IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v101y2010i5p1239-1251.html
   My bibliography  Save this article

On the structure of the quadratic subspace in discriminant analysis

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(10)00005-9
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. Schott, James R., 1993. "Dimensionality reduction in quadratic discriminant analysis," Computational Statistics & Data Analysis, Elsevier, vol. 16(2), pages 161-174, August.
    4. 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.
    5. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wang, Qin & Xue, Yuan, 2021. "An ensemble of inverse moment estimators for sufficient dimension reduction," Computational Statistics & Data Analysis, Elsevier, vol. 161(C).
    2. Wang, Tao & Zhu, Lixing, 2013. "Sparse sufficient dimension reduction using optimal scoring," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 223-232.
    3. repec:jss:jstsof:39:i03 is not listed on IDEAS
    4. Forzani, Liliana & Rodriguez, Daniela & Smucler, Ezequiel & Sued, Mariela, 2019. "Sufficient dimension reduction and prediction in regression: Asymptotic results," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 339-349.
    5. 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.
    6. Yin, Xiangrong & Li, Bing & Cook, R. Dennis, 2008. "Successive direction extraction for estimating the central subspace in a multiple-index regression," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1733-1757, September.
    7. Yasushi Narushima & Shummin Nakayama & Masashi Takemura & Hiroshi Yabe, 2023. "Memoryless Quasi-Newton Methods Based on the Spectral-Scaling Broyden Family for Riemannian Optimization," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 639-664, May.
    8. Xinchao Luo & Lixing Zhu & Hongtu Zhu, 2016. "Single‐index varying coefficient model for functional responses," Biometrics, The International Biometric Society, vol. 72(4), pages 1275-1284, December.
    9. Li, Junlan & Wang, Tao, 2021. "Dimension reduction in binary response regression: A joint modeling approach," Computational Statistics & Data Analysis, Elsevier, vol. 156(C).
    10. Scrucca, Luca, 2011. "Model-based SIR for dimension reduction," Computational Statistics & Data Analysis, Elsevier, vol. 55(11), pages 3010-3026, November.
    11. Velilla Cerdan, Santiago & Hernández, Adolfo, 2001. "Dimension reduction transformations in discriminant analysis," DES - Working Papers. Statistics and Econometrics. WS ws016614, Universidad Carlos III de Madrid. Departamento de Estadística.
    12. Kim, Kyongwon, 2022. "On principal graphical models with application to gene network," Computational Statistics & Data Analysis, Elsevier, vol. 166(C).
    13. Schott, James R., 2012. "A note on maximum likelihood estimation for covariance reducing models," Statistics & Probability Letters, Elsevier, vol. 82(9), pages 1629-1631.
    14. Xu Guo & Tao Wang & Lixing Zhu, 2016. "Model checking for parametric single-index models: a dimension reduction model-adaptive approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(5), pages 1013-1035, November.
    15. Wang, Pei & Yin, Xiangrong & Yuan, Qingcong & Kryscio, Richard, 2021. "Feature filter for estimating central mean subspace and its sparse solution," Computational Statistics & Data Analysis, Elsevier, vol. 163(C).
    16. Sheng, Wenhui & Yin, Xiangrong, 2013. "Direction estimation in single-index models via distance covariance," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 148-161.
    17. Alessandro Barbarino & Efstathia Bura, 2015. "Forecasting with Sufficient Dimension Reductions," Finance and Economics Discussion Series 2015-74, Board of Governors of the Federal Reserve System (U.S.).
    18. Diego Tomassi & Liliana Forzani & Efstathia Bura & Ruth Pfeiffer, 2017. "Sufficient dimension reduction for censored predictors," Biometrics, The International Biometric Society, vol. 73(1), pages 220-231, March.
    19. Guochang Wang & Jianjun Zhou & Wuqing Wu & Min Chen, 2017. "Robust functional sliced inverse regression," Statistical Papers, Springer, vol. 58(1), pages 227-245, March.
    20. Kapla, Daniel & Fertl, Lukas & Bura, Efstathia, 2022. "Fusing sufficient dimension reduction with neural networks," Computational Statistics & Data Analysis, Elsevier, vol. 168(C).
    21. Bouveyron, C. & Girard, S. & Schmid, C., 2007. "High-dimensional data clustering," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 502-519, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:101:y:2010:i:5:p:1239-1251. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.