# Earth mover's distance

## Distance between probability distributions / From Wikipedia, the free encyclopedia

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In statistics, the **earth mover's distance** (**EMD**) is a measure of the distance between two probability distributions over a region *D*. In mathematics, this is known as the Wasserstein metric. Informally, if the distributions are interpreted as two different ways of piling up a certain amount of earth (dirt) over the region *D*, the EMD is the minimum cost of turning one pile into the other; where the cost is assumed to be the amount of dirt moved times the distance by which it is moved.[1]

The above definition is valid only if the two distributions have the same integral (informally, if the two piles have the same amount of dirt), as in normalized histograms or probability density functions. In that case, the EMD is equivalent to the 1st Mallows distance or 1st Wasserstein distance between the two distributions.[2][3]