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Enriched biplots for canonical correlation analysis

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  • Jan Graffelman

Abstract

This paper discusses biplots of the between-set correlation matrix obtained by canonical correlation analysis. It is shown that these biplots can be enriched with the representation of the cases of the original data matrices. A representation of the cases that is optimal in the generalized least squares sense is obtained by the superposition of a scatterplot of the canonical variates on the biplot of the between-set correlation matrix. Goodness of fit statistics for all correlation and data matrices involved in canonical correlation analysis are discussed. It is shown that adequacy and redundancy coefficients are in fact statistics that express the goodness of fit of the original data matrices in the biplot. The within-set correlation matrix that is represented in standard coordinates always has a better goodness of fit than the within-set correlation matrix that is represented in principal coordinates. Given certain scalings, the scalar products between variable vectors approximate correlations better than the cosines of angles between variable vectors. Several data sets are used to illustrate the results.

Suggested Citation

  • Jan Graffelman, 2005. "Enriched biplots for canonical correlation analysis," Journal of Applied Statistics, Taylor & Francis Journals, vol. 32(2), pages 173-188.
    • Handle: RePEc:taf:japsta:v:32:y:2005:i:2:p:173-188
    DOI: 10.1080/02664760500054202
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    References listed on IDEAS

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    1. Cajo Braak, 1990. "Interpreting canonical correlation analysis through biplots of structure correlations and weights," Psychometrika, Springer;The Psychometric Society, vol. 55(3), pages 519-531, September.
    2. K. Ruben Gabriel, 2002. "Goodness of fit of biplots and correspondence analysis," Biometrika, Biometrika Trust, vol. 89(2), pages 423-436, June.
    3. Arnold Wollenberg, 1977. "Redundancy analysis an alternative for canonical correlation analysis," Psychometrika, Springer;The Psychometric Society, vol. 42(2), pages 207-219, June.
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