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The Canonical Analysis of Distance

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Abstract

Canonical Variate Analysis (CVA) is one of the most useful of multivariate methods. It is concerned with separating between and within group variation among N samples from K populations with respect to p measured variables. Mahalanobis distance between the K group means can be represented as points in a (K - 1) dimensional space and approximated in a smaller space, with the variables shown as calibrated biplot axes. Within group variation may also be shown, together with circular confidence regions and other convex prediction regions, which may be used to discriminate new samples. This type of representation extends to what we term Analysis of Distance (AoD), whenever a Euclidean inter-sample distance is defined. Although the N × N distance matrix of the samples, which may be large, is required, eigenvalue calculations are needed only for the much smaller K × K matrix of distances between group centroids. All the ancillary information that is attached to a CVA analysis is available in an AoD analysis. We outline the theory and the R programs we developed to implement AoD by presenting two examples.

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Authors and Affiliations

  1. The Open University, Buckinghamshire, UK

    John C. Gower

  2. Stellenbosch University, Stellenbosch, South Africa

    Niel J. le Roux

  3. University of Cape Town, Cape Town, South Africa

    Sugnet Gardner-Lubbe

Authors
  1. John C. Gower
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  2. Niel J. le Roux
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  3. Sugnet Gardner-Lubbe
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Corresponding author

Correspondence to John C. Gower.

Additional information

We would like to thank the anonymous reviewers and the editor for their helpful comments and suggestions. This work is based upon research supported by the National Research Foundation of South Africa. Any opinions, findings and conclusions, or recommendations expressed in this material are those of the authors and therefore the NRF does not accept any liability in regard thereof.

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Gower, J.C., le Roux, N.J. & Gardner-Lubbe, S. The Canonical Analysis of Distance. J Classif 31, 107–128 (2014). https://doi.org/10.1007/s00357-014-9149-8

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  • DOI: https://doi.org/10.1007/s00357-014-9149-8

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