Link between CCA and PCA

Question

I have two datasets, $X$ and $Y$. I calculate the PCA components of $X$ and also perform CCA on $X$ and $Y$. If I create a model with all the PCA components of $X$, and another model with all the CCA components of $X$, are these models identical?

Answer 1

CCA of X on X is the same as PCA of X. PCA can be thought of rotating your data such that the axis in the new rotation (i.e., PC components) are uncorrelated, and order by the variance. CCA finds rotations of X and rotations of Y so that the components you get from X and Y are correlated to each other. So if you run PCA first, and keep all the components, i.e., rotate your data, CCA will just rotate the data back in the solution that makes the components correlate with each other the most. So you will get the same results, only now you have weights on PCA components and not on the original variables, but the canonical correlations are the same, and you can rotate these PCA weights to get the weights on the original variables using the PCA solution anyway (I don't know the formula), so yes you will get basically identical results.