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 analysed. 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, i.e. not dependent on the scale of measurements.

Formally, the Mahalanobis distance from a group of values with mean $\backslash mu\; =\; (\; \backslash mu\_1,\; \backslash mu\_2,\; \backslash mu\_3,\; \backslash dots\; ,\; \backslash mu\_p\; )$ and covariance matrix $\backslash Sigma$ for a multivariate vector $x\; =\; (\; x\_1,\; x\_2,\; x\_3,\; \backslash dots,\; x\_p\; )$ is defined as:

$D\_M(x)\; =\; \backslash sqrt\{(x\; -\; \backslash mu)^T\; \backslash Sigma^\{-1\}\; (x-\backslash mu)\}.\backslash ,$

Mahalanobis distance can also be defined as dissimilarity measure between two random vectors $\backslash vec\{x\}$ and $\backslash vec\{y\}$ of the same distribution with the covariance matrix $\backslash Sigma$ :

$d(\backslash vec\{x\},\backslash vec\{y\})=\backslash sqrt\{(\backslash vec\{x\}-\backslash vec\{y\})^T\backslash Sigma^\{-1\}\; (\backslash vec\{x\}-\backslash vec\{y\})\}.\backslash ,$

If the covariance matrix is the identity matrix then it is the same as Euclidean distance. If the covariance matrix is diagonal, then it is called normalized Euclidean distance:

$d(\backslash vec\{x\},\backslash vec\{y\})=$

\sqrt{\sum_{i=1}^p {(x_i - y_i)^2 \over \sigma_i^2}},

where $\backslash sigma\_i$ is the standard deviation of the $x\_i$ over the sample set.

statistics

Distància de Mahalanobis | Mahalanobis-Distanz | Distancia de Mahalanobis | Distance de Mahalanobis | 马氏距离