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# distance measure

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A measure of the distance between points in multidimensional space (also called a metric). Distance measures are used with techniques such as cluster analysis and multidimensional scaling. The two measures most commonly used are Euclidean distance and the city-block metric.

Euclidean distance is the straight-line distance between two points, also called the l2-metric. In d dimensions, if the positions of P and Q are given by the coordinates (p1, p2,…, pd) and (q1, q2,…, qd) then the Euclidean distance between P and Q is given by . The city-block metric in two dimensions measures the distance between two points in a city if, for example, the only directions in which one could travel were north-south and east-west. It is also called the l1-metric or Manhattan distance. In d dimensions the city-block distance between P and Q is given by . A generalization of the previous measures is the Minkowski distance where k is a positive integer. This is also called the lk-metric.

The Chebyshev distance (the l-metric) is the largest difference over the d dimensions , Two other measures that have been proposed are the Canberra distance, given by . and, in the context of counts of organisms, the Bray-Curtis distance (also called the Sorensen distance):

Distance measure. These are the two simplest of an infinite number of possible measures of distance.

Subjects: Probability and Statistics.