Editing Canonical correlation (section)

==Whitening and probabilistic canonical correlation analysis==

CCA can also be viewed as a special [[whitening transformation]] where the random vectors <math>X</math> and <math>Y</math> are simultaneously transformed in such a way that the cross-correlation between the whitened vectors <math>X^{CCA}</math> and <math>Y^{CCA}</math> is diagonal.<ref>{{cite journal | last1 = Jendoubi | first1 = T. | last2 = Strimmer | first2 = K. | title = A whitening approach to probabilistic canonical correlation analysis for omics data integration | journal = BMC Bioinformatics | volume = 20 | issue = 1 | pages = 15 | year = 2018 | arxiv = 1802.03490 | doi = 10.1186/s12859-018-2572-9 | pmid = 30626338 | pmc = 6327589 | doi-access = free }}</ref>
The canonical correlations are then interpreted as regression coefficients linking <math>X^{CCA}</math> and <math>Y^{CCA}</math> and may also be negative.  The regression view of CCA also provides a way to construct a latent variable probabilistic generative model for CCA, with uncorrelated hidden variables representing shared and non-shared variability.<ref>{{cite journal |last1=Jendoubi |first1=Takoua |last2=Strimmer |first2=Korbinian |title=A whitening approach to probabilistic canonical correlation analysis for omics data integration |journal=BMC Bioinformatics |date=9 January 2019 |volume=20 |issue=1 |pages=15 |doi=10.1186/s12859-018-2572-9 |doi-access=free |pmid=30626338 |pmc=6327589 |language=en |issn=1471-2105}}</ref>