analysis I," Mathematical Social Sciences, Elsevier, vol. 43(1), pages 79-113, January.
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    6. Herden, Gerhard & Pallack, Andreas, 2005. "Adequateness and interpretability of objective functions in ordinal data analysis," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 19-69, May.
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