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A data-based power transformation for compositional data

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  • T. Tsagris, Michail
  • Preston, Simon
  • T.A. Wood, Andrew

Abstract

Compositional data analysis is carried out either by neglecting the compositional constraint and applying standard multivariate data analysis, or by transforming the data using the logs of the ratios of the components. In this work we examine a more general transformation which includes both approaches as special cases. It is a power transformation and involves a single parameter�. The transformation has two equivalent versions. The �first is the stay-in-the-simplex version. This expression is the power transformation as de�fined by Aitchison (1986). The second version, which is a linear transformation of the stay-in-the-simplex, is a Box-Cox type transformation. We call the second version the isometric �alpha-transformation because of the multiplication with the Helmert sub-matrix. We discuss a parametric way of estimating the value of alpha�, which is maximization of its pro�le like-lihood (assuming multivariate normality of the transformed data) and the equivalence between the two versions is exhibited. Other ways include maximization of the correct classi�cation probability in discriminant analysis and maximization of the pseudo-R2 in linear regression. We examine the relationship between the transformation, the raw data approach and the isometric log-ratio transformation. Furthermore, we also de�fine a suitable family of metrics corresponding to the family of �alpha-transformation and consider the corresponding family of Fr�echet means.

Suggested Citation

  • T. Tsagris, Michail & Preston, Simon & T.A. Wood, Andrew, 2011. "A data-based power transformation for compositional data," MPRA Paper 53068, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:53068
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    References listed on IDEAS

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    1. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(4), pages 691-705, August.
    2. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(1), pages 225-228, February.
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    4. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(2), pages 411-413, April.
    5. M. J. Baxter, 1995. "Standardization and Transformation in Principal Component Analysis, with Applications to Archaeometry," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 44(4), pages 513-527, December.
    6. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(6), pages 1195-1198, December.
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    Cited by:

    1. Tsagris, Michail & Preston, Simon & T.A. Wood, Andrew, 2016. "Improved classi cation for compositional data using the $\alpha$-transformation," MPRA Paper 67657, University Library of Munich, Germany.
    2. Michail Tsagris & Simon Preston & Andrew T. A. Wood, 2016. "Improved Classification for Compositional Data Using the α-transformation," Journal of Classification, Springer;The Classification Society, vol. 33(2), pages 243-261, July.
    3. Tsagris, Michail, 2015. "Regression analysis with compositional data containing zero values," MPRA Paper 67868, University Library of Munich, Germany.
    4. Yannis Pantazis & Michail Tsagris & Andrew T. A. Wood, 2019. "Gaussian Asymptotic Limits for the α-transformation in the Analysis of Compositional Data," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 63-82, February.
    5. Tsagris, Michail & Preston, Simon & T.A. Wood, Andrew, 2016. "Nonparametric hypothesis testing for equality of means on the simplex," MPRA Paper 72771, University Library of Munich, Germany.

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    More about this item

    Keywords

    Compositional data; power transformation; alpha; Frechet mean;
    All these keywords.

    JEL classification:

    • C89 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Other

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