# Can CCA be used to replace expensive data with available data?

I have two sets of variables in the same dataset. Say DATA_free and DATA_exp. DATA_exp, however, consists of variables which are very expensive/difficult to obtain whereas DATA_free are always available easily.

I was wondering if it was possible to use Canonical Correlation Analysis between these two sets of variables so that I can find a linear combination of variables in DATA_free that best explains what DATA_exp does, in hopes that, if the correlation between the two linear combinations is high enough I could stop relying on DATA_exp to compute my results.

Is this reasonable? Any advice/example you could give me? Is this theoretically correct?

However: 1) The largest canonical correlation will rarely be 1 (and all the others will be smaller), and 2) If DATA_free is of dimension $p$ and DATA_exp is of dimension $q$ with $p < q$, the very best you can hope for is to replicate a $p$-dimensional projection of DATA_exp.