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I am bit confused between two terms Canonical Correpondence Analysis and Canonical Correlation Analysis.

Are the two some how related or they are entirely different techniques?

Do they point to the same outcome or different outcomes (if they are applied to same datasets)?

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Canonical correspondence analysis is a technique developed, I believe, by the community ecology people. A founding paper is Canonical correspondence analysis: a new eigenvector technique for multivariate direct gradient analysis by Cajo J.F. Ter Braak (1986). The method involves a canonical correlation analysis and a direct gradient analysis. The idea is to relate the prevalences of a set of species to a collection of environmental variables.

Traditionally CCA (correlation) seeks to find that linear combination of the X variables and that linear combination of the Y variables that have the greatest correlation with each other. It relies on the eigen decomposition of $\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}$, where the Sigma matrices are correlation matrices of the variables. See Mardia, Kent and Bibby (Multivariate Analysis).

CCA thus assumes a linear relationship between the two sets of variables. The correspondence analysis assumes a different relationship: The species have a gaussian distribution along a direction determined by the environmental factors.

Note that CCA is symmetric in the X variables and the Y variables. Correspondence analysis presumes no symmetry, since we want to explain the species in terms of their environment - not the other way around.

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  • $\begingroup$ Shouldn't the eigen decomposition be: $\Sigma_{11}^{-1} \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21}$ ? $\endgroup$
    – Kiran K.
    Feb 19, 2016 at 0:06

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