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Weighted Euclidean Biplots

Author

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  • Michael Greenacre
  • Patrick J.F. Groenen

Abstract

We construct a weighted Euclidean distance that approximates any distance or dissimilarity measure between individuals that is based on a rectangular cases-by-variables data matrix. In contrast to regular multidimensional scaling methods for dissimilarity data, the method leads to biplots of individuals and variables while preserving all the good properties of dimension-reduction methods that are based on the singular-value decomposition. The main benefits are the decomposition of variance into components along principal axes, which provide the numerical diagnostics known as contributions, and the estimation of nonnegative weights for each variable. The idea is inspired by the distance functions used in correspondence analysis and in principal component analysis of standardized data, where the normalizations inherent in the distances can be considered as differential weighting of the variables. In weighted Euclidean biplots we allow these weights to be unknown parameters, which are estimated from the data to maximize the fit to the chosen distances or dissimilarities. These weights are estimated using a majorization algorithm. Once this extra weight-estimation step is accomplished, the procedure follows the classical path in decomposing the matrix and displaying its rows and columns in biplots.

Suggested Citation

  • Michael Greenacre & Patrick J.F. Groenen, 2013. "Weighted Euclidean Biplots," Working Papers 708, Barcelona Graduate School of Economics.
  • Handle: RePEc:bge:wpaper:708
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    File URL: http://www.barcelonagse.eu/sites/default/files/working_paper_pdfs/708.pdf
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    References listed on IDEAS

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    1. Farré Lidia & González Libertad & Ortega Francesc, 2011. "Immigration, Family Responsibilities and the Labor Supply of Skilled Native Women," The B.E. Journal of Economic Analysis & Policy, De Gruyter, pages 1-48.
    2. Jose Apesteguia & Miguel Angel Ballester, 2007. "On The Complexity of Rationalizing Behavior," Working Papers 320, Barcelona Graduate School of Economics.
    3. 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.
    4. Greenacre Michael, 2010. "Biplots in Practice," Books, Fundacion BBVA / BBVA Foundation, number 2011113.
    5. de Leeuw, Jan & Mair, Patrick, 2009. "Multidimensional Scaling Using Majorization: SMACOF in R," Journal of Statistical Software, Foundation for Open Access Statistics.
    6. Jan Leeuw, 1988. "Convergence of the majorization method for multidimensional scaling," Journal of Classification, Springer;The Classification Society, vol. 5(2), pages 163-180, September.
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    1. repec:cog:socinc:v:5:y:2017:i:4:p:38-47 is not listed on IDEAS

    More about this item

    Keywords

    Biplot; correspondence analysis; Distance; majorization; multidimensional scaling; singular-value decomposition; weighted least squares;

    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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