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One of the basic problems in data analysis lies in choosing the optimalrescaling (change of coordinate system) to study properties of a given data-set$Y$. The classical Mahalanobis approach has its basis in the classicalnormalization/rescaling formula $Y \ni y \to \Sigma_Y^{-1/2} \cdot(y-\mathrm{m}_Y)$, where $\mathrm{m}_Y$ denotes the mean of $Y$ and $\Sigma_Y$the covariance matrix .  Based on the cross-entropy we generalize this approach and define theparameter which measures the fit of a given affine rescaling of $Y$ compared tothe Mahalanobis one. This allows in particular to find an optimal change ofcoordinate system which satisfies some additional conditions. In particular weshow that in the case when we put origin of coordinate system in $ \mathrm{m} $the optimal choice is given by the transformation $Y \ni y \to \Sigma_Y^{-1/2}\cdot (y-\mathrm{m}_Y)$, where $$\Sigma=\Sigma_Y(\Sigma_Y-\frac{(\mathrm{m}-\mathrm{m}_Y)(\mathrm{m}-\mathrm{m}_Y)^T}{1+\|\mathrm{m}-\mathrm{m}_Y\|_{\Sigma_Y}^2})^{-1}\Sigma_Y.$$

Optimal Rescaling and the Mahalanobis Distance