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:

    “According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
    —Ana Castillo (b. 1953)

    “Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.”
    —Willard Van Orman Quine (b. 1908)

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)