Editing Standard deviation (section)

=== Properties ===

* The eigenvectors and eigenvalues of <math>\mathbf{S}</math> correspond to the axes of the 1 sd error ellipsoid of the multivariate normal distribution. See ''[[Multivariate normal distribution#Geometric interpretation|Multivariate normal distribution: geometric interpretation]]''.[[File:MultivariateNormal.png|thumb|The standard deviation ellipse (green) of a two-dimensional normal distribution]]
* The standard deviation of the ''projection'' of the multivariate distribution (i.e. the marginal distribution) on to a line in the direction of the unit vector <math>\hat{\boldsymbol{\eta}}</math> equals <math>\sqrt{\hat{\boldsymbol{\eta}}' \mathbf{\Sigma} \hat{\boldsymbol{\eta}}} = \lVert \mathbf{S} \hat{\boldsymbol{\eta}} \rVert</math>.<ref name="Das"/> 
* The standard deviation of a ''slice'' of the multivariate distribution (i.e. the conditional distribution) along the line in the direction of the unit vector <math>\hat{\boldsymbol{\eta}}</math> equals <math>\frac{1}{\lVert \mathbf{S}^{-1}\hat{\boldsymbol{\eta}} \rVert}</math>.<ref name="Das"/> 
* The [[Sensitivity index | discriminability index]] between two equal-covariance distributions is their [[Mahalanobis distance]], which can also be expressed in terms of the sd matrix: <math>d'=\sqrt{(\boldsymbol{\mu}_a-\boldsymbol{\mu}_b)'\boldsymbol{\Sigma}^{-1}(\boldsymbol{\mu}_a-\boldsymbol{\mu}_b)} = \lVert \mathbf{S}^{-1}\boldsymbol{d} \rVert</math>, where <math>\boldsymbol{d}=\boldsymbol{\mu}_a-\boldsymbol{\mu}_b</math> is the mean-difference vector.<ref name="Das"/>
* Since <math>\mathbf{S}</math> scales a normalized variable, it can be used to invert the transformation, and make it decorrelated and unit-variance: <math>\boldsymbol{z}=\mathbf{S}^{-1} (\boldsymbol{x}-\boldsymbol{\mu})</math> has zero mean and identity covariance. This is called the [[Whitening transformation|Mahalanobis whitening transform]].