Editing Generalized canonical correlation

{{Short description|Generalized CCA}}

In [[statistics]], the '''generalized [[canonical correlation]] analysis''' (gCCA), is a way of making sense of [[cross-correlation]] matrices between the sets of random variables when there are more than two sets. While a conventional CCA generalizes [[principal component analysis]] (PCA) to two sets of random variables, a gCCA  generalizes PCA to more than two sets of random variables. The '''canonical variables''' represent those '''common factors''' that can be found by a large PCA of all of the transformed random variables after each set underwent its own PCA.

== Applications ==
The [[Helmert-Wolf blocking]] (HWB) method of estimating [[linear regression]] parameters can find an optimal solution only if all cross-correlations between the data blocks are zero. They can always be made to vanish by introducing a new regression parameter for each common factor. The gCCA method can be used for finding those harmful common factors that create cross-correlation between the blocks. However, no optimal HWB solution exists if the random variables do not contain enough information on all of the new regression parameters.

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==References==
* Afshin-Pour, B.; Hossein-Zadeh, G.A. Strother, S.C.; Soltanian-Zadeh, H. (2012), [http://www.sciencedirect.com/science/article/pii/S1053811912001644 "Enhancing reproducibility of fMRI statistical maps using generalized canonical correlation analysis in NPAIRS framework"], NeuroImage 60(4): 1970–1981. {{doi|10.1016/j.neuroimage.2012.01.137}}
*Sun, Q.S., Liu, Z.D., Heng, P.A., Xia, D.S. (2005) "A Theorem on the Generalized Canonical Projective Vectors". ''Pattern Recognition'' 38 (3) 449
*Kettenring, J. R. (1971) "Canonical analysis of several sets of variables". "Biometrika" 58 (3) 433

==External links==
*[http://factominer.free.fr/ FactoMineR] (free exploratory multivariate data analysis software linked to [[R programming language|R]])

[[Category:Covariance and correlation]]
[[Category:Dimension reduction]]