Editing Standard deviation (section)

== Standard deviation matrix ==
The standard deviation matrix <math>\mathbf{S}</math> is the extension of the standard deviation to multiple dimensions.  It is the symmetric square root of the covariance matrix <math>\mathbf{\Sigma}</math>.<ref name="Das">{{cite arXiv |eprint=2012.14331 |last1=Das |first1=Abhranil |author2=Wilson S Geisler |title=Methods to integrate multinormals and compute classification measures |date=2020 |class=stat.ML }}</ref>

<math>\mathbf{S}</math> linearly scales a random vector in multiple dimensions in the same way that <math>\sigma</math> does in one dimension. A scalar random variable <math>x</math> with variance <math>\sigma^2</math> can be written as <math>x=\sigma z</math>, where <math>z</math> has unit variance. In the same way, a random vector <math>\boldsymbol{x}</math> in several dimensions with covariance <math>\mathbf{\Sigma}</math> can be written as <math>\boldsymbol{x}=\mathbf{S}\boldsymbol{z}</math>, where <math>\boldsymbol{z}</math> is a normalized variable with identity covariance <math>\mathbf{1}</math>. This requires that <math>\mathbf{S}\mathbf{S'} = \mathbf{\Sigma}</math>. There are then infinite solutions for <math>\mathbf{S}</math>, and consequently there are multiple ways to whiten the distribution.<ref name="kessy">{{cite journal|last1=Kessy|first1=A.|last2=Lewin|first2=A.|last3=Strimmer|first3=K.|title=Optimal whitening and decorrelation|year=2018|journal=The American Statistician| volume=72|issue=4| pages=309–314|doi=10.1080/00031305.2016.1277159|arxiv=1512.00809|s2cid=55075085 }}</ref> The symmetric square root of <math>\mathbf{\Sigma}</math> is one of the solutions.

For example, a multivariate normal vector <math>\boldsymbol{x} \sim N(\boldsymbol{\mu}, \mathbf{\Sigma})</math> can be defined as <math>\boldsymbol{x}=\mathbf{S}\boldsymbol{z}+\boldsymbol{\mu}</math>, where <math>\boldsymbol{z} \sim N(\boldsymbol{0}, \mathbf{1})</math> is the multivariate standard normal.<ref name="Das"/>

=== 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]].