bundles / scipy 2.0.0.dev0+git20260907.41eeb59 / scipy / stats / _stats_py / wasserstein_distance
function
scipy.stats._stats_py:wasserstein_distance
source: build-install/usr/lib/python3.14/site-packages/scipy/stats/_stats_py.py:9960
Signature
def wasserstein_distance(u_values, v_values, u_weights = None, v_weights = None)Summary
Compute the Wasserstein-1 distance between two 1D 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: 1d array_likeA sample from a probability distribution or the support (set of all possible values) of a probability distribution. Each element is an observation or possible value.
v_values: 1d array_likeA sample from or the support of a second distribution.
u_weights, v_weights: 1d array_like, optionalWeights 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: floatThe computed distance between the distributions.
Notes
Given two 1D probability mass functions, and , the first Wasserstein distance between the distributions is:
where is the set of (probability) distributions on whose marginals are and on the first and second factors respectively. For a given value , gives the probability of at position , and the same for .
If and are the respective CDFs of and , this distance also equals to:
See [3] for a proof of the equivalence of both definitions.
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.
Examples
from scipy.stats import wasserstein_distance
✓wasserstein_distance([0, 1, 3], [5, 6, 8])
✗np.float64(5.0)When weights are provided, they represent the probability mass (e.g. "shovelsful of dirt" in the Earth mover's distance analogy [2]_) at each location. Weights are normalized internally to sum to 1:
wasserstein_distance([0, 1], [0, 1], u_weights=[3, 1], v_weights=[2, 2]) wasserstein_distance([3.4, 3.9, 7.5, 7.8], [4.5, 1.4], [1.4, 0.9, 3.1, 7.2], [3.2, 3.5])✗
np.float64(4.078133143804785)
See also
- wasserstein_distance_nd
Compute the Wasserstein-1 distance between two N-D discrete distributions.
Aliases
scipy.stats.wasserstein_distance