2019-01-28[Stay Sharp]Mahalanobi

2019-01-27 本文已影响1人三千雨点

Mahalanobis distance is the distance between a point and a distribution, it's a measure of how many standard deviations away the point is from the mean the distribution.

$\Delta ^ { 2 } = ( \mathbf { x } - \boldsymbol { \mu } ) ^ { \mathrm { T } } \boldsymbol { \Sigma } ^ { - 1 } ( \mathbf { x } - \boldsymbol { \mu } )$
where $\Delta$ is the Mahalanobis distance of the point $\mathbf{x}$ from a distribution with mean $\mu$ , and $\boldsymbol \Sigma$ is the covariance matrix.

It also can be defined as a dissimilarity measure between two random vectors $\underline x$ and $\underline y$ of the same distribution with the covariance matrix $\Sigma$

$d ( \mathbf { x } , \mathbf { y } ) = \sqrt { ( \mathbf { x } - \mathbf { y } ) ^ { T } \Sigma ^ { - 1 } ( \mathbf { x } - \mathbf { y } ) }$
The Mahalanobis distance will reduce to the Euclidean distance when $\Sigma$ is the identity matrix, and become the standardized Euclidean distance when $\Sigma$ is diagonal.

2019-01-28[Stay Sharp]Mahalanobi

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