`scipy.stats._stats_py:energy_distance`

source: /scipy/stats/_stats_py.py:10011

Signature

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

Summary

Compute the energy distance between two 1D distributions.

Extended Summary

Parameters

u_values, v_values : array_like: Values observed in the (empirical) distribution.
u_weights, v_weights : array_like, optional: Weight for each value. If unspecified, each value is assigned the same weight. u_weights (resp. v_weights) must have the same length as u_values (resp. v_values). If the weight sum differs from 1, it must still be positive and finite so that the weights can be normalized to sum to 1.

Returns

distance : float: The computed distance between the distributions.

Notes

The energy distance between two distributions $u$ and $v$ , whose respective CDFs are $U$ and $V$ , equals to:

D (u, v) = (2 E ∣ X - Y ∣ - E ∣ X - X^{'} ∣ - E ∣ Y - Y^{'} ∣)^{1/2}

where $X$ and $X^{'}$ (resp. $Y$ and $Y^{'}$ ) are independent random variables whose probability distribution is $u$ (resp. $v$ ).

Sometimes the square of this quantity is referred to as the "energy distance" (e.g. in ^[2], ^[4]), but as noted in ^[1] and ^[3], only the definition above satisfies the axioms of a distance function (metric).

As shown in ^[2], for one-dimensional real-valued variables, the energy distance is linked to the non-distribution-free version of the Cramér-von Mises distance:

D (u, v) = 2 l_{2} (u, v) = (2 \int_{- \infty}^{+ \infty} (U - V)^{2})^{1/2}

Note that the common Cramér-von Mises criterion uses the distribution-free version of the distance. See ^[2] (section 2), for more details about both versions of the distance.

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 energy_distance

✓

energy_distance([0], [2])
energy_distance([0, 8], [0, 8], [3, 1], [2, 2])
energy_distance([0.7, 7.4, 2.4, 6.8], [1.4, 8. ],
                [2.1, 4.2, 7.4, 8. ], [7.6, 8.8])

✗

np.float64(0.8800334097615822)