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Assessment of the number of components in Gaussian mixture models in the presence of multiple local maximizers

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  • Kim, Daeyoung
  • Seo, Byungtae

Abstract

Gaussian mixtures are very flexible in representing the underlying structure in the data. However, the likelihood inference for Gaussian mixtures with unrestricted covariance matrices is theoretically and practically challenging because the likelihood function is unbounded and often has multiple local maximizers. As shown in the numerical studies of this paper, the presence of multiple local maximizers including spurious local maximizers affects the performances of model selection criteria used to choose the number of components. In this paper we propose a new type of likelihood-based estimator, a gradient-based k-deleted maximum likelihood estimator, for Gaussian mixture models. The proposed estimator is designed to avoid spurious local maximizers and choose a statistically desirable local maximizer in the presence of multiple local maximizers. We first prove the consistency of the proposed estimator and then examine, by a real-data example and simulation studies, the performance of the proposed method in the likelihood-based model selection criteria commonly used to assess the number of components in Gaussian mixture models.

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  • Kim, Daeyoung & Seo, Byungtae, 2014. "Assessment of the number of components in Gaussian mixture models in the presence of multiple local maximizers," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 100-120.
    • Handle: RePEc:eee:jmvana:v:125:y:2014:i:c:p:100-120
    DOI: 10.1016/j.jmva.2013.11.018
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    1. Seo, Byungtae & Lindsay, Bruce G., 2010. "A computational strategy for doubly smoothed MLE exemplified in the normal mixture model," Computational Statistics & Data Analysis, Elsevier, vol. 54(8), pages 1930-1941, August.
    2. Biernacki, Christophe & Chrétien, Stéphane, 2003. "Degeneracy in the maximum likelihood estimation of univariate Gaussian mixtures with EM," Statistics & Probability Letters, Elsevier, vol. 61(4), pages 373-382, February.
    3. Chen, Jiahua & Tan, Xianming, 2009. "Inference for multivariate normal mixtures," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1367-1383, August.
    4. Wilfried Seidel & Hana Ševčíková, 2004. "Types of likelihood maxima in mixture models and their implication on the performance of tests," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(4), pages 631-654, December.
    5. Gabriela Ciuperca & Andrea Ridolfi & Jérôme Idier, 2003. "Penalized Maximum Likelihood Estimator for Normal Mixtures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(1), pages 45-59, March.
    6. Gilles Celeux & Gilda Soromenho, 1996. "An entropy criterion for assessing the number of clusters in a mixture model," Journal of Classification, Springer;The Classification Society, vol. 13(2), pages 195-212, September.
    7. Dankmar Böhning & Ekkehart Dietz & Rainer Schaub & Peter Schlattmann & Bruce Lindsay, 1994. "The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(2), pages 373-388, June.
    8. Hamparsum Bozdogan, 1987. "Model selection and Akaike's Information Criterion (AIC): The general theory and its analytical extensions," Psychometrika, Springer;The Psychometric Society, vol. 52(3), pages 345-370, September.
    9. Chris Fraley & Adrian E. Raftery, 2007. "Bayesian Regularization for Normal Mixture Estimation and Model-Based Clustering," Journal of Classification, Springer;The Classification Society, vol. 24(2), pages 155-181, September.
    10. Christian Hennig, 2010. "Methods for merging Gaussian mixture components," 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. 4(1), pages 3-34, April.
    11. Fraley C. & Raftery A.E., 2002. "Model-Based Clustering, Discriminant Analysis, and Density Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 611-631, June.
    12. Seo, Byungtae & Kim, Daeyoung, 2012. "Root selection in normal mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 56(8), pages 2454-2470.
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    Cited by:

    1. Shiyao Liu & Huaiqing Wu & William Q. Meeker, 2015. "Understanding and Addressing the Unbounded "Likelihood" Problem," The American Statistician, Taylor & Francis Journals, vol. 69(3), pages 191-200, August.
    2. Roberto Rocci & Stefano Antonio Gattone & Roberto Di Mari, 2018. "A data driven equivariant approach to constrained Gaussian mixture modeling," 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. 12(2), pages 235-260, June.
    3. Heather Shappell & Sean L. Simpson, 2022. "Discussion on “Distributional independent component analysis for diverse neuroimaging modalities” by Ben Wu, Subhadip Pal, Jian Kang, and Ying Guo," Biometrics, The International Biometric Society, vol. 78(3), pages 1106-1108, September.
    4. Xuwen Zhu & Volodymyr Melnykov, 2015. "Probabilistic assessment of model-based clustering," 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 395-422, December.
    5. Sangkon Oh & Byungtae Seo, 2023. "Merging Components in Linear Gaussian Cluster-Weighted Models," Journal of Classification, Springer;The Classification Society, vol. 40(1), pages 25-51, April.
    6. Roberto Mari & Roberto Rocci & Stefano Antonio Gattone, 2020. "Scale-constrained approaches for maximum likelihood estimation and model selection of clusterwise linear regression models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(1), pages 49-78, March.
    7. Derek S. Young & Xi Chen & Dilrukshi C. Hewage & Ricardo Nilo-Poyanco, 2019. "Finite mixture-of-gamma distributions: estimation, inference, and model-based clustering," 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. 13(4), pages 1053-1082, December.

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