Talk:Whitening transformation

Page Creation

This is definitely worth of its own page. The White Noise page has information about how to whiten a random variable, but this should be explicitly stated. daviddoria (talk) 11:45, 12 September 2008 (UTC)

Images

There are very standard images describing the whitening transform (ellipsoidal distributions being rotated and made spherical). Can someone please post them? daviddoria (talk) 19:39, 11 September 2008 (UTC)

Linking from Whitening Transform

I put a redirect to here on Whitening Transform, but it doesn't work, when you click "goto Whitening Transformation", it goes to the edit page instead of the article itself. Can someone tell me how to fix that? daviddoria (talk) 11:45, 12 September 2008 (UTC)

Variances

The whitened variables definately not have the same variances as the original random variables, but have unit variances, which is explicitly implied by "whitening transformation is a decorrelation method that converts the covariance matrix S of a set of samples into the identity matrix I". There are simply variable variances on the diagonal of covariance matrix. —Preceding unsigned comment added by 178.36.138.217 (talk) 14:37, 26 September 2010 (UTC)

Yes. I've changed the article. StevenBell (talk) 23:09, 3 February 2012 (UTC)

Page updates

I've attempted to add derivations/proofs for whitening. They're not very good, but hopefully someone with a better mathematical foundation can improve them. The formatting should also be improved. When I have time, I will create some figures demonstrating the whitening/coloring concept. StevenBell (talk) 23:06, 3 February 2012 (UTC)

I think that in the steps from Cov(Y) = I there is a mistake in the transposing of the product of Sigma to the minus half and X to the product of the transpose of X and sigma to the minus half. In my view it should be the product of the transpose of X and the transpose of Sigma to the minus half. Do you agree? I would also add an extra line expanding Sigma to the product of Sigma to the half and the transpose of Sigma to the half. From that it should be clear that the product of Sigma to the minus half and Sigma to the half "cancel" to I and similarly the product of the transpose of Sigma to the half and the transpose of Sigma to the minus half also "cancel" to I to generate the product of I and I which of course equal I. If you agree I will attempt to make the changes. Thanks AgentStylites — Preceding unsigned comment added by Agentstylites (talk • contribs) 11:01, 25 June 2012 (UTC)

Unfortunately, proofs generally do not belong in Wikipedia; it is supposed to be an encyclopedia, not a textbook. At most one should give a hint ("this result can be derived from ..."), and provide a reference if the proof is not trivial. The proof of the whitening transformation would perhaps be of use in some Wikibook on statistics; if so, it can be recovered from the article's History.

Is the notation of $\Sigma ^{1/2}$ somewhat confusing since it looks like the matrix square root?

Usually $A^{1/2}$ is defined such that

$A^{1/2}A^{1/2}=A$

But here it is

$\Sigma ^{1/2}(\Sigma ^{1/2})^{T}=\Sigma$

I don't know if this is the standard notation, but it confused me on first read. If I understand it correctly, the square root of $\Sigma$ should be $\Phi \Lambda ^{1/2}\Phi ^{T}$. 112.120.54.104 (talk) 09:07, 15 September 2012 (UTC)

https://en.wikipedia.org/wiki/Talk:Square_root_of_a_matrix#Cholesky_vs_square_root contains a (somewhat meandering and unclear) discussion of the issue of what is meant by square root of a matrix—it has sometimes been used to denote part of the Cholesky decomposition. However, the Cholesky usage seems now to be vanishingly rare and should be removed from Wikipedia pages. In particular, it is not in accord with current conventions to use the transpose, and the previous contributor to this talk subsection is correct that a matrix $B$ is a square root of $A$ if $B^{2}=A$. For any real number $\lambda$ and symmetric positive-definite matrix $A$, $A^{\lambda }$ is a well-defined symmetric, positive-definite matrix. And there is only one such choice satisfying the natural arithmetical laws. Remember that the square root of a positive real number is ambiguous, unless one insists on a positive answer. David.B.A.Epstein (talk) 14:44, 1 July 2015 (UTC)

Original research?

I see nothing wrong with the contents of the article, but is this topic really known out there, with this name? Or is it "original research"? (Granted that there may not be such thing as "original research" in mathematics, but the name and popularity of the concept still matter.) One--Jorge Stolfi (talk) 13:34, 27 February 2013 (UTC) reference to a respectable print publication would settle the issue. --Jorge Stolfi (talk) 13:34, 27 February 2013 (UTC)

PS. indeed the old "Limitation" section seems to have been added as an afterthought when it was realized that the tranformation does not exist if the correlation matrix is singular. It also fails to point out that the transformation is not unique. I have tried to fix these problems, but the question about original research still stands. --Jorge Stolfi (talk) 14:19, 27 February 2013 (UTC)

The transformation is unique if one requires it to be symmetric positive-definite. However, I can't see any obvious reason to make this requirement. And such a transformation is available if and only if the covariance matrix is non-singular. (Theoretically, the correlation matrix may not exist, because its definition requires division by quantities that might theoretically be zero, though in practice they never are zero.) The operation of whitening is widely used and is discussed in diverse places under this name (Google gives many hits), so it's good that Wikipedia mentions it. David.B.A.Epstein (talk) 15:04, 1 July 2015 (UTC)

Potential Error in Cholesky decomposition expression

Where the article suggests the Cholesky decomposition of $\Sigma ^{-1}$, I believe this is wrong. It is actually the inverse of the Cholesky decomposition of $\Sigma$. I'm not confident enough in the subject to make the change, but I was misled by the article as-is. 141.168.75.76 (talk) 01:44, 24 June 2020 (UTC)

I believe you can get a whitening matrix from either one. $W^{T}W=\Sigma ^{-1}=(L_{1}L_{1}^{T})^{-1}=(L_{1}^{T})^{-1}L_{1}^{-1}\implies W=L_{1}^{-1}$, where $L_{1}$ is the cholesky decomp of $\Sigma$, or $W^{T}W=\Sigma ^{-1}=L_{2}L_{2}^{T}\implies W=L_{2}^{T}$ were $L_{2}$ is the cholesky decomp of $\Sigma ^{-1}$. Either way, it isn't directly the chol. decomp. You need to either invert or transpose it to get the whitening matrix. Mysticdan (talk) 14:29, 18 August 2021 (UTC)