# Mahalanobis distance

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In statistics, Mahalanobis distance is a distance measure introduced by P. C. Mahalanobis in 1936. It is based on correlations between variables by which different patterns can be identified and analysed. It is a useful way of determining similarity of an unknown sample set to a known one. It differs from Euclidean distance in that it takes into account the correlations of the data set.

Formally, the Mahalanobis distance from a group of values with mean $\mu = ( \mu_1, \mu_2, \mu_3, \dots , \mu_p )$

and covariance matrix $\Sigma$



for a multivariate vector $x = ( x_1, x_2, x_3, \dots, x_p )$

is defined as:

$D_M(x) = \sqrt{(x - \mu)^T \Sigma^{-1} (x-\mu)}.\,$

Mahalanobis distance can also be defined as dissimilarity measure between two random vectors $\vec{x}$

and $\vec{y}$
of the same distribution with the covariance matrix


$\Sigma$

:

$d(\vec{x},\vec{y})=\sqrt{(\vec{x}-\vec{y})^T\Sigma^{-1} (\vec{x}-\vec{y})}.\,$

If the covariance matrix is the identity matrix then it is the same as Euclidean distance. If covariance matrix is diagonal, then it is called normalized Euclidean distance:

$d(\vec{x},\vec{y})= \sqrt{\sum_{i=1}^p {(x_i - y_i)^2 \over \sigma_i^2}},$

where $\sigma_i$

is the standard deviation of the $x_i$
over the sample set.