A generalization for the expected value of the earth mover's distance
arXiv:2009.12723 · doi:10.2140/astat.2021.12.139
Abstract
The earth mover's distance (EMD), also called the first Wasserstein distance, can be naturally extended to compare arbitrarily many probability distributions, rather than only two, on the set . We present the details for this generalization, along with a highly efficient algorithm inspired by combinatorics; it turns out that in the special case of three distributions, the EMD is half the sum of the pairwise EMD's. Extending the methods of Bourn and Willenbring (arXiv:1903.03673), we compute the expected value of this generalized EMD on random -tuples of distributions, using a generating function which coincides with the Hilbert series of the Segre embedding. We then use the EMD to analyze a real-world data set of grade distributions.
23 pages, 2 figures; corrected typos, simplified notation, added proof of Proposition 6, added discussion of even vs. odd d-values, updated real-world example to compare 7 distributions instead of 3, rewrote Section 7 for clarity