Robust cluster analysis via mixtures of multivariate t-distributions

  • Geoffrey J. McLachlan
  • David Peel
Statistical Pattern Recognition
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1451)


Normal mixture models are being increasingly used as a way of clustering sets of continuous multivariate data. They provide a probabilistic (soft) clustering of the data in terms of their fitted posterior probabilities of membership of the mixture components corresponding to the clusters. An outright (hard) clustering can be subsequently obtained by assigning each observation to the component to which it has the highest fitted posterior probability of belonging. However, outliers in the data can affect the estimates of the parameters in the normal component densities, and hence the implied clustering. A more robust approach is to fit mixtures of multivariate t-distributions, which have longer tails than the normal components. The expectation-maximization (EM) algorithm can be used to fit mixtures of t-distributions by maximum likelihood. The application of this model to provide a robust approach to clustering is illustrated on a real data set. It is demonstrated how the use of t-components provides less extreme estimates of the posterior probabilities of cluster membership.


Posterior Probability Normal Mixture Finite Mixture Model Normal Mixture Model Minimum Covariance Determinant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Campbell, N.A. (1994). Mixture models and atypical values. Mathematical Geology 16, 465–477.CrossRefGoogle Scholar
  2. Campbell, N.A. and Mahon, R.J. (1974). A multivariate study of variation in two species of rock crab of genus Leptograpsus. Australian Journal of Zoology 22, 417–425.CrossRefGoogle Scholar
  3. Davé, R.N. and Krishnapuram, R. (1995). Robust clustering methods: a unified view. IEEE Transactions on Fuzzy Systems 5, 270–293.CrossRefGoogle Scholar
  4. Dempster, A.P., Laird, N.M., and and Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society B 39, 1–38.Google Scholar
  5. De Veaux, R.D. and Kreiger, A.M. (1990). Robust estimation of a normal mixture. Statistics & Probability Letters 10, 1–7.Google Scholar
  6. Frigui, H. and Krishnapuram, R. (1996). A robust algorithm for automatic extraction of an unknown number of clusters from noisy data. Pattern Recognition Letters 17, 1223–1232.CrossRefGoogle Scholar
  7. Hawkins, D.M. (1981). A new test for multivariate normality and homoscedasticity. Technometrics 23, 105–110.Google Scholar
  8. Hampel, F.R. (1973). Robust estimation: a condensed partial survey. Z. Wahrscheinlickeitstheorie verw. Gebiete 27, 87–104.CrossRefGoogle Scholar
  9. Hawkins, D.M. and McLachlan, G.J. (1997). High-breakdown linear discriminant analysis. Journal of the American Statistical Association 92, 136–143.MathSciNetGoogle Scholar
  10. Huber, P.J. (1964). Robust estimation of a location parameter. Annals of Mathematical Statistics 35, 73–101.Google Scholar
  11. Kharin, Y. (1996). Robustness in Statistical Pattern Recognition. Dordrecht: Kluwer.Google Scholar
  12. Liu, C. and Rubin, D.B. (1995). ML estimation of the t distribution using EM and its extensions, ECM and ECME. Statistica Sinica, 5, 19–39.Google Scholar
  13. McLachlan, G.J. (1992). Discriminant Analysis and Statistical Pattern Recognition. New York: Wiley.Google Scholar
  14. McLachlan, G.J. (1999). Finite Mixture Models. New York: Wiley.Google Scholar
  15. McLachlan, G.J. and Basford, K.E. (1988). Mixture Models: Inference and Applications to Clustering. New York: Marcel Dekker.Google Scholar
  16. McLachlan, G.J. and Krishnan, T. (1997). The EM Algorithm and Extensions. New York: Wiley.Google Scholar
  17. McLachlan, G.J., Peel, D., Basford, K.E., and Adams, P. (1997). MIXFIT: an algorithm for the automatic fitting and testing of normal mixture models. Unpublished manuscript.Google Scholar
  18. Ripley, B.D. (1996). Pattern Recognition and Neural Networks. Cambridge: Cambridge University Press.Google Scholar
  19. Rocke, D.M. and Woodruff, D.L. (1997). Robust estimation of multivariate location and shape. Journal of Statistical Planning and Inference 57, 245–255.CrossRefMathSciNetGoogle Scholar
  20. Rousseeuw, P.J., Kaufman, L., and Trauwaert, E. (1996). Fuzzy clustering using scatter matrices. Computational Statistics and Data Analysis 23, 135–151.CrossRefGoogle Scholar
  21. Zhuang, X., Huang, Y., Palaniappan, K., and Zhao, Y. (1996). Gaussian density mixture modeling, decomposition and applications. IEEE Transactions on Image Processing 5, 1293–1302.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Geoffrey J. McLachlan
    • 1
  • David Peel
    • 1
  1. 1.Department of MathematicsUniversity of QueenslandSt. LuciaAustralia

Personalised recommendations