Timeline for Condition number of covariance matrix
Current License: CC BY-SA 3.0
8 events
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| May 8, 2015 at 6:15 | comment | added | usεr11852 | I still believe that the really really unlikely is a bit strong but OK, that's a minor thing. For me the post as it stands it is useful. (+1) | |
| May 8, 2015 at 5:57 | history | edited | Sid | CC BY-SA 3.0 |
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| May 8, 2015 at 5:41 | comment | added | Sid | Thanks for the help. I realize I had linked the wrong paper. The truth is the result holds 'in general'. It is unbelievably unlikely you have a random $N*N$ matrix with poor condition number (See theorem P2 of the new Tao and Vu link), let alone symmetric positive definite ones. Hence moral of the story: if I do see an ill-conditioned matrix, I generally mistrust something about its computation | |
| May 8, 2015 at 5:38 | history | edited | Sid | CC BY-SA 3.0 |
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| May 8, 2015 at 5:36 | comment | added | usεr11852 | I appreciate the time replying. Having said that you write: "In general" and not Given you have a finite condition number to your correlation matrix. To that extend the OP specifically comments about "small standard deviations as well as large standard deviations", ie. eigenvalues of different magnitudes which renders the findings of the Tao and Vu paper inapplicable (directly at least). I will have not problem retracting my vote (and possibly upvoting) if you elaborate on how Marcenko-Pastur would solve the OP's issue in your post. | |
| May 8, 2015 at 5:16 | comment | added | Sid | I have also seen covariance matrices with small condition numbers. That is irrelevant considering the reader has seen a well-conditioned (which is what I think he means by finite) correlation matrix. My note was conditioning on his observation. Marcenko-Pastur gives a very general framework to regulate condition numbers of covariance matrices (though the result is asymptotic, it has been shown to work extremely well for large samples. Given a condition number one desires and a correlation matrix, back-tracking becomes a lot easier | |
| May 8, 2015 at 4:51 | comment | added | usεr11852 | I downvoted because I think this answer is too short for it's own good and gives a false sense of assurance. I have seen numerous covariance matrices being ill-conditioned; especially when real data are involved it is far from uncommon to have 0 eigenvalues due to finite precision arithmetic in large covariance matrices. Additionally the paper linked talks about matrices whose entries are that IID RV, $N(0,1)$; the OP clearly states he is interested on $\Gamma$ distributions; generalizing from the paper's results is far from obvious. | |
| May 8, 2015 at 3:42 | history | answered | Sid | CC BY-SA 3.0 |