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A latent Gaussian model for compositional data with zeros

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  • Adam Butler
  • Chris Glasbey

Abstract

Summary. Compositional data record the relative proportions of different components within a mixture and arise frequently in many fields. Standard statistical techniques for the analysis of such data assume the absence of proportions which are genuinely zero. However, real data can contain a substantial number of zero values. We present a latent Gaussian model for the analysis of compositional data which contain zero values, which is based on assuming that the data arise from a (deterministic) Euclidean projection of a multivariate Gaussian random variable onto the unit simplex. We propose an iterative algorithm to simulate values from this model and apply the model to data on the proportions of fat, protein and carbohydrate in different groups of food products. Finally, evaluation of the likelihood involves the calculation of difficult integrals if the number of components is more than 3, so we present a hybrid Gibbs rejection sampling scheme that can be used to draw inferences about the parameters of the model when the number of components is arbitrarily large.

Suggested Citation

  • Adam Butler & Chris Glasbey, 2008. "A latent Gaussian model for compositional data with zeros," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 57(5), pages 505-520, December.
    • Handle: RePEc:bla:jorssc:v:57:y:2008:i:5:p:505-520
    DOI: 10.1111/j.1467-9876.2008.00627.x
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    References listed on IDEAS

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    1. Jane Fry & Tim Fry & Keith McLaren, 2000. "Compositional data analysis and zeros in micro data," Applied Economics, Taylor & Francis Journals, vol. 32(8), pages 953-959.
    2. David J. Allcroft & Chris A. Glasbey, 2003. "A latent Gaussian Markov random‐field model for spatiotemporal rainfall disaggregation," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 52(4), pages 487-498, October.
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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. Jacob Fiksel & Scott Zeger & Abhirup Datta, 2022. "A transformation‐free linear regression for compositional outcomes and predictors," Biometrics, The International Biometric Society, vol. 78(3), pages 974-987, September.
    4. Michail Tsagris, 2018. "Modelling Structural Zeros in Compositional Data," Working Papers 1803, University of Crete, Department of Economics.
    5. Paulo Rodrigues & Ana Lima, 2009. "Analysis of an European union election using principal component analysis," Statistical Papers, Springer, vol. 50(4), pages 895-904, August.
    6. Ganggang Xu & Marc G. Genton, 2017. "Tukey -and- Random Fields," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1236-1249, July.

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