#### Vol. 12, No. 2, 2021

 Recent Issues Volume 12, Issue 2 Volume 12, Issue 1 Volume 11, Issue 2 Volume 11, Issue 1
 The Journal About the Journal Editorial Board Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN (electronic): 2693-3004 ISSN (print): 2693-2997 To Appear Other MSP Journals
A generalization for the expected value of the earth mover's distance

### William Q. Erickson

Vol. 12 (2021), No. 2, 139–166
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 $\left[n\right]=\left\{1,\dots ,n\right\}$. 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 EMDs. Extending the methods of Bourn and Willenbring (2020), 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.

##### Keywords
earth mover's distance, generating functions, Wallach representations
##### Mathematical Subject Classification
Primary: 05E14, 13P25
Secondary: 05E40