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Weighted Euclidean Biplots

Author

Listed:
  • Michael J. Greenacre

    (Universitat Pompeu Fabra)

  • Patrick J. F. Groenen

    (Erasmus University)

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, our approach 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 J. Greenacre & Patrick J. F. Groenen, 2016. "Weighted Euclidean Biplots," Journal of Classification, Springer;The Classification Society, vol. 33(3), pages 442-459, October.
    • Handle: RePEc:spr:jclass:v:33:y:2016:i:3:d:10.1007_s00357-016-9213-7
    DOI: 10.1007/s00357-016-9213-7
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    References listed on IDEAS

    as
    1. Michael Greenacre, 2009. "Contribution biplots," Economics Working Papers 1162, Department of Economics and Business, Universitat Pompeu Fabra, revised Jan 2011.
    2. Michael Greenacre, 2008. "Correspondence analysis of raw data," Economics Working Papers 1112, Department of Economics and Business, Universitat Pompeu Fabra, revised Jul 2009.
    3. Greenacre Michael, 2010. "Biplots in Practice," Books, Fundacion BBVA / BBVA Foundation, number 2011113, October.
    4. 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.
    5. de Leeuw, Jan & Mair, Patrick, 2009. "Multidimensional Scaling Using Majorization: SMACOF in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 31(i03).
    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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    Cited by:

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    2. Federico Gobbo, 2017. "Beyond the Nation-State? The Ideology of the Esperanto Movement between Neutralism and Multilingualism," Social Inclusion, Cogitatio Press, vol. 5(4), pages 38-47.
    3. Fry, J.T. & Slifko, Matt & Leman, Scotland, 2018. "Generalized biplots for stress-based multidimensionally scaled projections," Computational Statistics & Data Analysis, Elsevier, vol. 128(C), pages 340-353.

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    More about this item

    Keywords

    Biplot; Correspondence analysis; Distance; Majorization; Multidimensional scaling; Singular value decomposition; Weighted least squares;
    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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