Canonical Analysis: Ranks, Ratios and Fits

Casper Albers () and John Gower

Journal of Classification, 2014, vol. 31, issue 1, 2-27

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

Keywords: Canonical analysis; Ratio form; Fit form; Mahalonobis distance; Discriminant analysis (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s00357-014-9146-y

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