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Biplots of compositional data

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Abstract

The singular value decomposition and its interpretation as a linear biplot has proved to be a powerful tool for analysing many forms of multivariate data. Here we adapt biplot methodology to the speciffic case of compositional data consisting of positive vectors each of which is constrained to have unit sum. These relative variation biplots have properties relating to special features of compositional data: the study of ratios, subcompositions and models of compositional relationships. The methodology is demonstrated on a data set consisting of six-part colour compositions in 22 abstract paintings, showing how the singular value decomposition can achieve an accurate biplot of the colour ratios and how possible models interrelating the colours can be diagnosed.

Suggested Citation

  • J. Aitchison & Michael Greenacre, 2001. "Biplots of compositional data," Economics Working Papers 557, Department of Economics and Business, Universitat Pompeu Fabra.
  • Handle: RePEc:upf:upfgen:557
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    Cited by:

    1. Michael Greenacre, 2009. "Contribution biplots," Economics Working Papers 1162, Department of Economics and Business, Universitat Pompeu Fabra, revised Jan 2011.
    2. Jonas Schöley & Frans Willekens, 2017. "Visualizing compositional data on the Lexis surface," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 36(21), pages 627-658, February.
    3. Aerni, Philipp, 2009. "What is sustainable agriculture? Empirical evidence of diverging views in Switzerland and New Zealand," Ecological Economics, Elsevier, vol. 68(6), pages 1872-1882, April.
    4. Violetta Simonacci & Michele Gallo, 2017. "Statistical tools for student evaluation of academic educational quality," Quality & Quantity: International Journal of Methodology, Springer, vol. 51(2), pages 565-579, March.
    5. Michael Greenacre & Paul Lewi, 2009. "Distributional Equivalence and Subcompositional Coherence in the Analysis of Compositional Data, Contingency Tables and Ratio-Scale Measurements," Journal of Classification, Springer;The Classification Society, vol. 26(1), pages 29-54, April.
    6. Udina, Frederic, 2005. "Interactive Biplot Construction," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 13(i05).
    7. Michael Greenacre, 2003. "Singular value decomposition of matched matrices," Journal of Applied Statistics, Taylor & Francis Journals, vol. 30(10), pages 1101-1113.
    8. Gardner-Lubbe, Sugnet, 2016. "A triplot for multiclass classification visualisation," Computational Statistics & Data Analysis, Elsevier, vol. 94(C), pages 20-32.
    9. Michael Greenacre, 2002. "Ratio maps and correspondence analysis," Economics Working Papers 598, Department of Economics and Business, Universitat Pompeu Fabra.
    10. 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.
    11. Juan Manuel Larrosa, 2003. "A Compositional Statistical Analysis of Capital per Worker," Macroeconomics 0301006, EconWPA.
    12. Peter Filzmoser & Karel Hron & Matthias Templ, 2012. "Discriminant analysis for compositional data and robust parameter estimation," Computational Statistics, Springer, vol. 27(4), pages 585-604, December.
    13. Huiwen Wang & Liying Shangguan & Rong Guan & Lynne Billard, 2015. "Principal component analysis for compositional data vectors," Computational Statistics, Springer, vol. 30(4), pages 1079-1096, December.
    14. repec:spr:advdac:v:11:y:2017:i:2:d:10.1007_s11634-016-0245-y is not listed on IDEAS
    15. Michael Greenacre & Paul Lewi, 2005. "Distributional equivalence and subcompositional coherence in the analysis of contingency tables, ratio-scale measurements and compositional data," Economics Working Papers 908, Department of Economics and Business, Universitat Pompeu Fabra, revised Aug 2007.
    16. Greenacre, Michael, 2009. "Power transformations in correspondence analysis," Computational Statistics & Data Analysis, Elsevier, vol. 53(8), pages 3107-3116, June.
    17. Michael Greenacre & Anna Torres, 2002. "Measuring asymmetries in brand associations using correspondence analysis," Economics Working Papers 630, Department of Economics and Business, Universitat Pompeu Fabra.
    18. Michael Greenacre, 2006. "Tying up the loose ends in simple correspondence analysis," Economics Working Papers 940, Department of Economics and Business, Universitat Pompeu Fabra.

    More about this item

    Keywords

    Logratio transformation; principal component analysis; relative variation biplot; singular value decomposition; subcomposition;

    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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