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 all have variance 1. The transformation is called "whitening" because it changes the input vector into a white noise vector.
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Several other transformations are closely related to whitening: 1) the decorrelation transform removes only the correlations but leaves variances intact, 2) the standardization transform sets variances to 1 but leaves correlations intact, and 3) a coloring transformation transforms a vector of white random variables into a random vector with a specified covariance matrix.
Definition
Suppose
There are infinitely many possible whitening matrices
Kessy et al. (2015) demonstrate that optimal whitening transforms can be singled out by investigating the cross-covariance and cross-correlation of
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).