## What (general)? #

- In statistics, the earth mover's distance (EMD) is a measure of the distance between two probability distributions over a region D.
^{[ref]} - In stats or computer science, it's "
*Earth mover's distance*". - In maths, it's "
*Wasserstein metric*" - The Wasserstein distance is the minimum cost of transporting mass in converting the data distribution q to the data distribution p.

## What (math way)? #

The idea borrowed from this. The first Wasserstein distance between the distributions $u$ and $v$ is:

where $\Gamma(u,v)$ is the set of (probability) distributions on $\mathbb{R}\times \mathbb{R}$ whose marginals are and on the first and second factors respectively.

If $U$ and $V$ are the respective CDFs of $u$ and $v$, this distance also equals to:

## Example of metric #

Suppose we wanna move the blocks on the left to dotted-blocks on the right, we wanna find the "energy" (or metric) to do that.

*Energy = $\Sigma$ weight of block x distance to move that block*.

Suppose that weight of each block is 1. All below figures are copied from this.

There are 2 ways to do that,

*2 ways of moving blocks from left to right.*

Above example gives the same energies ($42$) but there are usually different as below example,

## Coding #

`from scipy.stats import wasserstein_distance`

`arr1 = [1,2,3,4,5,6]`

arr2 = [1,2,3,4,5,6]

wasserstein_distance(arr1, arr2)

`0.0`

# they are exactly the same!

`arr1 = [1,2,3]`

arr2 = [4,5,6]

wasserstein_distance(arr1, arr2)

# 3.0000000000000004

import seaborn as sns

sns.distplot(arr1, kde=False, hist_kws={"histtype": "step", "linewidth": 3, "alpha": 1, "color": "b"})

sns.distplot(arr2, kde=False, hist_kws={"histtype": "step", "linewidth": 3, "alpha": 1, "color": "r"})

## References #

- What is an intuitive explanation of the Wasserstein distance?
- GAN — Wasserstein GAN & WGAN-GP
- An example of why we need to use EMD instead of Kolmogorov–Smirnov distance (video).

^{•}Notes with this notation aren't good enough. They are being updated.