A Regression Model for Compositional Data Based on the Shifted-Dirichlet Distribution

  • G. S. Monti
  • G. Mateu-Figueras
  • V. Pawlowsky-Glahn
  • J. J. Egozcue
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 187)

Abstract

Using an approach based on the Aitchison geometry of the simplex, a Shifted-Dirichlet covariate model is obtained. Allowing the parameters to change linearly with a set of covariates, their effects on the relative contributions of different components in a composition are assessed. An application of this model to sedimentary petrography is given.

Keywords

Dirichlet regression Simplicial regression Model selection 

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1964)MATHGoogle Scholar
  2. 2.
    Aitchison, J.: The Statistical Analysis of Compositional Data. Chapman & Hall Ltd. (Reprinted in 2003 with additional material by The Blackburn Press), London (1986)Google Scholar
  3. 3.
    Aitchison, J., Shen, S.M.: Logistic-normal distributions. Some properties and uses. Biometrika 67, 261–272 (1980)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Billheimer, D., Guttorp, P., Fagan, W.F.: Statistical interpretation of species composition. J. Am. Stat. Assoc. 96, 1205–1214 (2001)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cameron, A.C., Windmeijer, F.A.G.: R-squared measures for count data regression models with applications to health-care utilization. J. Busin. Econ. Stat. 14(2), 209–220 (1996)Google Scholar
  6. 6.
    Campbell, G., Mosimann, J.: Multivariate methods for proportional shape. In: ASA Proceedings of the Section on Statistical Graphics (1987)Google Scholar
  7. 7.
    Coakley, J.P., Rust, B.R.: Sedimentation in an Arctic lake. J. Sediment. Petrol. 38, 1290–1300 (1968)Google Scholar
  8. 8.
    Egozcue, J.J., Daunis-i-Estadella, J., Pawlowsky-Glahn, V., Hron, K., Filzmoser, P.: Simplicial regression. The normal model. J. App. Prob. Stat. 6, 87–108 (2012)MATHGoogle Scholar
  9. 9.
    Egozcue, J.J., Pawlowsky-Glahn, V.: Groups of parts and their balances in compositional data analysis. Math. Geol. 37(7), 795–828 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ferrari, S., Cribari-Neto, F.: Beta regression for modelling rates and proportions. J. App. Stat. 31, 799–815 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Garzanti, E., Vezzoli, G., Andó, S.: Paleogeographic and paleodrainage changes during Pleistocene glaciations (Po Plain, Northern Italy). Earth-Sci. Rev. 105, 25–48 (2011)CrossRefGoogle Scholar
  12. 12.
    Gueorguieva, R., Rosenheck, R., Zelterman, D.: Dirichlet component regression and its applications to psychiatric data. Comput. Stat. Data Anal. 52, 5344–5355 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hijazi, R.H.: Residuals and Diagnostics in Dirichlet regression. Tech Report of United Arab Emirates University, Department of Statistics (2006)Google Scholar
  14. 14.
    Hijazi, R.H., Jernigan, R.W.: Modelling compositional data using dirichlet regression models. J. App. Prob. Stat. 4, 77–91 (2009)MathSciNetMATHGoogle Scholar
  15. 15.
    Mateu-Figueras, G., Pawlowsky-Glahn, V., Egozcue, J.J.: The principle of working on coordinates. In: Compositional Data Analysis, pp. 29–42. John Wiley & Sons, Ltd (2011)Google Scholar
  16. 16.
    Mateu-Figueras, G., Pawlowsky-Glahn, V., Egozcue, J.J.: The normal distribution in some constrained sample spaces. SORT 37(2), 231–252 (2011)MathSciNetMATHGoogle Scholar
  17. 17.
    Melo, T.F.N., Vasconcellos, K.L.P., Lemonte, A.J.: Some restriction tests in a new class of regression models for proportions. Comput. Stat. Data Anal. 53, 3972–3979 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Monti, G.S., Mateu-Figueras, G., Pawlowsky-Glahn, V.: Notes on the scaled Dirichlet distribution. In: Compositional Data Analysis, pp. 128–138. John Wiley & Sons, Ltd (2011)Google Scholar
  19. 19.
    Monti, G.S., Mateu-Figueras, G., Pawlowsky-Glahn, V., Egozcue, J.J.: The shifted-scaled Dirichlet distribution in the simplex. In: Proceedings of The 4th International Workshop on Compositional Data Analysis (2011)Google Scholar
  20. 20.
    Monti, G.S., Mateu-Figueras, G., Pawlowsky-Glahn, V. and Egozcue, J.J.: Scaled-Dirichlet covariate models for compositional data. In: Proceedings of 47th Scientific Meeting of the Italian Statistical Society (2014)Google Scholar
  21. 21.
    Pawlowsky-Glahn, V.: Statistical modelling on coordinates. In: Proceedings of Compositional Data Analysis Workshop—CoDaWork’03 (2003)Google Scholar
  22. 22.
    Pawlowsky-Glahn, V., Egozcue, J.J.: Geometric approach to statistical analysis on the simplex. Stoch Env. Res. Risk A. 15, 384–398 (2001)CrossRefMATHGoogle Scholar
  23. 23.
    Pawlowsky-Glahn, V., Egozcue, J.J., Tolosana-Delgado, R.: Modelling and analysis of compositional data, p. 272. Statistics in practice. John Wiley & Sons, Chichester UKGoogle Scholar
  24. 24.
    R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing (2014)Google Scholar
  25. 25.
    Rayens, W.S., Srinivasan, C.: Dependence properties of generalized Liouville distributions on the simplex. J. Am. Stat. Ass. 89, 1465–1470 (1994)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • G. S. Monti
    • 1
  • G. Mateu-Figueras
    • 2
  • V. Pawlowsky-Glahn
    • 2
  • J. J. Egozcue
    • 3
  1. 1.Department of Economics, Management and StatisticsUniversity of Milano-BicoccaMilanoItaly
  2. 2.Department of Computer Science, Applied Mathematics and StatisticsUniversity of GironaGironaSpain
  3. 3.Department of Civil and Environmental EngineeringTechnical University of CataloniaBarcelonaSpain

Personalised recommendations