# What is the difference between whitening and PCA?

• Wenlong

#### Wenlong

Hi, all

I am looking into whitening transformation. According to the definition and explanation of Wikipedia, whitening transformation is a decorrelating process and it can be done by eigenvalue decomposition (EVD).

As far as I know, EVD is one of the solutions of principal component analysis (PCA). And the results of both whitening and PCA are uncorrelated(vectors, if the input are matrices). Thus I am being confused by these two methods.

May I say that whitening is equivalent to PCA? If not, may I know why?

Thank you very much for your kindly help.

Best regards
Wenlong

Hi,

Matrix (M,N)*(M,1)=(M,M) = whitening is the passage (M,N)-> (M,M)

The whitening transformation is a decorrelation method which transforms a set of random variables having the covariance matrix Σ into a set of new random variables whose covariance is aI, where a is a constant and I is the identity matrix. The new random variables are uncorrelated and all have variance 1.
http://en.wikipedia.org/wiki/Whitening_transformation

Standard PCA is often used for whitening because information can be optimally
compressed in the mean-square error sense and some possible noise is filtered out. The
PCA whitening matrix can be expressed in the form:

V=D^(-1/2)*E^T
where EDET = E{xxT }is the eigenvector decomposition of the covariance matrix of the
(zero mean) data x, implying that D = diag [d1 ,d2 ,...,dM] is a M*M diagonal matrix
containing the eigenvalues, and E = [c1 ,c2 ,...,cM] is an orthogonal N * M matrix
having the eigenvectors as columns.
http://rrp.infim.ro/2004_56_1/Mutihac.pdf