# Expected value of the one-dimensional Earth Mover's Distance

## TL;DR

This paper derives a method to compute the expected Earth Mover's Distance between probability distributions on a finite line, linking combinatorics and algebraic geometry, with applications in clustering analysis.

## Contribution

It introduces an easy computation method for the EMD generating function on a finite line, connecting it to algebraic geometry and enabling expected value calculations.

## Key findings

- Derived a generating function for EMD values on [n]
- Computed the expected EMD in the one-dimensional case
- Applied EMD in clustering analysis for a data set

## Abstract

From a combinatorial point of view, we consider the Earth Mover's Distance (EMD) associated with a metric measure space. The specific case considered is deceptively simple: Let the finite set [n] = {1,...,n} be regarded as a metric space by restricting the usual Euclidean distance on the real numbers. The EMD is defined on ordered pairs of probability distributions on [n]. We provide an easy method to compute a generating function encoding the values of EMD in its coefficients, which is related to the Segre embedding from projective algebraic geometry. As an application we use the generating function to compute the expected value of EMD in this one-dimensional case. The EMD is then used in clustering analysis for a specific data set.