Mahalanobis distance

The Mahalanobis distance is a measure of the distance between a point P and a distribution D, introduced by P. C. Mahalanobis in 1936. It is a multi-dimensional generalization of the idea of measuring how many standard deviations away P is from the mean of D. This distance is zero if P is at the mean of D, and grows as P moves away from the mean: along each principal component axis, it measures the number of standard deviations from P to the mean of D. If each of these axes is rescaled to have unit variance, then Mahalanobis distance corresponds to standard Euclidean distance in the transformed space. Mahalanobis distance is thus unitless and scale-invariant, and takes into account the correlations of the data set.

The Mahalanobis distance of an observation ${\displaystyle {\vec {x}}=(x_{1},x_{2},x_{3},\dots ,x_{N})^{T}}$ from a set of observations with mean ${\displaystyle {\vec {\mu }}=(\mu _{1},\mu _{2},\mu _{3},\dots ,\mu _{N})^{T}}$ and covariance matrix S is defined as:

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