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On the upper bound of the number of modes of a multivariate normal mixture

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  • Ray, Surajit
  • Ren, Dan

Abstract

The main result of this article states that one can get as many as D+1 modes from just a two component normal mixture in D dimensions. Multivariate mixture models are widely used for modeling homogeneous populations and for cluster analysis. Either the components directly or modes arising from these components are often used to extract individual clusters. Although in lower dimensions these strategies work well, our results show that high dimensional mixtures are often very complex and researchers should take extra precautions when using mixture models for cluster analysis. Further our analysis shows that the number of modes depends on the component means and eigenvalues of the ratio of the two component covariance matrices, which in turn provides a clear guideline as to when one can use mixture analysis for clustering high dimensional data.

Suggested Citation

  • Ray, Surajit & Ren, Dan, 2012. "On the upper bound of the number of modes of a multivariate normal mixture," Journal of Multivariate Analysis, Elsevier, vol. 108(C), pages 41-52.
    • Handle: RePEc:eee:jmvana:v:108:y:2012:i:c:p:41-52
    DOI: 10.1016/j.jmva.2012.02.006
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    References listed on IDEAS

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    1. Chen, Jiahua & Tan, Xianming, 2009. "Inference for multivariate normal mixtures," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1367-1383, August.
    2. Jörn Dannemann & Hajo Holzmann, 2008. "Likelihood Ratio Testing for Hidden Markov Models Under Non‐standard Conditions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(2), pages 309-321, June.
    3. Hajo Holzmann & Sebastian Vollmer, 2008. "A likelihood ratio test for bimodality in two-component mixtures with application to regional income distribution in the EU," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 92(1), pages 57-69, February.
    4. M.‐Y. Cheng & P. Hall, 1998. "Calibrating the excess mass and dip tests of modality," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(3), pages 579-589.
    5. 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.
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    Cited by:

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    4. Redivo, Edoardo & Nguyen, Hien D. & Gupta, Mayetri, 2020. "Bayesian clustering of skewed and multimodal data using geometric skewed normal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 152(C).

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