Advertisement

Journal of Classification

, Volume 13, Issue 2, pp 195–212 | Cite as

An entropy criterion for assessing the number of clusters in a mixture model

  • Gilles Celeux
  • Gilda Soromenho
Article

Abstract

In this paper, we consider an entropy criterion to estimate the number of clusters arising from a mixture model. This criterion is derived from a relation linking the likelihood and the classification likelihood of a mixture. Its performance is investigated through Monte Carlo experiments, and it shows favorable results compared to other classical criteria.

Keywords

Cluster analysis Gaussian mixture Entropy Bayesian criteria 

Résumé

Nous proposons un critère d'entropie pour évaluer le nombre de classes d'une partition en nous fondant sur un modèle de mélange de lois de probabilité. Ce critère se déduit d'une relation liant la vraisemblance et la vraisemblance classifiante d'un mélange. Des simulations de Monte Carlo illustrent ses qualités par rapport à des critères plus classiques.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. AITKIN, M., and RUBIN, D. B. (1985), “Estimation and Hypothesis Testing in Finite Mixture Models,”Journal of the Royal Statistical Society, Series B, 47, 67–75.Google Scholar
  2. AITKIN, M., and TUNNICLIFFE WILSON, G. (1980), “Mixture Models, Outliers and the EM Algorithm,”Technometrics, 22, 325–332.Google Scholar
  3. AKAIKE, H. (1974), “A New Look at the Statistical Identification Model,”IEEE Transactions on Automatic Control, 19, 716–723.Google Scholar
  4. BANFIELD, J. D., and RAFTERY, A. E. (1993), “Model-Based Gaussian and non Gaussian Clustering,”Biometrics, 49, 803–821.Google Scholar
  5. BEZDEK, J. C. (1981),Pattern Recognition with Fuzzy Objective Function Algorithms, New York: Plenum.Google Scholar
  6. BOCK, H. H. (1985), “On Tests Concerning the Existence of a Classification,”Journal of Classification, 2, 77–108.Google Scholar
  7. BOCK, H. H. (1989), “Probabilistic Aspects in Cluster Analysis,” inConceptual and Numerical Analysis of Data, Ed., O. Opitz, Springer-Verlag, Heidelberg, pp. 12–44.Google Scholar
  8. BOZDOGAN, H. (1990), “On the Information-Based Measure of Covariance Complexity and its Application to the Evaluation of Multivariate Linear Models,”Communications in Statistics, Theory and Methods, 19, 221–278.Google Scholar
  9. BOZDOGAN, H. (1993), “Choosing the Number of Component Clusters in the Mixture-Model Using a New Informational Complexity Criterion of the Inverse-Fisher Information Matrix,” inInformation and Classification, Eds., O. Optiz, B. Lausen, and R. Klar, Heidelberg: Springer-Verlag, pp. 40–54.Google Scholar
  10. BOZDOGAN, H., and SCLOVE, S. L. (1984), “Multi-Sample Cluster Analysis using Akaike 's Information Criterion,”Annals of Institute of Statistical Mathematics, 36, 163–180.Google Scholar
  11. BRYANT, P. G. (1991), “Large-Sample Results for Optimization Based Clustering Methods,”Journal of Classification, 8, 31–44.Google Scholar
  12. BRYANT, P. G. (1993), “On Detecting the Numbers of Clusters Using the MDL Principle,” Unpublished Manuscript.Google Scholar
  13. BRYANT, P. G., and WILLIAMSON, J. A. (1978), “Asymptotic Behavior of Classification Maximum Likelihood Estimates,”Biometrika, 65, 273–281.Google Scholar
  14. BRYANT, P. G., and WILLIAMSON, J. A. (1986), “Maximum Likelihood and Classification: a Comparison of Three Approaches,” inClassification as a tool of research, Eds., W. Gaul and M. Schader, North-Holland, pp. 33–45.Google Scholar
  15. CELEUX, G. (1986), “Validity Tests in Cluster Analysis Using a Probabilistic Teacher Algorithm,”COMPSTAT 90, Eds., F. de Antoni, N. Lauro and A. Rizzi, Heidelberg: Springer-Verlag, pp. 163–169.Google Scholar
  16. CELEUX, G. and GOVAERT, G. (1991), “Clustering Criteria for Discrete Data and Latent Class Models,”Journal of Classification, 8, 157–176.Google Scholar
  17. CELEUX, G. and GOVAERT, G. (1993), “Comparison of the Mixture and the Classification Maximum Likelihood in Cluster Analysis,”Journal of Statistical Computation and Simulation, 47, 127–146.Google Scholar
  18. CUTLER, A., and WINDHAM, M. P. (1993), “Information-Based Validity Functionals for Mixture Analysis,”Proceedings of the first US-Japan Conference on the Frontiers of Statistical Modeling, Ed., H. Bozdogan, Amsterdam: Kluwer, pp. 149–170.Google Scholar
  19. DEMPSTER, A. P., LAIRD, N. M., and RUBIN, D. B. (1977), “Maximum Likelihood from Incomplete Data via the EM Algorithm (with discussion),”Journal of the Royal Statistical Society, Series B, 39, 1–38.Google Scholar
  20. GANESALINGAM, S. (1989), “Clasification and Mixture Approaches to Clustering via Maximum Likelihood,”Applied Statistics, 38, 455–466.Google Scholar
  21. HATHAWAY, R. J. (1986), “Another Interpretation of the EM Algorithm for Mixture Distributions,”Statistics and Probability Letters, 4, 53–56.Google Scholar
  22. KOEHLER, A. B., and MURPHREE, E. H. (1988), “A comparison of the Akaike and Schwarz Criteria for Selecting Model Order,”Applied Statistics, 37, 187–195.Google Scholar
  23. MACQUEEN, J. (1967), “Some Methods for Classification and Analysis of Multivariate Observations,”Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, Eds., L. M. Le Cam and J. Neyman, Berkeley: University of California Press, Vol. 1 pp. 281–297.Google Scholar
  24. MCLACHLAN, G. J. (1987), “On Bootstraping the Likelihood Ratio Test Statistic for the Number of Components in a Normal Mixture,”Applied Statistics, 36, 318–324.Google Scholar
  25. MCLACHLAN, G. J., and BASFORD, K. E. (1988),Mixture Models, Inference and Applications to Clustering, New York. Marcel Dekker.Google Scholar
  26. MARRIOTT, F. H. C. (1975), “Separating Mixtures of Normal Distributions,”Biometrics, 31, 767–769.Google Scholar
  27. RISSANEN, J. (1989),Stochastic Complexity in Statistical Inquiry, Teaneck, New Jersey: World Scientific.Google Scholar
  28. SCHWARZ, G. (1978), “Estimating the Dimension of a Model,”Annals of Statistics, 6, 461–464.Google Scholar
  29. SOROMENHO, G. (1994), “Comparing Approaches for Testing the Number of Components in a Finite Mixture Model,”Computational Statistics, 9, 65–78.Google Scholar
  30. TITTERINGTON, D. M., SMITH, A. F., and MAKOV, U. E. (1985),Statistical Analysis of Finite Mixture Distributions, New York: Wiley.Google Scholar
  31. WINDHAM, M. P., and CUTLER, A. (1992), “Information Ratios for Validating Cluster Analyses,”Journal of the American Statistical Association, 87, 1188–1192.Google Scholar
  32. WOLFE, J. H. (1970), “Pattern Clustering by Multivariate Mixture Analysis,”Multivariate Behavioral Research, 5, 329–350.Google Scholar
  33. WOLFE, J. H. (1971), “A Monte Carlo Study of the Sampling Distribution of the Likelihood Ratio for Mixtures of Multinormal Distributions,” US Naval Personnel Research Activity.Technical Bulletin STB 72-2, San Diego, California.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1996

Authors and Affiliations

  • Gilles Celeux
    • 1
  • Gilda Soromenho
    • 2
  1. 1.INRIA Rhône-AlpesMartinFrance
  2. 2.Lisbon UniversityLisboaPortugal

Personalised recommendations