IDEAS home Printed from https://ideas.repec.org/a/spr/jclass/v31y2014i1p2-27.html
   My bibliography  Save this article

Canonical Analysis: Ranks, Ratios and Fits

Author

Listed:
  • Casper Albers
  • John Gower

Abstract

Measurements of p variables for n samples are collected into a n×p matrix X, where the samples belong to one of k groups. The group means are separated by Mahalanobis distances. CVA optimally represents the group means of X in an r-dimensional space. This can be done by maximizing a ratio criterion (basically one- dimensional) or, more flexibly, by minimizing a rank-constrained least-squares fitting criterion (which is not confined to being one-dimensional but depends on defining an appropriate Mahalanobis metric). In modern n > p problems, where W is not of full rank, the ratio criterion is shown not to be coherent but the fit criterion, with an attention to associated metrics, readily generalizes. In this context we give a unified generalization of CVA, introducing two metrics, one in the range space of W and the other in the null space of W, that have links with Mahalanobis distance. This generalization is computationally efficient, since it requires only the spectral decomposition of a n×n matrix. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Casper Albers & John Gower, 2014. "Canonical Analysis: Ranks, Ratios and Fits," Journal of Classification, Springer;The Classification Society, vol. 31(1), pages 2-27, April.
  • Handle: RePEc:spr:jclass:v:31:y:2014:i:1:p:2-27
    DOI: 10.1007/s00357-014-9146-y
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00357-014-9146-y
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1007/s00357-014-9146-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    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. W. J. Krzanowski & P. Jonathan & W. V. McCarthy & M. R. Thomas, 1995. "Discriminant Analysis with Singular Covariance Matrices: Methods and Applications to Spectroscopic Data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 44(1), pages 101-115, March.
    2. Queen, Catriona M. & Albers, Casper J., 2009. "Intervention and Causality: Forecasting Traffic Flows Using a Dynamic Bayesian Network," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 669-681.
    3. Albers, C.J. & Critchley, F. & Gower, J.C., 2011. "Applications of quadratic minimisation problems in statistics," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 714-722, March.
    4. Albers, C.J. & Critchley, F. & Gower, J.C., 2011. "Quadratic minimisation problems in statistics," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 698-713, March.
    Full references (including those not matched with items on IDEAS)

    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. Stefan Jakubek & Elisabeth Luchini & Alexander Oberhummer & Felix Pfister, 2016. "A model-based interfacing concept for accurate power hardware-in-the-loop systems," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 22(1), pages 1-20, January.
    2. Albers, C.J. & Critchley, F. & Gower, J.C., 2011. "Applications of quadratic minimisation problems in statistics," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 714-722, March.
    3. S. Robin & M. Lecomte & H. Hofte & G. Mouille, 2003. "A procedure for the clustering of cell wall mutants in the model plant Arabidopsis based on Fourier-transform infrared (FT-IR) spectroscopy," Journal of Applied Statistics, Taylor & Francis Journals, vol. 30(6), pages 669-681.
    4. W. J. Krzanowski, 1999. "Antedependence models in the analysis of multi-group high-dimensional data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 26(1), pages 59-67.
    5. Brendan P. W. Ames & Mingyi Hong, 2016. "Alternating direction method of multipliers for penalized zero-variance discriminant analysis," Computational Optimization and Applications, Springer, vol. 64(3), pages 725-754, July.
    6. Duintjer Tebbens, Jurjen & Schlesinger, Pavel, 2007. "Improving implementation of linear discriminant analysis for the high dimension/small sample size problem," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 423-437, September.
    7. John Gower & Casper Albers, 2011. "Between-Group Metrics," Journal of Classification, Springer;The Classification Society, vol. 28(3), pages 315-326, October.
    8. Bouveyron, C. & Girard, S. & Schmid, C., 2007. "High-dimensional data clustering," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 502-519, September.
    9. Nickolay T. Trendafilov & Tsegay Gebrehiwot Gebru, 2016. "Recipes for sparse LDA of horizontal data," METRON, Springer;Sapienza Università di Roma, vol. 74(2), pages 207-221, August.
    10. Lei-Hong Zhang & Li-Zhi Liao & Michael K. Ng, 2013. "Superlinear Convergence of a General Algorithm for the Generalized Foley–Sammon Discriminant Analysis," Journal of Optimization Theory and Applications, Springer, vol. 157(3), pages 853-865, June.
    11. Mkhadri, A. & Celeux, G. & Nasroallah, A., 1997. "Regularization in discriminant analysis: an overview," Computational Statistics & Data Analysis, Elsevier, vol. 23(3), pages 403-423, January.
    12. Chin-Yi Chen & Jih-Jeng Huang, 2023. "Integrating Dynamic Bayesian Networks and Analytic Hierarchy Process for Time-Dependent Multi-Criteria Decision-Making," Mathematics, MDPI, vol. 11(10), pages 1-12, May.

    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:spr:jclass:v:31:y:2014:i:1:p:2-27. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.