Mahalanobis Distance

Mahalanobis distance

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Description:
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 [itex]x = ( x_1, x_2, x_3, dots, x_N )^T[/itex] from a group of values with mean [itex]mu = ( mu_1, mu_2, mu_3, dots , mu_N )^T[/itex] and covariance matrix [itex]S[/itex] is defined as:

[itex]D_M(x) = sqrt., [/itex]

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

[itex]......
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