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This distance measure was designed by P. C. Mahalanobis in 1936. It differs from Euclidean distance in that it takes into account the correlations between variables in the data set and is scale-invariant.
It is defined by the equation:
μ = (μ1,μ2,μ3,...,μp)T and covariance matrix Σ for a multivariate vector x = (x1,x2,x3,...,xp)T is defined as:
where x and μ are the two vectors of variables between which the distance is measured and S-1 is the inverse of the covariance matrix between the variables.
If the covariance matrix is the identity matrix, the Mahalanobis distance reduces to the Euclidean distance.