r callable, optional
    The distance metric to use. See the documentation
    for `scipy.spatial.distance.pdist` for the available metrics and
    the default.

Returns
-------
discrepancy : float
    Discrepancy (higher values correspond to greater sample uniformity).

See Also
--------
discrepancy

Notes
-----
The discrepancy can serve as a simple measure of quality of a random sample.
This measure is based on the geometric properties of the distribution of points
in the sample, such as the minimum distance between any pair of points, or
the mean edge length in a minimum spanning tree.

The higher the value is, the better the coverage of the parameter space is.
Note that this is different from `scipy.stats.qmc.discrepancy`, where lower
values correspond to higher quality of the sample.

Also note that when comparing different sampling strategies using this function,
the sample size must be kept constant.

It is possible to calculate two metrics from the minimum spanning tree:
the mean edge length and the standard deviation of edges lengths. Using
both metrics offers a better picture of uniformity than either metric alone,
with higher mean and lower standard deviation being preferable (see [1]_
for a brief discussion). This function currently only calculates the mean
edge length.

References
----------
.. [1] Franco J. et al. "Minimum Spanning Tree: A new approach to assess the quality
   of the design of computer experiments." Chemometrics and Intelligent Laboratory
   Systems, 97 (2), pp. 164-169, 2009.

Examples
--------
Calculate the quality of the sample using the minimum euclidean distance
(the defaults):

>>> import numpy as np
>>> from scipy.stats import qmc
>>> rng = np.random.default_rng(191468432622931918890291693003068437394)
>>> sample = qmc.LatinHypercube(d=2, rng=rng).random(50)
>>> qmc.geometric_discrepancy(sample)
0.03708161435687876

Calculate the quality using the mean edge length in the minimum
spanning tree:

>>> qmc.geometric_discrepancy(sample, method='mst')
0.1105149978798376

Display the minimum spanning tree and the points with
the smallest distance:

>>> import matplotlib.pyplot as plt
>>> from matplotlib.lines import Line2D
>>> from scipy.sparse.csgraph import minimum_spanning_tree
>>> from scipy.spatial.distance import pdist, squareform
>>> dist = pdist(sample)
>>> mst = minimum_spanning_tree(squareform(dist))
>>> edges = np.where(mst.toarray() > 0)
>>> edges = np.asarray(edges).T
>>> min_dist = np.min(dist)
>>> min_idx = np.argwhere(squareform(dist) == min_dist)[0]
>>> fig, ax = plt.subplots(figsize=(10, 5))
>>> _ = ax.set(aspect='equal', xlabel=r'$x_1$', ylabel=r'$x_2$',
...            xlim=[0, 1], ylim=[0, 1])
>>> for edge in edges:
...     ax.plot(sample[edge, 0], sample[edge, 1], c='k')
>>> ax.scatter(sample[:, 0], sample[:, 1])
>>> ax.add_patch(plt.Circle(sample[min_idx[0]], min_dist, color='red', fill=False))
>>> markers = [
...     Line2D([0], [0], marker='o', lw=0, label='Sample points'),
...     Line2D([0], [0], color='k', label='Minimum spanning tree'),
...     Line2D([0], [0], marker='o', lw=0, markerfacecolor='w', markeredgecolor='r',
...            label='Minimum point-to-point distance'),
... ]
>>> ax.legend(handles=markers, loc='center left', bbox_to_anchor=(1, 0.5));
>>> plt.show()

r