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On the structure of the quadratic subspace in discriminant analysis

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  • Velilla, Santiago

Abstract

The concept of quadratic subspace is introduced as a helpful tool for dimension reduction in quadratic discriminant analysis (QDA). It is argued that an adequate representation of the quadratic subspace may lead to better methods for both data representation and classification. Several theoretical results describe the structure of the quadratic subspace, that is shown to contain some of the subspaces previously proposed in the literature for finding differences between the class means and covariances. A suitable assumption of orthogonality between location and dispersion subspaces allows us to derive a convenient reduced version of the full QDA rule. The behavior of these ideas in practice is illustrated with three real data examples.

Suggested Citation

  • Velilla, Santiago, 2010. "On the structure of the quadratic subspace in discriminant analysis," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1239-1251, May.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:5:p:1239-1251
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    References listed on IDEAS

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    1. Schott, James R., 1993. "Dimensionality reduction in quadratic discriminant analysis," Computational Statistics & Data Analysis, Elsevier, vol. 16(2), pages 161-174, August.
    2. Ye Z. & Weiss R.E., 2003. "Using the Bootstrap to Select One of a New Class of Dimension Reduction Methods," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 968-979, January.
    3. Cook, R. Dennis & Forzani, Liliana, 2009. "Likelihood-Based Sufficient Dimension Reduction," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 197-208.
    4. Wei‐Chien Chang, 1987. "A Graph for Two Training Samples in a Discriminant Analysis," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 36(1), pages 82-91, March.
    5. R. Dennis Cook & Liliana Forzani, 2008. "Covariance reducing models: An alternative to spectral modelling of covariance matrices," Biometrika, Biometrika Trust, vol. 95(4), pages 799-812.
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    Cited by:

    1. Luca Scrucca, 2014. "Graphical tools for model-based mixture discriminant analysis," 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. 8(2), pages 147-165, June.
    2. Velilla, Santiago, 2012. "A note on the structure of the quadratic subspace in discriminant analysis," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 739-747.

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