def test_discrepancy_hierarchy(self):
seed = 68756348576543
lhs = qmc.LatinHypercube(d=2, seed=seed)
sample_ref = lhs.random(n=20)
disc_ref = qmc.discrepancy(sample_ref)
optimal_ = qmc.LatinHypercube(d=2, seed=seed, optimization="random-CD")
sample_ = optimal_.random(n=20)
disc_ = qmc.discrepancy(sample_)
assert disc_ < disc_ref
def test_discrepancy_hierarchy(self):
seed = np.random.RandomState(123456)
lhs = qmc.LatinHypercube(d=2, seed=seed)
sample_ref = lhs.random(n=20)
disc_ref = qmc.discrepancy(sample_ref)
seed = np.random.RandomState(123456)
optimal_ = qmc.LatinHypercube(d=2, seed=seed, optimization="random-CD")
sample_ = optimal_.random(n=20)
disc_ = qmc.discrepancy(sample_)
assert disc_ < disc_ref
def test_discrepancy_parallel(self, monkeypatch):
sample = np.array([[2, 1, 1, 2, 2, 2],
[1, 2, 2, 2, 2, 2],
[2, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 2, 2],
[1, 2, 2, 2, 1, 1],
[2, 2, 2, 2, 1, 1],
[2, 2, 2, 1, 2, 2]])
sample = (2.0 * sample - 1.0) / (2.0 * 2.0)
assert_allclose(qmc.discrepancy(sample, method='MD', workers=8),
2.5000,
atol=1e-4)
assert_allclose(qmc.discrepancy(sample, method='WD', workers=8),
1.3680,
atol=1e-4)
assert_allclose(qmc.discrepancy(sample, method='CD', workers=8),
0.3172,
atol=1e-4)
# From Tim P. et al. Minimizing the L2 and Linf star discrepancies
# of a single point in the unit hypercube. JCAM, 2005
# Table 1 on Page 283
for dim in [2, 4, 8, 16, 32, 64]:
ref = np.sqrt(3 ** (-dim))
assert_allclose(qmc.discrepancy(np.array([[1] * dim]),
method='L2-star', workers=-1), ref)
monkeypatch.setattr(os, 'cpu_count', lambda: None)
with pytest.raises(NotImplementedError, match="Cannot determine the"):
qmc.discrepancy(sample, workers=-1)
with pytest.raises(ValueError, match="Invalid number of workers..."):
qmc.discrepancy(sample, workers=-2)
def test_perm_discrepancy(self):
rng = np.random.default_rng(46449423132557934943847369749645759997)
qmc_gen = qmc.LatinHypercube(5, seed=rng)
sample = qmc_gen.random(10)
disc = qmc.discrepancy(sample)
for i in range(100):
row_1 = rng.integers(10)
row_2 = rng.integers(10)
col = rng.integers(5)
disc = _perturb_discrepancy(sample, row_1, row_2, col, disc)
sample[row_1, col], sample[row_2, col] = (
sample[row_2, col], sample[row_1, col])
disc_reference = qmc.discrepancy(sample)
assert_allclose(disc, disc_reference)
def test_update_discrepancy(self):
space_1 = np.array([[1, 3], [2, 6], [3, 2], [4, 5], [5, 1], [6, 4]])
space_1 = (2.0 * space_1 - 1.0) / (2.0 * 6.0)
disc_init = qmc.discrepancy(space_1[:-1], iterative=True)
disc_iter = update_discrepancy(space_1[-1], space_1[:-1],
disc_init)
assert_allclose(disc_iter, 0.0081, atol=1e-4)
# errors
with pytest.raises(ValueError, match=r"Sample is not in unit "
r"hypercube"):
update_discrepancy(space_1[-1], space_1[:-1] + 1, disc_init)
with pytest.raises(ValueError, match=r"Sample is not a 2D array"):
update_discrepancy(space_1[-1], space_1[0], disc_init)
x_new = [1, 3]
with pytest.raises(ValueError, match=r"x_new is not in unit "
r"hypercube"):
update_discrepancy(x_new, space_1[:-1], disc_init)
x_new = [[0.5, 0.5]]
with pytest.raises(ValueError, match=r"x_new is not a 1D array"):
update_discrepancy(x_new, space_1[:-1], disc_init)
x_new = [0.3, 0.1, 0]
with pytest.raises(ValueError, match=r"x_new and sample must be "
r"broadcastable"):
update_discrepancy(x_new, space_1[:-1], disc_init)
def test_perm_discrepancy(self):
seed = np.random.RandomState(123456)
qmc_gen = qmc.LatinHypercube(5, seed=seed)
sample = qmc_gen.random(10)
disc = qmc.discrepancy(sample)
for i in range(100):
row_1 = np.random.randint(10)
row_2 = np.random.randint(10)
col = np.random.randint(5)
disc = _perturb_discrepancy(sample, row_1, row_2, col, disc)
sample[row_1, col], sample[row_2,
col] = (sample[row_2,
col], sample[row_1, col])
disc_reference = qmc.discrepancy(sample)
assert_allclose(disc, disc_reference)
def test_update_discrepancy(self):
# From Fang et al. Design and modeling for computer experiments, 2006
space_1 = np.array([[1, 3], [2, 6], [3, 2], [4, 5], [5, 1], [6, 4]])
space_1 = (2.0 * space_1 - 1.0) / (2.0 * 6.0)
disc_init = qmc.discrepancy(space_1[:-1], iterative=True)
disc_iter = update_discrepancy(space_1[-1], space_1[:-1], disc_init)
assert_allclose(disc_iter, 0.0081, atol=1e-4)
# n<d
rng = np.random.default_rng(241557431858162136881731220526394276199)
space_1 = rng.random((4, 10))
disc_ref = qmc.discrepancy(space_1)
disc_init = qmc.discrepancy(space_1[:-1], iterative=True)
disc_iter = update_discrepancy(space_1[-1], space_1[:-1], disc_init)
assert_allclose(disc_iter, disc_ref, atol=1e-4)
# errors
with pytest.raises(ValueError,
match=r"Sample is not in unit "
r"hypercube"):
update_discrepancy(space_1[-1], space_1[:-1] + 1, disc_init)
with pytest.raises(ValueError, match=r"Sample is not a 2D array"):
update_discrepancy(space_1[-1], space_1[0], disc_init)
x_new = [1, 3]
with pytest.raises(ValueError,
match=r"x_new is not in unit "
r"hypercube"):
update_discrepancy(x_new, space_1[:-1], disc_init)
x_new = [[0.5, 0.5]]
with pytest.raises(ValueError, match=r"x_new is not a 1D array"):
update_discrepancy(x_new, space_1[:-1], disc_init)
x_new = [0.3, 0.1, 0]
with pytest.raises(ValueError,
match=r"x_new and sample must be "
r"broadcastable"):
update_discrepancy(x_new, space_1[:-1], disc_init)
def test_discrepancy_errors(self):
sample = np.array([[1, 3], [2, 6], [3, 2], [4, 5], [5, 1], [6, 4]])
with pytest.raises(ValueError,
match=r"Sample is not in unit hypercube"):
qmc.discrepancy(sample)
with pytest.raises(ValueError, match=r"Sample is not a 2D array"):
qmc.discrepancy([1, 3])
sample = [[0, 0], [1, 1], [0.5, 0.5]]
with pytest.raises(ValueError, match=r"'toto' is not a valid ..."):
qmc.discrepancy(sample, method="toto")
def test_discrepancy(self):
space_1 = np.array([[1, 3], [2, 6], [3, 2], [4, 5], [5, 1], [6, 4]])
space_1 = (2.0 * space_1 - 1.0) / (2.0 * 6.0)
space_2 = np.array([[1, 5], [2, 4], [3, 3], [4, 2], [5, 1], [6, 6]])
space_2 = (2.0 * space_2 - 1.0) / (2.0 * 6.0)
# From Fang et al. Design and modeling for computer experiments, 2006
assert_allclose(qmc.discrepancy(space_1), 0.0081, atol=1e-4)
assert_allclose(qmc.discrepancy(space_2), 0.0105, atol=1e-4)
# From Zhou Y.-D. et al. Mixture discrepancy for quasi-random point
# sets. Journal of Complexity, 29 (3-4), pp. 283-301, 2013.
# Example 4 on Page 298
sample = np.array([[2, 1, 1, 2, 2, 2],
[1, 2, 2, 2, 2, 2],
[2, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 2, 2],
[1, 2, 2, 2, 1, 1],
[2, 2, 2, 2, 1, 1],
[2, 2, 2, 1, 2, 2]])
sample = (2.0 * sample - 1.0) / (2.0 * 2.0)
assert_allclose(qmc.discrepancy(sample, method='MD'), 2.5000,
atol=1e-4)
assert_allclose(qmc.discrepancy(sample, method='WD'), 1.3680,
atol=1e-4)
assert_allclose(qmc.discrepancy(sample, method='CD'), 0.3172,
atol=1e-4)
# From Tim P. et al. Minimizing the L2 and Linf star discrepancies
# of a single point in the unit hypercube. JCAM, 2005
# Table 1 on Page 283
for dim in [2, 4, 8, 16, 32, 64]:
ref = np.sqrt(3**(-dim))
assert_allclose(qmc.discrepancy(np.array([[1]*dim]),
method='L2-star'), ref)
def test_discrepancy_alternative_implementation(self):
"""Alternative definitions from Matt Haberland."""
def disc_c2(x):
n, s = x.shape
xij = x
disc1 = np.sum(np.prod((1
+ 1/2*np.abs(xij-0.5)
- 1/2*np.abs(xij-0.5)**2), axis=1))
xij = x[None, :, :]
xkj = x[:, None, :]
disc2 = np.sum(np.sum(np.prod(1
+ 1/2*np.abs(xij - 0.5)
+ 1/2*np.abs(xkj - 0.5)
- 1/2*np.abs(xij - xkj), axis=2),
axis=0))
return (13/12)**s - 2/n * disc1 + 1/n**2*disc2
def disc_wd(x):
n, s = x.shape
xij = x[None, :, :]
xkj = x[:, None, :]
disc = np.sum(np.sum(np.prod(3/2
- np.abs(xij - xkj)
+ np.abs(xij - xkj)**2, axis=2),
axis=0))
return -(4/3)**s + 1/n**2 * disc
def disc_md(x):
n, s = x.shape
xij = x
disc1 = np.sum(np.prod((5/3
- 1/4*np.abs(xij-0.5)
- 1/4*np.abs(xij-0.5)**2), axis=1))
xij = x[None, :, :]
xkj = x[:, None, :]
disc2 = np.sum(np.sum(np.prod(15/8
- 1/4*np.abs(xij - 0.5)
- 1/4*np.abs(xkj - 0.5)
- 3/4*np.abs(xij - xkj)
+ 1/2*np.abs(xij - xkj)**2,
axis=2), axis=0))
return (19/12)**s - 2/n * disc1 + 1/n**2*disc2
def disc_star_l2(x):
n, s = x.shape
return np.sqrt(
3 ** (-s) - 2 ** (1 - s) / n
* np.sum(np.prod(1 - x ** 2, axis=1))
+ np.sum([
np.prod(1 - np.maximum(x[k, :], x[j, :]))
for k in range(n) for j in range(n)
]) / n ** 2
)
rng = np.random.default_rng(117065081482921065782761407107747179201)
sample = rng.random((30, 10))
disc_curr = qmc.discrepancy(sample, method='CD')
disc_alt = disc_c2(sample)
assert_allclose(disc_curr, disc_alt)
disc_curr = qmc.discrepancy(sample, method='WD')
disc_alt = disc_wd(sample)
assert_allclose(disc_curr, disc_alt)
disc_curr = qmc.discrepancy(sample, method='MD')
disc_alt = disc_md(sample)
assert_allclose(disc_curr, disc_alt)
disc_curr = qmc.discrepancy(sample, method='L2-star')
disc_alt = disc_star_l2(sample)
assert_allclose(disc_curr, disc_alt)
