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Weighted metric multidimensional scaling

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Abstract

This paper establishes a general framework for metric scaling of any distance measure between individuals based on a rectangular individuals-by-variables data matrix. The method allows visualization of both individuals and variables as well as preserving all the good properties of principal axis methods such as principal components and correspondence analysis, based on the singular-value decomposition, including the decomposition of variance into components along principal axes which provide the numerical diagnostics known as contributions. The idea is inspired from the chi-square distance in correspondence analysis which weights each coordinate by an amount calculated from the margins of the data table. In weighted metric multidimensional scaling (WMDS) we allow these weights to be unknown parameters which are estimated from the data to maximize the fit to the original distances. Once this extra weight-estimation step is accomplished, the procedure follows the classical path in decomposing a matrix and displaying its rows and columns in biplots.

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  • Michael Greenacre, 2004. "Weighted metric multidimensional scaling," Economics Working Papers 777, Department of Economics and Business, Universitat Pompeu Fabra.
    • Handle: RePEc:upf:upfgen:777
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    References listed on IDEAS

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    1. Michael Greenacre, 2008. "Correspondence analysis of raw data," Economics Working Papers 1112, Department of Economics and Business, Universitat Pompeu Fabra, revised Jul 2009.
    2. J. Gower & P. Legendre, 1986. "Metric and Euclidean properties of dissimilarity coefficients," Journal of Classification, Springer;The Classification Society, vol. 3(1), pages 5-48, March.
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    Cited by:

    1. Michael Greenacre & Rafael Pardo, 2005. "Multiple correspondence analysis of a subset of response categories," Economics Working Papers 881, Department of Economics and Business, Universitat Pompeu Fabra.
    2. Michael Greenacre & Rafael Pardo, 2004. "Subset correspondence analysis: Visualizing relationships among a selected set of response categories from a questionnaire survey," Economics Working Papers 791, Department of Economics and Business, Universitat Pompeu Fabra.

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    More about this item

    Keywords

    Biplot; correspondence analysis; distance; multidimensional scaling; singular-value decomposition;
    All these keywords.

    JEL classification:

    • C19 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Other
    • C88 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Other Computer Software

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