Mahalanobis Distance - Definition

Definition

Formally, the Mahalanobis distance of a multivariate vector from a group of values with mean and covariance matrix is defined as:

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

d(\vec{x},\vec{y})=\sqrt{(\vec{x}-\vec{y})^T S^{-1} (\vec{x}-\vec{y})}.\,

If the covariance matrix is the identity matrix, the Mahalanobis distance reduces to the Euclidean distance. If the covariance matrix is diagonal, then the resulting distance measure is called the normalized Euclidean distance:

d(\vec{x},\vec{y})= \sqrt{\sum_{i=1}^N {(x_i - y_i)^2 \over s_{i}^2}},

where is the standard deviation of the and over the sample set.

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