Computational Statistics

, Volume 27, Issue 3, pp 427–457 | Cite as

Copula analysis of mixture models

  • M. Vrac
  • L. BillardEmail author
  • E. Diday
  • A. Chédin
Original Paper


Contemporary computers collect databases that can be too large for classical methods to handle. The present work takes data whose observations are distribution functions (rather than the single numerical point value of classical data) and presents a computational statistical approach of a new methodology to group the distributions into classes. The clustering method links the searched partition to the decomposition of mixture densities, through the notions of a function of distributions and of multi-dimensional copulas. The new clustering technique is illustrated by ascertaining distinct temperature and humidity regions for a global climate dataset and shows that the results compare favorably with those obtained from the standard EM algorithm method.


Classification of distributions Copulas Dynamical clustering Data distributions Estimation Mixture model 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Achard V (1991) Trois Problemes dés de d’Analyse 3D de la Structure Thermodynamique de l’Atmosphére par Satellite: Mesure du Contenu en Ozone; Classification des Masses d’Air; Modélisation Hyper Rapide du Transfert Radiatif. Ph.D. Dissertation, University of ParisGoogle Scholar
  2. Ali MM, Mikhail NN, Haq MS (1978) A class of bivariate distributions including the bivariate logistic. J Multivar Anal 8: 405–412MathSciNetzbMATHCrossRefGoogle Scholar
  3. Arabie P, Carroll JD (1980) MAPCLUS: a mathematical programming approach to fitting the ADCLUS model. Psychometrika 45: 211–235zbMATHCrossRefGoogle Scholar
  4. Banfield JD, Raftery AE (1993) Model-based Gaussian and non-Gaussian clustering. Biometrics 49: 803–821MathSciNetzbMATHCrossRefGoogle Scholar
  5. Bishop CM (1995) Neural networks for pattern recognition. Clarendon Press, OxfordGoogle Scholar
  6. Bock HH (1998) Clustering and neural networks. In: Rizzi A, Vichi M, Bock HH (eds) Advances in data science and classification. Springer, Berlin, pp 265–277CrossRefGoogle Scholar
  7. Bock RD, Gibbons RD (1996) High-dimensional multivariate probit analysis. Biometrics 52: 1183–1194MathSciNetzbMATHCrossRefGoogle Scholar
  8. Brossier G (1990) Piecewise hierarchical clustering. J Classif 7: 197–216MathSciNetzbMATHCrossRefGoogle Scholar
  9. Celeux G, Diday E, Govaert G, Lechevallier Y, Ralambondrainy H (1989) Classification automatique des données. Dunod Informatique, ParisGoogle Scholar
  10. Celeux G, Diebolt J (1986) L’Algorithme SEM: Un algorithme d’apprentissage probabiliste pour la reconnaissance de mélange de densities. Revue de Statistiques Appliquées 34: 35–51zbMATHGoogle Scholar
  11. Celeux G, Govaert G (1992) A classification EM algorithm for clustering and two stochastic versions. Comput Stat Data Anal 14: 315–332MathSciNetzbMATHCrossRefGoogle Scholar
  12. Celeux G, Govaert G (1993) Comparison of the mixture and the classification maximum likelihood in cluster analysis. J Stat Comput Simul 47: 127–146CrossRefGoogle Scholar
  13. Chédin A, Scott N, Wahiche C, Moulinier P (1985) The improved initialization inversion method: a high resolution physical method for temperature retrievals from satellites of tiros-n series. J Appl Meteorol 24: 128–143CrossRefGoogle Scholar
  14. Chan JSK, Kuk AYC (1997) Maximum likelihood estimation for probit-linear mixed models with correlated random effects. Biometrics 53: 86–97MathSciNetzbMATHCrossRefGoogle Scholar
  15. Clayton DG (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65: 141–151MathSciNetzbMATHCrossRefGoogle Scholar
  16. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc B 39: 1–38MathSciNetzbMATHGoogle Scholar
  17. Diday E (1984) Une représentation visuelle des classes empietantes: les pyramides. Rapport de Recherche 291 INRIAGoogle Scholar
  18. Diday E (2001) A generalization of the mixture decomposition problem in the symbolic data analysis framework. Rapport de Recherche, CEREMADE 112: 1–14Google Scholar
  19. Diday E, Schroeder A, Ok Y (1974) The dynamic clusters method in pattern recognition. In: Proceedings of international federation for information processing congress. Elsevier, New York, pp 691–697Google Scholar
  20. Diday E, Vrac M (2005) Mixture decomposition of distributions by copulas in the symbolic data analysis framework. Discrete Appl Math 147: 27–41MathSciNetzbMATHCrossRefGoogle Scholar
  21. Fraley C, Raftery AE (2002) Model-based clustering, discriminant analysis and density estimation. J Am Stat Assoc 97: 611–631MathSciNetzbMATHCrossRefGoogle Scholar
  22. Frank MJ (1979) On the simultaneous associativity of F(x, y) and x + yF(x, y). Aequationes Mathematicae 19: 194–226MathSciNetzbMATHCrossRefGoogle Scholar
  23. Genest C, Ghoudi K (1994) Une famille de lois bidimensionelles insolite. Compte Rendus Academy Sciences Paris I 318: 351–354MathSciNetzbMATHGoogle Scholar
  24. Genest C, MacKay J (1986) The joy of copulas: bivariate distributions with uniform marginals. Am Stat 40: 280–283MathSciNetGoogle Scholar
  25. Genest C, Rivest LP (1993) Statistical inference procedures for bivariate Archimedean copulas. J Am Stat Assoc 88: 1034–1043MathSciNetzbMATHCrossRefGoogle Scholar
  26. Gordon A (1999) Classification. 2nd edn. Chapman and Hall, Boca RatonzbMATHGoogle Scholar
  27. Hartigan JA, Wong MA (1979) Algorithm AS136. A k-means clustering algorithm. Appl Stat 28: 100–108zbMATHCrossRefGoogle Scholar
  28. Hillali Y (1998) Analyse et modélisation des données probabilistes: Capacités et lois multidimensionelles. Ph.D. Dissertation, University of ParisGoogle Scholar
  29. Jain AK, Dubes RC (1988) Algorithms for clustering data. Prentice Hall, New JerseyzbMATHGoogle Scholar
  30. James GM, Sugar CA (2003) Clustering for sparsely sampled functional data. J Am Stat Assoc 98: 397–408MathSciNetzbMATHCrossRefGoogle Scholar
  31. Kuk AYC, Chan JSK (2001) Three ways of implementing the EM algorithm when parameters are not identifiable. Biometric J 43: 207–218MathSciNetzbMATHCrossRefGoogle Scholar
  32. Li LA, Sedransk N (1988) Mixtures of distributions: a topological approach. Ann Stat 16: 1623–1634MathSciNetzbMATHCrossRefGoogle Scholar
  33. McLachlan G, Peel D (2000) Finite mixture models. Wiley, New YorkzbMATHCrossRefGoogle Scholar
  34. Meng XL, Rubin DB (1991) Using EM to obtain asymptotic variance-covariance matrices: the SEM algorithm. J Am Stat Assoc 86: 899–909CrossRefGoogle Scholar
  35. Nelsen RB (1999) An introduction to copulas. Springer, New YorkzbMATHGoogle Scholar
  36. Parzen E (1962) On estimation of a probability density function and mode. Ann Math Stat 33: 1065–1076MathSciNetzbMATHCrossRefGoogle Scholar
  37. Prakasa Rao BLS (1983) Nonparametric functional estimation. Academic Press, New YorkzbMATHGoogle Scholar
  38. Redner RA, Walker H (1984) Mixture densities, maximum likelihood and the EM algorithm. SIAM Rev 26: 195–239MathSciNetzbMATHCrossRefGoogle Scholar
  39. Richardson S, Green PJ (1997) On Bayesian analysis of mixtures with an unknown number of components. J R Stat Soc Ser B 59: 731–792MathSciNetzbMATHCrossRefGoogle Scholar
  40. Schroeder A (1976) Analyse d’un mélange de distributions de probabilité de měme type. Revue de Statistiques Appliquées 24: 39–62MathSciNetGoogle Scholar
  41. Schweizer B, Sklar A (1983) Probabilistic metric spaces. North-Holland, New YorkzbMATHGoogle Scholar
  42. Schweizer B (1984) Distributions are the numbers of the future. In: diNola A, Ventre A (eds) Proceedings of the mathematics of fuzzy systems meeting, Naples, Italy. University of Naples, Naples, pp 137–149Google Scholar
  43. Scott AJ, Symons MJ (1971) Clustering methods based on likelihood ratio criteria. Biometrics 27: 387–397CrossRefGoogle Scholar
  44. Silverman BW (1986) Density estimation for statistics and data analysis. Chapman and Hall, LondonzbMATHGoogle Scholar
  45. Sklar A (1959) Fonction de répartition a n dimensions et leurs marges. Institute Statistics Université de Paris 8: 229–231MathSciNetGoogle Scholar
  46. Symons MJ (1981) Clustering criteria and multivariate normal mixtures. Biometrics 37: 35–43MathSciNetzbMATHCrossRefGoogle Scholar
  47. Tanner MA, Wong WH (1987) The calculation of posterior distribution by data augmentation (with discussion). J Am Stat Assoc 82: 528–550MathSciNetzbMATHCrossRefGoogle Scholar
  48. Titterington DM, Smith AFM, Makov UE (1985) Statistical analysis of finite mixture distributions. Wiley, New YorkzbMATHGoogle Scholar
  49. Vrac M (2002) Analyse et modélisation de données probabilistes par decomposition de mélange de copules et application á une base de données climatologiques. Ph.D. Dissertation, University of ParisGoogle Scholar
  50. Vrac M, Chédin A, Diday E (2005) Clustering a global field of atmospheric profiles by mixture decomposition of copulas. J Atmos Ocean Technol 22: 1445–1459CrossRefGoogle Scholar
  51. Wei GCG, Tanner MA (1990) A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithms. J Am Stat Assoc 85: 699–704CrossRefGoogle Scholar
  52. Winsberg S, DeSoete G (1999) Latent class models for time series analysis. Appl Stoch Models Bus Ind 15: 183–194zbMATHCrossRefGoogle Scholar
  53. Yakowitz SJ, Spragins LD (1968) On the identifiability of finite mixtures. Ann Math Stat 39: 209–214MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Laboratoire de Sciences du Climat et de l’Environment, IPSL-CNRS/CEA/UVSQCentre d’Etudes de SaclayGif-sur-YvetteFrance
  2. 2.Department of StatisticsUniversity of GeorgiaAthensUSA
  3. 3.CEREMADEUniversity of Paris DauphineParisFrance
  4. 4.Laboratoire de Meteorologie, Dynamique/IPSLEcole PolytechniquePalaiseauFrance

Personalised recommendations