Talk:Principal component analysis
We probably need a small article on the arg max and arg min notations.
The article seems to be missing crucial details. I can't see where the actual dimension reduction is happening. Is the idea that you have several samples of the measurement vector x and you use these to estimate the expectations? 130.188.8.9 16:49, 20 Aug 2003 (UTC)
- There should now be a clue. However, the article still needs work
Principle components analysis is better known as Principle component analysis (singular). This should be the main title and the plural form a synonym referring to this page (Unfortunately I do not know how to do it).
- I've always heard it with the plural. I have a PhD in statistics. I'm not saying the singular could never be used, but the plural is certainly the one that's frequently heard. Michael Hardy 21:18, 22 Mar 2004 (UTC)
- The only monography solely dedicated to PCA is from Jolliffe to my knowledge and is titled "Principal component analysis". The naming issue is discussed in the introduction otherwise than you indicate. Then again naming issues are conventions and vary across the globe. Sboehringer
- Google says: "Principal component analysis": 103,000 hits, "Principal components analysis": 46,300 hits. MH 13:48, 25 Mar 2004 (UTC)
- I have that monograph and you are correct. It seems, however, that the analysis elucidates the principal components, plural, and so unless one is only interested in one principal component at a time, the plural appears to be more appropriate.
Moving Michael Hardy's comments to Talk:
This article needs some serious revamping, to say the least. One cannot assume without loss of generality that the expectation is zero. If the expectation were observable, one could subtract it from x and get something with zero expectation, and so no generality would be lost by this assumption. In practice the expecation is never observable, and one must consider the probability distribution of the difference between x and an estimate, based on data, of the expectation of x.
Excuse me, but that is absurd. If the mean were observable, then one could simply subtract the mean from X, getting something with zero mean, and then indeed no generality would be lost by assuming that. In practice, one must use a data-based and therefore uncertain estimate of the mean, and one must therefore consider the probability distribution of the difference between X and the estimate of the mean of X.
- If I may respond --- PCA is a technique that is applied to empirical data sets. PCA eigendecomposes the maximum likelihood covariance matrix. Indeed, there is a distribution of PCA decompositions about the "true" decomposition that you would get in the infinite data limit. But, that does not make it absurd. Or rather, no more absurd than any other maximum likelihood estimate. Any ML technique will have a variance around the estimate from infinite data.
- Are you objecting because ML is not mentioned in the article? Or is it something else? -- hike395 04:39, 5 May 2004 (UTC)
- Something else. Several something elses. It doesn't seem like that good an article. I'll probably drastically edit it within a few months; it's on my list. Michael Hardy 16:31, 5 May 2004 (UTC)
PCR and PLS?
would it be redundant to include some discussion of principal components regression? i don't think so, but i don't feel qualified to explain it.
It would also be nice to have a piece on Partial Least Squares. Geladi and Kowalski Analytica Chimica Acta 185 (1986) 1-17 may serve as a starting point.
- I disagree --- PLS and PCR are both forms of linear regression, which is supervised learning. PCA is density estimation, which is unsupervised learning. Very different sorts of algorithms --- hike395 04:35, 22 Mar 2005 (UTC)
PCA & Least Squares
Is PCA the same as a least squares fit? (Furthermore, is either the same as finding the principle moment of inertia of an n-dimensional body?) —BenFrantzDale 23:53, August 3, 2005 (UTC)
- No. A least-squares fit minimizes (the squares of) the residuals, the vertical distances from the fit line (hyperplane) to the data. PCA minimizes the orthogonal projections to the hyperplane. (Or something like that; I don't really know what I'm talking about.) As for moments of inertia, well, physics isn't exactly my area of expertise. —Caesura(t) 18:44, 14 December 2005 (UTC)
- PCA is equivalent to finding the principal axes of inertia for N point masses in m dimensions, and then throwing all but l of the new transformed co-ordinates away. It's also mathematically the same problem as Total Least Squares (errors in all variables), rather than Ordinary Least Squares (errors only in y, not x), if you can scale it so the errors in all the variables are uncorrelated and the same size. You're then finding the best l dimensional hyperplane through the m dimensional space that your data ought to sit on. The real power tool behind all of this to get a feel for is Singular Value Decomposition. PCA is just SVD applied to your data. -- Jheald 19:40, 12 January 2006 (UTC).
Derivation of PCA
Shouldn't the constraint that we are looking for the maximum variance appear somewhere in that derivation ? I cannot understand it clearly as it is right now. --Raistlin 12:49, 24 August 2005 (UTC)
Conjugate transpose
- and * T represents the conjugate transpose operation.
Why conjugate transpose instead of a normal transpose ? Does it even work with complex numbers ? Taw 04:18, 31 December 2005 (UTC)
- As you probably know, conjugate transpose is a generalization of plain old transpose that allows these operations to work on complex numbers instead of just real numbers. If the source data X consists entirely of real numbers, then the conjugate operation is completely transparent, since the conjugate of a real number is the number itself. But if the source data includes complex numbers, then the conjugate operations is absolutely essential for the matrix operations to yield meaningful results. As far as I can tell, it does work on complex numbers. As an example where you might have complex numbers as source data, you might want to use PCA on the Fourier components of a real, discrete-time signal, which are in general complex. -- Metacomet 18:59, 1 January 2006 (UTC)
Computation
The section on computation looks to make a real meal of things, IMO; and to be pretty dubious too, as regards its numerical analysis. As soon as you square the data matrix, you're going to reduce the accuracy of your SVD from double precision to single precision.
Is there any reason to prefer either of the methods in the text, compared to choosing which bits of the SVD you actually want to keep, and then just wheeling out R-SVD ? (Which I imagine is quicker, too). -- Jheald 19:05, 12 January 2006 (UTC).