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Canonical Analysis: Ranks, Ratios and Fits

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  • 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
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    References listed on IDEAS

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    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. "Quadratic minimisation problems in statistics," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 698-713, March.
    4. 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.
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