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Eigenvectors of a kurtosis matrix as interesting directions to reveal cluster structure

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  • Peña, Daniel
  • Prieto, Francisco J.
  • Viladomat, Júlia

Abstract

In this paper we study the properties of a kurtosis matrix and propose its eigenvectors as interesting directions to reveal the possible cluster structure of a data set. Under a mixture of elliptical distributions with proportional scatter matrix, it is shown that a subset of the eigenvectors of the fourth-order moment matrix corresponds to Fisher's linear discriminant subspace. The eigenvectors of the estimated kurtosis matrix are consistent estimators of this subspace and its calculation is easy to implement and computationally efficient, which is particularly favourable when the ratio n/p is large.

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  • Peña, Daniel & Prieto, Francisco J. & Viladomat, Júlia, 2010. "Eigenvectors of a kurtosis matrix as interesting directions to reveal cluster structure," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 1995-2007, October.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:9:p:1995-2007
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    References listed on IDEAS

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    1. Kollo, Tõnu, 2008. "Multivariate skewness and kurtosis measures with an application in ICA," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2328-2338, November.
    2. Oja, Hannu, 1983. "Descriptive statistics for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 327-332, October.
    3. Pena D. & Prieto F.J., 2001. "Cluster Identification Using Projections," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1433-1445, December.
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    Cited by:

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    2. Loperfido, Nicola, 2021. "Some theoretical properties of two kurtosis matrices, with application to invariant coordinate selection," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    3. Loperfido, Nicola, 2013. "Skewness and the linear discriminant function," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 93-99.
    4. Javed, Farrukh & Loperfido, Nicola & Mazur, Stepan, 2024. "The Method of Moments for Multivariate Random Sums," Working Papers 2024:6, Örebro University, School of Business.
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    7. Becquart, Colombe & Archimbaud, Aurore & Ruiz-Gazen, Anne & Prilé, Luka & Nordhausen, Klaus, 2024. "Invariant Coordinate Selection and Fisher Discriminant Subspace Beyond The Case of Two Groups," TSE Working Papers 24-1579, Toulouse School of Economics (TSE).
    8. Alashwali, Fatimah & Kent, John T., 2016. "The use of a common location measure in the invariant coordinate selection and projection pursuit," Journal of Multivariate Analysis, Elsevier, vol. 152(C), pages 145-161.
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