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Journal of Classification

, Volume 31, Issue 1, pp 2–27 | Cite as

Canonical Analysis: Ranks, Ratios and Fits

  • Casper J. AlbersEmail author
  • John C. Gower
Article

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.

Keywords

Canonical analysis Ratio form Fit form Mahalonobis distance Discriminant analysis 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.University of GroningenGroningenThe Netherlands
  2. 2.The Open UniversityBuckinghamshireUK

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