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.

Read more about this topic:  Mahalanobis Distance

Famous quotes containing the word definition:

    “The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
    —Samuel Taylor Coleridge (1772–1834)

    “It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
    —Jane Adams (20th century)

    “The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
    —William James (1842–1910)