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The Canonical Analysis of Distance

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  • John Gower
  • Niel Roux
  • Sugnet Gardner-Lubbe

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. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • John Gower & Niel Roux & Sugnet Gardner-Lubbe, 2014. "The Canonical Analysis of Distance," Journal of Classification, Springer;The Classification Society, vol. 31(1), pages 107-128, April.
    • Handle: RePEc:spr:jclass:v:31:y:2014:i:1:p:107-128
    DOI: 10.1007/s00357-014-9149-8
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    References listed on IDEAS

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    1. Gower, John C. & Ngouenet, Roger F., 2005. "Nonlinearity effects in multidimensional scaling," Journal of Multivariate Analysis, Elsevier, vol. 94(2), pages 344-365, June.
    2. W. Krzanowski, 1994. "Ordination in the presence of group structure, for general multivariate data," Journal of Classification, Springer;The Classification Society, vol. 11(2), pages 195-207, September.
    3. J. Gower & P. Legendre, 1986. "Metric and Euclidean properties of dissimilarity coefficients," Journal of Classification, Springer;The Classification Society, vol. 3(1), pages 5-48, March.
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    1. John C. Gower & Niël J. Le Roux & Sugnet Gardner-Lubbe, 2022. "Properties of individual differences scaling and its interpretation," Statistical Papers, Springer, vol. 63(4), pages 1221-1245, August.
    2. Le Roux, Niël J. & Gardner-Lubbe, Sugnet & Gower, John C., 2014. "The analysis of distance of grouped data with categorical variables: Categorical canonical variate analysis," Journal of Multivariate Analysis, Elsevier, vol. 132(C), pages 9-24.

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