Advertisement

Testing for Genuine Multimodality in Finite Mixture Models: Application to Linear Regression Models

  • Bettina Grün
  • Friedrich Leisch
Conference paper
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)

Abstract

Identifiability problems can be encountered when fitting finite mixture models and their presence should be investigated by model diagnostics. In this paper we propose diagnostic tools to check for identifiability problems based on the fact that they induce multiple (global) modes in the distribution of the parameterizations of the maximum likelihood models depending on the data generating process. The parametric bootstrap is used to approximate this distribution. In order to investigate the presence of multiple (global) modes the congruence between the results of information-based methods based on asymptotic theory and those derived using the models fitted to the bootstrap samples with initalization in the solution as well as random initialization is assessed. The methods are illustrated using a finite mixture of Gaussian regression models on data from a study on spread of viral infection.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BARINGHAUS, L. and FRANZ, C. (2004): On a New Multivariate Two-sample Test. Journal of Multivariate Analysis, 88, 190–206.CrossRefMathSciNetMATHGoogle Scholar
  2. BASFORD, K.E., GREENWAY, D.R., MCLACHLAN, G.J. and PEEL, D. (1997): Standard Errors of Fitted Means Under Normal Mixture Model. Computational Statistics, 12, 1–17.MATHGoogle Scholar
  3. DEMPSTER, A.P., LAIRD, N.M. and RUBIN, D.B. (1977): Maximum Likelihood from Incomplete Data Via the EM Algorithm. Journal of the Royal Statistical Society B, 39, 1–38.MathSciNetMATHGoogle Scholar
  4. GRÜN, B. and LEISCH, F. (2004): Bootstrapping Finite Mixture Models. In: J. Antoch (Ed.): Compstat 2004 — Proceedings in Computational Statistics. Physica, Heidelberg, 1115–1122.Google Scholar
  5. HENNIG, C. (2000): Identifiability of Models for Clusterwise Linear Regression. Journal of Classification, 17,2, 273–296.CrossRefMathSciNetMATHGoogle Scholar
  6. HOTHORN, T., LEISCH, F., ZEILEIS, A. and HORNIK, K. (2005): The Design and Analysis of Benchmark Experiments. Journal of Computational and Graphical Statistics, 14,3, 1–25.CrossRefMathSciNetGoogle Scholar
  7. LOUIS, T.A. (1982): Finding the Observed Information Matrix When Using the EM Algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 44,2, 226–233.MathSciNetMATHGoogle Scholar
  8. MCLACHLAN, G.J. and PEEL, D. (2000): Finite Mixture Models. Wiley.Google Scholar
  9. MINNOTTE, M. C. (1997): Nonparameteric Testing of the Existence of Modes. The Annals of Statistics, 25,4, 1646–1660.CrossRefMathSciNetMATHGoogle Scholar
  10. STEPHENS, M. (2000): Dealing with Label Switching in Mixture Models. Journal of the Royal Statistical Society B, 62,4, 795–809.CrossRefMathSciNetMATHGoogle Scholar
  11. TEICHER, H. (1963): Identifiability of Finite Mixtures. The Annals of Mathematical Statistics, 34, 1265–1269.CrossRefMathSciNetMATHGoogle Scholar
  12. TURNER, T.R. (2000): Estimating the Propagation Rate of a Viral Infection of Potato Plants Via Mixtures of Regressions. Journal of the Royal Statistical Society C, 49,3, 371–384.CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Bettina Grün
    • 1
  • Friedrich Leisch
    • 2
  1. 1.Institut für Statistik und WahrscheinlichkeitstheorieTechnische Universität WienWienAustria
  2. 2.Institut für StatistikLudwig-Maximilians-Universität MünchenMünchenGermany

Personalised recommendations