Statistics and Computing

, Volume 27, Issue 4, pp 963–983 | Cite as

A structured Dirichlet mixture model for compositional data: inferential and applicative issues

  • Sonia Migliorati
  • Andrea Ongaro
  • Gianna S. Monti
Article
  • 481 Downloads

Abstract

The flexible Dirichlet (FD) distribution (Ongaro and Migliorati in J. Multvar. Anal. 114: 412–426, 2013) makes it possible to preserve many theoretical properties of the Dirichlet one, without inheriting its lack of flexibility in modeling the various independence concepts appropriate for compositional data, i.e. data representing vectors of proportions. In this paper we tackle the potential of the FD from an inferential and applicative viewpoint. In this regard, the key feature appears to be the special structure defining its Dirichlet mixture representation. This structure determines a simple and clearly interpretable differentiation among mixture components which can capture the main features of a large variety of data sets. Furthermore, it allows a substantially greater flexibility than the Dirichlet, including both unimodality and a varying number of modes. Very importantly, this increased flexibility is obtained without sharing many of the inferential difficulties typical of general mixtures. Indeed, the FD displays the identifiability and likelihood behavior proper to common (non-mixture) models. Moreover, thanks to a novel non random initialization based on the special FD mixture structure, an efficient and sound estimation procedure can be devised which suitably combines EM-types algorithms. Reliable complete-data likelihood-based estimators for standard errors can be provided as well.

Keywords

Simplex distribution Dirichlet mixture Identifiability Multimodality EM type algorithms 

References

  1. Aitchison, J.: The Statistical Analysis of Compositional Data. Chapman & Hall, London (2003)MATHGoogle Scholar
  2. Azzalini A, Menardi G, Rosolin T (2012) R package pdfCluster: cluster analysis via nonparametric density estimation (version 1.0-0). Università di Padova, Italia. http://cran.r-project.org/web/packages/pdfCluster/index.html
  3. Banfield, J.D., Raftery, A.E.: Model-based gaussian and non-gaussian clustering. Biometrics 49, 803–821 (1993)MathSciNetCrossRefMATHGoogle Scholar
  4. Barndorff-Nielsen, O., Jørgensen, B.: Some parametric models on the simplex. J. Multivar. Anal. 39(1), 106–116 (1991)MathSciNetCrossRefMATHGoogle Scholar
  5. Biernacki, C., Celeux, G., Govaert, G.: Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models. Comput. Stat. Data Anal 41, 561–575 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. Celeux, G., Govaert, G.: A classification EM algorithm for clustering and two stochastic versions. Comput. Stat. Data Anal. 4, 315–332 (1992)MathSciNetCrossRefMATHGoogle Scholar
  7. Celeux, G., Chauveau, D., Diebolt, J.: Stochastic versions of the EM algorithm: an experimental study in the mixture case. J. Stat. Comput. Simul. 55, 287–314 (1996)CrossRefMATHGoogle Scholar
  8. Connor, R.J., Mosimann, J.E.: Concepts of independence for proportions with a generalization of the dirichlet distribution. J. Am. Stat. Assoc. 64(325), 194–206 (1969)MathSciNetCrossRefMATHGoogle Scholar
  9. Coxeter, H.: Regular Polytopes. Dover Publications, New York (1973)MATHGoogle Scholar
  10. Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser B 39(1), 1–38 (1977)MathSciNetMATHGoogle Scholar
  11. Diebolt, J., Ip, E.: Stochastic EM: method and application. In: WR Gilks, S.R., Spiegelhalter, D. (eds.) Markov Chain Monte Carlo in Practice, pp. 259–273. Chapman & Hall, London (1996)Google Scholar
  12. Efron, B.: Missing data, imputation, and the bootstrap. J. Am. Stat. Assoc. 89(426), 463–475 (1994)MathSciNetCrossRefMATHGoogle Scholar
  13. Favaro, S., Hadjicharalambous, G., Prunster, I.: On a class of distributions on the simplex. J. Stat. Plan. Inference 141(426), 2987–3004 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. Feng, Z., McCulloch, C.: Using bootstrap likelihood ratio in finite mixture models. J. R. Stat. Soc. B 58, 609–617 (1996)MATHGoogle Scholar
  15. Forina M, Armanino C, Lanteri S, Tiscornia E (1983) Classification of olive oils from their fatty acid composition. In: Martens, Russwurm (eds) Food Research and Data Anlysis, Dip. Chimica e Tecnologie Farmaceutiche ed Alimentari, University of GenovaGoogle Scholar
  16. Frühwirth-Schnatter, S.: Finite Mixture and Markov Switching Models. Springer, New York (2006)MATHGoogle Scholar
  17. Gupta, R.D., Richards, D.S.P.: Multivariate liouville distributions. J. Multivar. Anal. 23, 233–256 (1987)MathSciNetCrossRefMATHGoogle Scholar
  18. Gupta, R.D., Richards, D.S.P.: Multivariate liouville distributions, II. Probab. Math. Stat. 12, 291–309 (1991)MathSciNetMATHGoogle Scholar
  19. Gupta, R.D., Richards, D.S.P.: Multivariate liouville distributions, III. J. Multivar. Anal. 43, 29–57 (1992)MathSciNetCrossRefMATHGoogle Scholar
  20. Gupta, R.D., Richards, D.S.P.: Multivariate liouville distributions, IV. J. Multivar. Anal. 54, 1–17 (1995)MathSciNetCrossRefMATHGoogle Scholar
  21. Gupta, R.D., Richards, D.S.P.: Multivariate liouville distributions, V. In: NL Johnson, N.B. (ed.) Advances in the Theory and Practice of Statistics: A Volume in Honour of Samuel Kotz, pp. 377–396. Wiley, New York (1997)Google Scholar
  22. Gupta, R.D., Richards, D.S.P.: The covariance structure of the multivariate liouville distributions. Contemp. Math. 287, 125–138 (2001a)MathSciNetCrossRefMATHGoogle Scholar
  23. Gupta, R.D., Richards, D.S.P.: The history of the Dirichlet and Liouville distributions. Int. Stat. Rev. 69(3), 433–446 (2001b)Google Scholar
  24. Hathaway, R.J.: A constrained formulation of maximum-likelihood estimation for normal mixture distributions. Ann. Stat. 13(2), 795–800 (1985)MathSciNetCrossRefMATHGoogle Scholar
  25. Kiefer, J., Wolfowitz, J.: Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann. Math. Stat. 27(4), 887–906 (1956)MathSciNetCrossRefMATHGoogle Scholar
  26. Lehmann, E., Casella, G.: Theory of Point Estimation. Springer, New York (1998)MATHGoogle Scholar
  27. Louis, T.A.: Finding the observed information matrix when using the EM algorithm. J. R. Stat. Soc. Ser. B 44(2), 226–233 (1982)MathSciNetMATHGoogle Scholar
  28. McLachlan, G., Peel, D.: Finite Mixture Models. Wiley, New York (2000)CrossRefMATHGoogle Scholar
  29. Meilijson, I.: A fast improvement to the EM algorithm on its own terms. J. R. Stat. Soc. Ser. B 51(1), 127–138 (1989)MathSciNetMATHGoogle Scholar
  30. Meng, X.L., Rubin, D.B.: Using EM to obtain asymptotic variance-covariance matrices: the SEM algorithm. J. Am. Stat. Assoc. 86(416), 899–909 (1991)CrossRefGoogle Scholar
  31. O’Hagan, A., Murphy, T.B., Gormley, I.C.: Computational aspects of fitting mixture models via the expectation-maximization algorithm. Comput. Stat. Data Anal. 56(12), 3843–3864 (2012)MathSciNetCrossRefMATHGoogle Scholar
  32. Ongaro, A., Migliorati, S.: A generalization of the Dirichlet distribution. J. Multivar. Anal. 114, 412–426 (2013)MathSciNetCrossRefMATHGoogle Scholar
  33. Palarea-Albaladejo, J., Martín-Fernández, J., Soto, J.: Dealing with distances and transformations for fuzzy c-means clustering of compositional data. J. Classif. 29, 144–169 (2012)MathSciNetCrossRefMATHGoogle Scholar
  34. Pawlowsky-Glahn, V., Egozcue, J., Tolosana-Delgado, R.: Modeling and Analysis of Compositional Data. Wiley, New York (2015)Google Scholar
  35. Peters, B.C., Walker, H.F.: An iterative procedure for obtaining maximum-likelihood estimates of the parameters for a mixture of normal distributions. SIAM J. Appl. Math. 35(2), 362–378 (1978)MathSciNetCrossRefMATHGoogle Scholar
  36. R Development Core Team (2015) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/
  37. Rayens, W.S., Srinivasan, C.: Dependence properties of generalized Liouville distributions on the simplex. J. Am. Stat. Assoc. 89(428), 1465–1470 (1994)MathSciNetCrossRefMATHGoogle Scholar
  38. Redner, R.: Note on the consistency of the maximum likelihood estimate for non-identifiable distributions. Ann. Stat. 9, 225–228 (1981)CrossRefMATHGoogle Scholar
  39. Rothenberg, T.: Identification in parametric models. Econometrica 39(3), 577–591 (1971)MathSciNetCrossRefMATHGoogle Scholar
  40. Smith, B., Rayens, W.: Conditional generalized Liouville distributions on the simplex. Statistics 36(2), 185–194 (2002)MathSciNetCrossRefMATHGoogle Scholar
  41. Wald, A.: Note on the consistency of the maximum likelihood estimate. Ann. Math. Stat. 20, 595–601 (1949)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Economics, Management and StatisticsUniversity of Milano-BicoccaMilanItaly
  2. 2.NeuroMi - Milan Center for NeuroscienceMilanItaly

Personalised recommendations