Earth mover's distance
In computer science, 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 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 amount of dirt moved times the distance by which it is moved [1].
在计算机科学与技术中，地球移动距离(EMD)是一种在D区域两个概率分布距离的度量，就是被熟知的Wasserstein度量标准。不正式的说，如果两个分布被看作在D区域上两种不同方式堆积一定数量的山堆，那么EMD就是把一堆变成另一堆所需要移动单位小块最小的距离之和。
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 orprobability density functions. In that case, the EMD is equivalent to the 1st Mallows distance or 1st Wasserstein distance between the two distributions [2] [3].
上述的定义如果两个分布有着同样的整体（粗浅的说，就像两个堆有着同样的数量），在规范化的直方图或者概率密度函数上。在这基础上，EMD等同于两个分布的第一Mallows距离或者第一Wasserstein距离。
Extensions
Some applications may require the comparison of distributions with different total masses. One approach is to allow for a partial match, where dirt from the most massive distribution is rearranged to make the least massive, and any leftover "dirt" is discarded at no cost. Under this approach, the EMD is no longer a true distance between distributions. Another approach is to allow for mass to be created or destroyed, on a global and/or local level, as an alternative to transportation, but with a cost penalty. In that case one must specify a real parameter σ, the ratio between the cost of creating or destroying one unit of "dirt", and the cost of transporting it by a unit distance. This is equivalent to minimizing the sum of the earth moving cost plus σ times t