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I'm struggling on getting a good explanation for the unexpected signs of Principal Component for months. I tried to replicate a result and I got exactly the opposite signs for all components. While I searched through forums for an answer, I found this website I'm getting "jumpy" loadings in rollapply PCA in R. Can I fix it?
From what I have read from this website, I understand that we can reverse the sign of components based on other criteria - e.g. EURO trend - as what have been mentioned by @amoeba. I'm wondering if there is a book or academic literature that says that we can flip the sign in this way (i.e. based on external factor)? I need a strong support for my research paper if I flip the signs of all components in this way. Hence, I would greatly appreciate if someone can recommend me some books that talk about this issue?
And also @amoeba mentioned that the signs are consistent in sliding PCA. Does it mean that we should have the same combination for each window (for example +a, +b, -c in first window & -a, -b, +c in second window)? So, if I think the signs in second window are correct, then I will flip the 1st window's vector and both vectors will have the same sign by then. What if they have different combination (e.g. +a, -b, -c in second window)? I think their correlation could have changed from time to time and hence we will have different combinations in different windows?