I am exploring de-noising and cleansing of covariance matrices via Random Matrix Theory. RMT is a competitor to shrinkage methods of covariance estimation. There are various methods expressed usually by the names of the authors: LPCB, PG+, and so on.
For each method, one can start by filtering the covariance matrix directly, or filter the correlation matrix and then covert the cleansed correlation matrix into a covariance matrix. My question involves the latter case.
I have noticed that when cleaning a correlation matrix that the resulting diagonal is not a diagonal of 1s (as one would expect to see in a correlation matrix).
My question -- when constructing a cleansed covariance matrix by first filtering its corresponding correlation matrix, does one:
"Fix" the diagonals of the intermediate cleansed correlation matrix to a diagonal of 1s before finally converting it back to a covariance matrix? This seems to be the case with the PG+ method, but not the LPCB method.
Or does one convert the cleansed correlation matrix to a covariance matrix and then fix the diagonal of the resulting covariance matrix to the diagonal of the original covariance matrix?
In both cases the trace is preserved.