Mahalanobis distance

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 analyzed. 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 and is scale-invariant. In other words, it is a multivariate effect size.

Definition

Formally, the Mahalanobis distance of a multivariate vector <math>x = ( x_1, x_2, x_3, dots, x_N )^T</math> from a group of values with mean <math>mu = ( mu_1, mu_2, mu_3, dots , mu_N )^T</math> and covariance matrix <math>S</math> is defined as:

<math>D_M(x) = sqrt., </math>

Mahalanobis distance (or "generalized squared interpoint distance" for its squared value) can also be defined as a dissimilarity measure between two random vectors <math> vec</math> and <math> vec</math> of the same distribution with the covariance matrix<math>S</math> :

<math>......

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