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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.
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.

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