mahalanobis distance

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

\Sigma

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

\Sigma

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

\sigma_i

is the standard deviation of the

x_i

over the sample set.

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