Editing Mahalanobis distance (section)

==Normal distributions==

For a [[multivariate normal distribution|normal distribution]] in any number of dimensions, the probability density of an observation <math>\vec{x}</math> is uniquely determined by the Mahalanobis distance <math>d</math>:
: <math>
\begin{align}
\Pr[\vec x] \,d\vec x & = \frac 1 {\sqrt{\det(2\pi \mathbf{\Sigma})}} \exp \left(-\frac{(\vec x - \vec \mu)^\mathsf{T} \mathbf{\Sigma}^{-1} (\vec x - \vec \mu)} 2 \right) \,d\vec{x} \\[6pt]
& = \frac{1}{\sqrt{\det(2\pi \mathbf{\Sigma})}} \exp\left( -\frac{d^2} 2 \right) \,d\vec x.
\end{align}
</math>
Specifically, <math>d^2</math> follows the [[chi-squared distribution]] with <math>n</math> degrees of freedom, where <math>n</math> is the number of dimensions of the normal distribution. If the number of dimensions is 2, for example, the probability of a particular calculated <math>d</math> being less than some threshold <math>t</math> is <math>1 - e^{-t^2/2}</math>. To determine a threshold to achieve a particular probability, <math>p</math>, use  <math display="inline">t = \sqrt{-2\ln(1 - p)}</math>, for 2 dimensions. For number of dimensions other than 2, the cumulative chi-squared distribution should be consulted.

In a normal distribution, the region where the Mahalanobis distance is less than one (i.e. the region inside the ellipsoid at distance one) is exactly the region where the probability distribution is [[concave function|concave]].

The Mahalanobis distance is proportional, for a normal distribution, to the square root of the negative [[log-likelihood]] (after adding a constant so the minimum is at zero).