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Maybe I'm having a brainfart or something... but since PCA is eigenvector/eigenspace-based and essentially determines linear noncorrellation of the different ve
by binarysolo 13y ago
Maybe I'm having a brainfart or something... but since PCA is eigenvector/eigenspace-based and essentially determines linear noncorrellation of the different vectors, changing units of measurement shouldn't change which dimensions are most different about said vectors?
Edit: Ah right - http://en.wikipedia.org/wiki/Whitening_transformation http://en.wikipedia.org/wiki/Whitening_transformation - let covariance matrix be I.
- thedufer 13y agoThat's what I thought when I read this, but I haven't looked at PCA in awhile, so I wasn't sure. It's only relative differences on each axis that matter, right?
- robrenaud 13y agoNo. PCA tries to project to the subspace that preserves as much distance in the input space possible. If you multiply a coordinate in the input space by a factor of 2, it will contribute relatively more to the distances, and hence change the fitted projection beyond just a scaling factor.
- binarysolo 13y agoThanks for the refresher, and a followup basic question: what methods are best at just ranking relative variance of each dimension/component regardless of unit scaling? SVDs, ? Edit: Ah right - http://en.wikipedia.org/wiki/Whitening_transformation http://en.wikipedia.org/wiki/Whitening_transformation
- thisrod 13y agoSo we're talking about a very general problem. When I noticed that a system of equations including x=y can have a different least-squares solution when you change it to 2x=2y, I was quite surprised.