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Maximum likelihood estimation of Gaussian mixture models without matrix operations

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  • Hien Nguyen
  • Geoffrey McLachlan

Abstract

The Gaussian mixture model (GMM) is a popular tool for multivariate analysis, in particular, cluster analysis. The expectation–maximization (EM) algorithm is generally used to perform maximum likelihood (ML) estimation for GMMs due to the M-step existing in closed form and its desirable numerical properties, such as monotonicity. However, the EM algorithm has been criticized as being slow to converge and thus computationally expensive in some situations. In this article, we introduce the linear regression characterization (LRC) of the GMM. We show that the parameters of an LRC of the GMM can be mapped back to the natural parameters, and that a minorization–maximization (MM) algorithm can be constructed, which retains the desirable numerical properties of the EM algorithm, without the use of matrix operations. We prove that the ML estimators of the LRC parameters are consistent and asymptotically normal, like their natural counterparts. Furthermore, we show that the LRC allows for simple handling of singularities in the ML estimation of GMMs. Using numerical simulations in the R programming environment, we then demonstrate that the MM algorithm can be faster than the EM algorithm in various large data situations, where sample sizes range in the tens to hundreds of thousands and for estimating models with up to 16 mixture components on multivariate data with up to 16 variables. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • Hien Nguyen & Geoffrey McLachlan, 2015. "Maximum likelihood estimation of Gaussian mixture models without matrix operations," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 9(4), pages 371-394, December.
    • Handle: RePEc:spr:advdac:v:9:y:2015:i:4:p:371-394
    DOI: 10.1007/s11634-015-0209-7
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    References listed on IDEAS

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    1. Hunter D.R. & Lange K., 2004. "A Tutorial on MM Algorithms," The American Statistician, American Statistical Association, vol. 58, pages 30-37, February.
    2. Salvatore Ingrassia, 2004. "A likelihood-based constrained algorithm for multivariate normal mixture models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 13(2), pages 151-166, September.
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    7. Salvatore Ingrassia & Simona Minotti & Giorgio Vittadini, 2012. "Local Statistical Modeling via a Cluster-Weighted Approach with Elliptical Distributions," Journal of Classification, Springer;The Classification Society, vol. 29(3), pages 363-401, October.
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    Cited by:

    1. Marek Śmieja & Magdalena Wiercioch, 2017. "Constrained clustering with a complex cluster structure," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 11(3), pages 493-518, September.
    2. Hien D. Nguyen & Geoffrey J. McLachlan & Jeremy F. P. Ullmann & Andrew L. Janke, 2016. "Spatial clustering of time series via mixture of autoregressions models and Markov random fields," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 70(4), pages 414-439, November.
    3. Lloyd-Jones, Luke R. & Nguyen, Hien D. & McLachlan, Geoffrey J., 2018. "A globally convergent algorithm for lasso-penalized mixture of linear regression models," Computational Statistics & Data Analysis, Elsevier, vol. 119(C), pages 19-38.
    4. Xifen Huang & Chaosong Xiong & Tao Jiang & Junfeng Lu & Jinfeng Xu, 2022. "Efficient Estimation and Inference in the Proportional Odds Model for Survival Data," Mathematics, MDPI, vol. 10(18), pages 1-17, September.
    5. Nguyen, Hien D. & McLachlan, Geoffrey J., 2016. "Maximum likelihood estimation of triangular and polygonal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 102(C), pages 23-36.

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