A whitening transformation or sphering transformation is a linear transformation that transforms a vector of random variables with a known covariance matrix into a set of new variables whose covariance is the identity matrix, meaning that they are uncorrelated and each have variance 1.^{[1]} The transformation is called "whitening" because it changes the input vector into a white noise vector.
Several other transformations are closely related to whitening:
 the decorrelation transform removes only the correlations but leaves variances intact,
 the standardization transform sets variances to 1 but leaves correlations intact,
 a coloring transformation transforms a vector of white random variables into a random vector with a specified covariance matrix.^{[2]}
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Contents
Definition
Suppose is a random (column) vector with nonsingular covariance matrix and mean . Then the transformation with a whitening matrix satisfying the condition yields the whitened random vector with unit diagonal covariance.
There are infinitely many possible whitening matrices that all satisfy the above condition. Commonly used choices are (Mahalanobis or ZCA whitening), the Cholesky decomposition of (Cholesky whitening), or the eigensystem of (PCA whitening).^{[3]}
Optimal whitening transforms can be singled out by investigating the crosscovariance and crosscorrelation of and .^{[4]} For example, the unique optimal whitening transformation achieving maximal componentwise correlation between original and whitened is produced by the whitening matrix where is the correlation matrix and the variance matrix.
Whitening a data matrix
Whitening a data matrix follows the same transformation as for random variables. An empirical whitening transform is obtained by estimating the covariance (e.g. by maximum likelihood) and subsequently constructing a corresponding estimated whitening matrix (e.g. by Cholesky decomposition).
R Implementation
An implementation of several whitening procedures in R, including ZCAwhitening and PCA whitening but also CCA whitening, is available in the "whitening" R package ^{[5]} published on CRAN.
See also
References
 ^ Koivunen, A.C.; Kostinski, A.B. (1999). "The Feasibility of Data Whitening to Improve Performance of Weather Radar". American Meteorological Society. doi:10.1175/15200450(1999)038<0741:TFODWT>2.0.CO;2.
 ^ Hossain, Miliha. "Whitening and Coloring Transforms for Multivariate Gaussian Random Variables". Project Rhea. Retrieved 21 March 2016.
 ^ Friedman, J. (1987). "Exploratory Projection Pursuit". ISSN 01621459. JSTOR 2289161.
 ^ Kessy, A.; Lewin, A.; Strimmer, K. (2018). "Optimal whitening and decorrelation". The American Statistician. 72: 309–314. arXiv:1512.00809. doi:10.1080/00031305.2016.1277159.
 ^ "whitening R package". Retrieved 20181125.
External links
 http://courses.media.mit.edu/2010fall/mas622j/whiten.pdf
 The ZCA whitening transformation. Appendix A of Learning Multiple Layers of Features from Tiny Images by A. Krizhevsky.