Optimal whitening and decorrelation

Whitening, or sphering, is a common preprocessing step in statistical analysis to transform random variables to orthogonality. However, due to rotational freedom there are infinitely many possible whitening procedures. Consequently, there is a diverse range of sphering methods in use, for example based on principal component analysis, Cholesky matrix decomposition and Mahalanobis transformation, among others. Here we provide an overview of the underlying theory and discuss five natural whitening procedures. Subsequently, we demonstrate that investigating the cross-covariance and the cross-correlation matrix between sphered and original variables allows to break the rotational invariance of whitening and to identify optimal transformations. As a result we recommended two particular whitening approaches: CAT-CAR whitening to produce sphered variables that are maximally similar to the original variables, and PCA-whitening based on the correlation matrix to obtain maximally compressed whitened variables.