`scipy.stats._stats_py:wasserstein_distance_nd`

source: /scipy/stats/_stats_py.py :9387

Signature

   def     wasserstein_distance_nd (    u_values    ,    v_values    ,    u_weights    =  None   ,    v_weights    =  None     )

Summary

Compute the Wasserstein-1 distance between two N-D discrete distributions.

Extended Summary

The Wasserstein distance, also called the Earth mover's distance or the optimal transport distance, is a similarity metric between two probability distributions ^[1]. In the discrete case, the Wasserstein distance can be understood as the cost of an optimal transport plan to convert one distribution into the other. The cost is calculated as the product of the amount of probability mass being moved and the distance it is being moved. A brief and intuitive introduction can be found at ^[2].

Parameters

u_values : 2d array_like: A sample from a probability distribution or the support (set of all possible values) of a probability distribution. Each element along axis 0 is an observation or possible value, and axis 1 represents the dimensionality of the distribution; i.e., each row is a vector observation or possible value.
v_values : 2d array_like: A sample from or the support of a second distribution.
u_weights, v_weights : 1d array_like, optional: Weights or counts corresponding with the sample or probability masses corresponding with the support values. Sum of elements must be positive and finite. If unspecified, each value is assigned the same weight.

Returns

distance : float: The computed distance between the distributions.

Notes

Given two probability mass functions, $u$ and $v$ , the first Wasserstein distance between the distributions using the Euclidean norm is:

l_{1} (u, v) = π \in Γ (u, v) in f \int ∥ x - y ∥_{2} d π (x, y)

where $Γ (u, v)$ is the set of (probability) distributions on $R^{n} \times R^{n}$ whose marginals are $u$ and $v$ on the first and second factors respectively. For a given value $x$ , $u (x)$ gives the probability of $u$ at position $x$ , and the same for $v (x)$ .

This is also called the optimal transport problem or the Monge problem. Let the finite point sets ${x_{i}}$ and ${y_{j}}$ denote the support set of probability mass function $u$ and $v$ respectively. The Monge problem can be expressed as follows,

Let $Γ$ denote the transport plan, $D$ denote the distance matrix and,

x = vec (Γ) c = vec (D) b = [u v]

The $vec ()$ function denotes the Vectorization function that transforms a matrix into a column vector by vertically stacking the columns of the matrix. The transport plan $Γ$ is a matrix $[γ_{ij}]$ in which $γ_{ij}$ is a positive value representing the amount of probability mass transported from $u (x_{i})$ to $v (y_{i})$ . Summing over the rows of $Γ$ should give the source distribution $u$ : $\sum_{j} γ_{ij} = u (x_{i})$ holds for all $i$ and summing over the columns of $Γ$ should give the target distribution $v$ : $\sum_{i} γ_{ij} = v (y_{j})$ holds for all $j$ . The distance matrix $D$ is a matrix $[d_{ij}]$ , in which $d_{ij} = d (x_{i}, y_{j})$ .

Given $Γ$ , $D$ , $b$ , the Monge problem can be transformed into a linear programming problem by taking $A x = b$ as constraints and $z = c^{T} x$ as minimization target (sum of costs) , where matrix $A$ has the form

10 ⋮ 010 ⋮ 0 10 ⋮ 001 ⋮ 0 \dots \dots ⋱ \dots \dots \dots ⋱ \dots 10 ⋮ 000 ⋮ 1 01 ⋮ 010 ⋮ 0 01 ⋮ 001 ⋮ 0 \dots \dots ⋱ \dots \dots \dots ⋱ \dots 01 ⋮ 0 \dots \dots ⋮ 1 \dots \dots ⋮ \dots \dots \dots ⋮ \dots 00 ⋮ 110 ⋮ 0 00 ⋮ 101 ⋮ 0 \dots \dots ⋱ \dots \dots \dots ⋱ \dots 00 ⋮ 100 ⋮ 1

By solving the dual form of the above linear programming problem (with solution $y^{*}$ ), the Wasserstein distance $l_{1} (u, v)$ can be computed as $b^{T} y^{*}$ .

The above solution is inspired by Vincent Herrmann's blog ^[3] . For a more thorough explanation, see ^[4] .

The input distributions can be empirical, therefore coming from samples whose values are effectively inputs of the function, or they can be seen as generalized functions, in which case they are weighted sums of Dirac delta functions located at the specified values.

Array API Standard Support

wasserstein_distance_nd has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ⛔                   
PyTorch               ⛔                     ⛔                   
JAX                   ⛔                     ⛔                   
Dask                  ⛔                     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.

Examples

Compute the Wasserstein distance between two three-dimensional samples, each with two observations.

from scipy.stats import wasserstein_distance_nd
wasserstein_distance_nd([[0, 2, 3], [1, 2, 5]], [[3, 2, 3], [4, 2, 5]])

✓

Compute the Wasserstein distance between two two-dimensional distributions with three and two weighted observations, respectively.

wasserstein_distance_nd([[0, 2.75], [2, 209.3], [0, 0]],
                     [[0.2, 0.322], [4.5, 25.1808]],
                     [0.4, 5.2, 0.114], [0.8, 1.5])

✓

Aliases

scipy.stats.wasserstein_distance_nd

Referenced by

This package

release:1.13.0-notes