# Optimal Transport for 1D distributions

This example illustrates the computation of EMD and Sinkhorn transport plans and their visualization.

```# Author: Remi Flamary <remi.flamary@unice.fr>
#
# sphinx_gallery_thumbnail_number = 3

import numpy as np
import matplotlib.pylab as pl
import ot
import ot.plot
from ot.datasets import make_1D_gauss as gauss
```

## Generate data

```n = 100  # nb bins

# bin positions
x = np.arange(n, dtype=np.float64)

# Gaussian distributions
a = gauss(n, m=20, s=5)  # m= mean, s= std
b = gauss(n, m=60, s=10)

# loss matrix
M = ot.dist(x.reshape((n, 1)), x.reshape((n, 1)))
M /= M.max()
```

## Plot distributions and loss matrix

```pl.figure(1, figsize=(6.4, 3))
pl.plot(x, a, 'b', label='Source distribution')
pl.plot(x, b, 'r', label='Target distribution')
pl.legend()
```
```<matplotlib.legend.Legend object at 0x7f95d99d4100>
```
```pl.figure(2, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, M, 'Cost matrix M')
```

## Solve EMD

```# use fast 1D solver
G0 = ot.emd_1d(x, x, a, b)

# Equivalent to
# G0 = ot.emd(a, b, M)

pl.figure(3, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, G0, 'OT matrix G0')
```

## Solve Sinkhorn

```lambd = 1e-3
Gs = ot.sinkhorn(a, b, M, lambd, verbose=True)

pl.figure(4, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, Gs, 'OT matrix Sinkhorn')

pl.show()
```
```It.  |Err
-------------------
0|2.861463e-01|
10|1.860154e-01|
20|8.144529e-02|
30|3.130143e-02|
40|1.178815e-02|
50|4.426078e-03|
60|1.661047e-03|
70|6.233110e-04|
80|2.338932e-04|
90|8.776627e-05|
100|3.293340e-05|
110|1.235791e-05|
120|4.637176e-06|
130|1.740051e-06|
140|6.529356e-07|
150|2.450071e-07|
160|9.193632e-08|
170|3.449812e-08|
180|1.294505e-08|
190|4.857493e-09|
It.  |Err
-------------------
200|1.822723e-09|
```

Total running time of the script: (0 minutes 0.671 seconds)

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