I'm trying to implement Principal Component Analysis (PCA) in a portfolio replication procedure. (The replication procedure looks like regression: there is a vector representing payoffs of an asset under different ecocnomic scenarios, I need to find a linear combination of vectors of another assets that fits the most closely to an initial asset).
Coefficients I get from regression should tell me how many assets I need to buy and sell to get approximately the same payoff as the asset I'm trying to replicate.
There is a problem with regression as the candidate asset matrix is ill-conditioned: assets show high correlation. There's a hope that orthogonal principal components could resolve this problem.
I need to do a PCA decomposition of candidate assets matrix, take only first $n$ components, do optimization and get components coefficients. Then I need to transform the coefficients back into an original basis.
Now the problem: PCA usually works with mean-centered data, but if I subtract means from the original data, I don't know how to interpret resulting coefficients in my case and don't know how to reverse the operation.
So far I'm doing eigen-decomposition of a covariance matrix, then using eigenvectors to make an orthogonal transformation of the data that is not mean-centered. Then I'm running a regression (actually L1 norm optimization) to get coefficients and transform them back into an original basis. The results are not bad, but I can't stop thinking about the problem with mean-centering, if I'm doing it completely wrong.
I was hoping to find a detailed math reasoning for this problem, but unfortunately failed. I'm very much a noob in this and my math skills are far from being good, so I would really appreciate your help if you can share some insights on the problem of mean-centering in the PCA.