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Mahalanobis distance

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In statistics, Mahalanobis distance is a distance measure invented 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 $Σ$ for a multivariate vector $x = ( x_1, x_2, x_3, \dots, x_p )$ is defined as:

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

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 $Σ$ :

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

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 $σ i$ is the standard deviation of the $x i$ over the sample set.

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