- 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.

where is the set of (probability) distributions on whose marginals are and on the first and second factors respectively.

If and are the respective CDFs of and , this distance also equals to:

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 =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,

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

`1from scipy.stats import wasserstein_distance`

```
1arr1 = [1,2,3,4,5,6]
2arr2 = [1,2,3,4,5,6]
3wasserstein_distance(arr1, arr2)
```

```
1# output
20.0
3# they are exactly the same!
```

```
1arr1 = [1,2,3]
2arr2 = [4,5,6]
3wasserstein_distance(arr1, arr2)
4# 3.0000000000000004
5
6import seaborn as sns
7sns.distplot(arr1, kde=False, hist_kws={"histtype": "step", "linewidth": 3, "alpha": 1, "color": "b"})
8sns.distplot(arr2, kde=False, hist_kws={"histtype": "step", "linewidth": 3, "alpha": 1, "color": "r"})
```

- An example of why we need to use EMD instead of Kolmogorov–Smirnov distance (video).