Awhitening transformation orsphering transformation is alinear transformation that transforms a vector ofrandom variables with a knowncovariance matrix into a set of new variables whose covariance is theidentity matrix, meaning that they areuncorrelated and each havevariance 1.^[1] The transformation is called "whitening" because it changes the input vector into awhite noise vector.

Several other transformations are closely related to whitening:

1. thedecorrelation transform removes only the correlations but leaves variances intact,
2. thestandardization transform sets variances to 1 but leaves correlations intact,
3. acoloring transformation transforms a vector of white random variables into a random vector with a specified covariance matrix.^[2]

Suppose${\displaystyle X}$ is arandom (column) vector with non-singular covariance matrix${\displaystyle \mathrm{\Sigma }}$ and mean${\displaystyle 0}$. Then the transformation${\displaystyle Y=WX}$ with awhitening matrix${\displaystyle W}$ satisfying the condition${\displaystyle W^{\mathrm{T}}W={\mathrm{\Sigma }}^{-1}}$ yields the whitened random vector${\displaystyle Y}$ with unit diagonal covariance.

If${\displaystyle X}$ has non-zero mean${\displaystyle \mu }$, then whitening can be performed by${\displaystyle Y=W(X-\mu )}$.

There are infinitely many possible whitening matrices${\displaystyle W}$ that all satisfy the above condition. Commonly used choices are${\displaystyle W={\mathrm{\Sigma }}^{-1/2}}$ (Mahalanobis or ZCA whitening),${\displaystyle W=L^T}$ where${\displaystyle L}$ is theCholesky decomposition of${\displaystyle {\mathrm{\Sigma }}^{-1}}$ (Cholesky whitening),^[3] or the eigen-system of${\displaystyle \mathrm{\Sigma }}$ (PCA whitening).^[4]

Optimal whitening transforms can be singled out by investigating the cross-covariance and cross-correlation of${\displaystyle X}$ and${\displaystyle Y}$.^[3] For example, the unique optimal whitening transformation achieving maximal component-wise correlation between original${\displaystyle X}$ and whitened${\displaystyle Y}$ is produced by the whitening matrix${\displaystyle W=P^{-1/2}{V}^{-1/2}}$ where${\displaystyle P}$ is the correlation matrix and${\displaystyle V}$ the diagonal variance matrix.

Whitening a data matrix follows the same transformation as for random variables. An empirical whitening transform is obtained byestimating the covariance (e.g. bymaximum likelihood) and subsequently constructing a corresponding estimated whitening matrix (e.g. byCholesky decomposition).

This modality is a generalization of the pre-whitening procedure extended to more general spaces where${\displaystyle X}$ is usually assumed to be a random function or other random objects in aHilbert space${\displaystyle H}$. One of the main issues of extending whitening to infinite dimensions is that thecovariance operator has an unbounded inverse in${\displaystyle H}$, therefore only partial standardization is possible in infinite dimensions. A whitening operator can be then defined from the factorization of a degenerated covariance operator. High-dimensional features of the data can be exploited through kernel regressors or basis function systems.^[5]

An implementation of several whitening procedures inR, including ZCA-whitening and PCA whitening but alsoCCA whitening, is available in the "whitening" R package^[6] published onCRAN. The R package "pfica"^[7] allows the computation of high-dimensional whitening representations using basis function systems (B-splines,Fourier basis, etc.).

• Decorrelation
• Principal component analysis
• Weighted least squares
• Canonical correlation
• Mahalanobis distance (is Euclidean after W. transformation).

• http://courses.media.mit.edu/2010fall/mas622j/whiten.pdf
• The ZCA whitening transformation. Appendix A ofLearning Multiple Layers of Features from Tiny Images by A. Krizhevsky.

• Classification algorithms

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