# Generalized canonical correlation

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), "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

- FactoMineR (free exploratory multivariate data analysis software linked to R)

Original source: https://en.wikipedia.org/wiki/Generalized canonical correlation.
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