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 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 σi is the standard deviation of the xi over the sample set.

See also: Mahalanobis distance, Correlation, Covariance matrix, Distance, Euclidean, Euclidean distance, Mathematics, P. C. Mahalanobis