def test_regression_comparison_2(self):
# Regression over copies of the same environment should yield
# same results as regressing over that distribution
graphs = [
src.utils.eg1(),
src.utils.eg2(),
src.utils.eg3(),
src.utils.eg4()
]
for g, graph in enumerate(graphs):
W, _, _ = graph
p = len(W)
A = sampling_matrix(W)
dist = NormalDistribution(np.arange(p), A @ A.T)
envs = [dist, dist, dist]
for i in range(p):
for l in range(p + 1):
S = [k for k in range(l) if k != i]
#print("eg%d: y=%d, S=%s" % (g+1,i,S))
(true_coefs, true_intercept) = dist.regress(i, S)
true_mse = dist.mse(i, S)
(pooled_coefs, pooled_intercept,
pooled_mse) = pooled_regression(i, S, envs)
self.assertTrue(np.allclose(true_coefs, pooled_coefs))
self.assertTrue(
np.allclose(true_intercept, pooled_intercept))
self.assertTrue(np.allclose(true_mse, pooled_mse))
def test_sampling(self):
# Test sampling of the distribution
np.random.seed(42)
n = round(1e6)
covariance = 1e-2 * np.array([[1, 0, 1, 1], [0, 1, 1, 1], [1, 1, 3, 3],
[1, 1, 3, 4]])
mean = np.array([0, 0, 0, 0])
dist = NormalDistribution(mean, covariance)
samples = dist.sample(n)
self.assertTrue(np.allclose(np.mean(samples, axis=0), mean, atol=1e-1))
self.assertTrue(
np.allclose(np.cov(samples, rowvar=False), covariance, atol=1e-1))
def test_marginalization(self):
# Test marginalization logic
covariance = np.array([[11, 12, 13], [21, 22, 23], [31, 32, 33]])
mean = np.array([1, 2, 3])
distribution = NormalDistribution(mean, covariance)
# Marginalize X1
marginal = distribution.marginal([0])
self.assertTrue((marginal.mean == np.array([1])).all())
self.assertTrue((marginal.covariance == np.array([11])).all())
# Marginalize X1 and X3
marginal = distribution.marginal([0, 2])
self.assertTrue((marginal.mean == np.array([1, 3])).all())
self.assertTrue((marginal.covariance == np.array([[11, 13],
[31, 33]])).all())
def regression_properties_test(self, W, parents, markov_blankets):
# Test basic properties of the regression coefficients.
# PRE: Assumed [1..p] is a valid causal ordering
p = len(W)
np.random.seed(42)
W = W * (np.random.uniform(size=W.shape) + 1
) # avoid 0 weights (causal minimality)
A = sampling_matrix(W)
covariance = A @ A.T
mean = np.random.uniform(size=p)
joint = NormalDistribution(mean, covariance)
# Tests
# Regressing on the parents should return their weights as coefficients
for i, pa in enumerate(parents):
(coefs, intercept) = joint.regress(i, pa)
if not np.allclose(W[pa, i], coefs[pa]):
self.fail("%s ~ %s. Expected: %s Got: %s" %
(i, pa, W[pa, i], coefs[pa]))
# Regressing on all variables should yield weight 1 on the
# target and 0 intercept
for i in range(p):
(coefs, intercept) = joint.regress(i, range(p))
truth = np.zeros(p)
truth[i] = 1
if not np.allclose(truth, coefs):
self.fail("%s ~ %s. Expected: %s Got: %s" %
(i, pa, truth, coefs))
if not np.allclose(0, intercept):
self.fail("%s ~ %s. Expected: %s Got: %s" %
(i, pa, 0, intercept))
# Regressing on all variables except target should yield Markov blanket
for i, mb in enumerate(markov_blankets):
mb.sort()
(coefs, intercept) = joint.regress(i, all_but(i, p))
non_mb = [i for i in range(p) if i not in mb]
# Check vars. outside MB have 0 coefficient
truth = np.zeros(len(non_mb))
if not np.allclose(truth, coefs[non_mb]):
self.fail("%s ~ %s. Expected: %s Got: %s" %
(i, mb, truth, coefs[non_mb]))
# Check that vars. inside MB have non-zero coefficient (we
# assumed causal minimality above)
tol = 1e-10
if not (np.abs(coefs[mb]) > tol).all():
self.fail("%s ~ %s. Expected all non-zero. Got: %s" %
(i, mb, coefs[mb]))
def test_mse_1(self):
# Test a set of regressions
covariance = np.array([[1, 0, 1, 1], [0, 1, 1, 1], [1, 1, 3, 3],
[1, 1, 3, 4]])
mean = np.array([0, 0, 0, 0])
joint = NormalDistribution(mean, covariance)
tests = [
# Regressing X0
(0, [1], 1), # indep. var
(0, [2], 2 / 3), # children
(0, [1, 2], 0.5), # markov blanket
(0, [2, 1], 0.5), # markov blanket (test order does not matter)
# Regressing X1
(1, [0], 1), # indep. var
(1, [2], 2 / 3), # children
(1, [0, 2], 0.5), # markov blanket
(1, [2, 0], 0.5), # markov blanket (test order does not matter)
# Regressing X2
(2, [0], 2), # parent
(2, [1], 2), # parent
(2, [0, 1], 1), # parents
(2, [0, 1, 3], 0.5), # markov blanket
(2, [3], 0.75), # child
# Regressing X3
(3, [0], 3), # X0
(3, [0], 3), # X1
(3, [0, 1], 2), # X0 and X1
(3, [1, 0], 2), # X1 and X0 (test order does not matter)
(3, [2, 1], 1), # markov blanket + d-separated
(3, [1, 2],
1), # markov blanket + d-separated (test order does not matter)
(3, [2, 0], 1), # markov blanket + d-separated
(3, [2], 1)
] # markov blanket
for test in tests:
(y, Xs, true_mse) = test
self.assertEqual(true_mse, joint.mse(y, Xs))
# Changing the mean of the noise variables should not change
# the results
joint.mean = np.array([1, 2, 3, 4])
for test in tests:
(y, Xs, true_mse) = test
self.assertEqual(true_mse, joint.mse(y, Xs))
def test_parameters(self):
covariance = np.array([[11, 12, 13], [21, 22, 23], [31, 32, 33]])
mean = np.array([1, 2, 3])
dist = NormalDistribution(mean, covariance)
# Marginalization
dist_a = dist.marginal(0)
dist_b = dist.marginal([0])
self.assertTrue(dist_a.equal(dist_b))
# Conditioning 1
dist_a = dist.conditional(0, 1, 0)
dist_b = dist.conditional([0], [1], [0])
self.assertTrue(dist_a.equal(dist_b))
# Conditioning 2
dist_a = dist.conditional(0, [], [])
try:
dist.conditional(0, [], [1, 2])
self.fail("Exception should have been raised")
except ValueError as e:
print("OK:", e)
dist_c = dist.marginal(0)
self.assertTrue(dist_a.equal(dist_c))
def test_initialization(self):
# Test initialization and that arguments are copied (not
# stored by reference)
covariance = np.array([[11, 12, 13], [21, 22, 23], [31, 32, 33]])
mean = np.array([1, 2, 3])
distribution = NormalDistribution(mean, covariance)
self.assertTrue((mean == distribution.mean).all())
self.assertTrue((covariance == distribution.covariance).all())
mean[2] = 0
covariance[:, 2] = 0
self.assertTrue(not (mean == distribution.mean).all())
self.assertTrue(not (covariance == distribution.covariance).all())
def test_regression_arguments(self):
p = 5
covariance = np.eye(p)
mean = np.random.uniform(size=p)
joint = NormalDistribution(mean, covariance)
# Test can call with regressors as np.array (see issue #3)
joint.regress(0, [1, 2])
joint.regress(0, np.array([1, 2]))
def mse_properties_test(self,
W,
parents,
markov_blankets,
sample_weights=True):
# Given a weight matrix, parents of each node and markov
# blankets of each node, test basic properties of the MSE
# PRE: It is assumed that [1,...,p] is the causal ordering
p = len(W)
np.random.seed(42)
if sample_weights:
W = W * (np.random.uniform(size=W.shape) + 1
) # avoid 0 weights (causal minimality)
A = sampling_matrix(W)
covariance = A @ A.T
mean = np.random.uniform(size=p)
joint = NormalDistribution(mean, covariance)
# Tests
tests = []
# When no regressors, the MSE is just the marginal variance
tests += [(i, [], covariance[i, i]) for i in range(p)]
# Regressing a variable on itself should give 0 MSE
tests += [(i, [i], 0) for i in range(p)]
# Regressing on all variables (including itself) should also
# give 0 MSE
tests += [(i, range(p), 0) for i in range(p)]
# Regressing on the parents should yield MSE equal to
# the variance of the noise variable (1 in this case)
tests += [(i, pa, 1) for i, pa in enumerate(parents)]
# Regressing on additional variables to the Markov Blanket
# should give the same result
tests += [(i, all_but(i, p), joint.mse(i, mb))
for i, mb in enumerate(markov_blankets)]
self.run_mse_tests(joint, tests)
# Changing the mean of the noise variables should not change
# the MSE
joint.mean = np.random.uniform(size=p)
self.run_mse_tests(joint, tests)
def test_markov_blanket(self):
graphs = [
src.utils.eg1(),
src.utils.eg2(),
src.utils.eg3(),
src.utils.eg4()
]
for g, graph in enumerate(graphs):
W, _, mb = graph
p = len(W)
A = sampling_matrix(W)
dist = NormalDistribution(np.arange(p), A @ A.T)
for i in range(p):
truth = set(mb[i])
result = set(markov_blanket(i, dist, tol=1e-10))
# print(truth, result)
self.assertEqual(truth, result)
def test_conditioning_2(self):
covariance = np.array([[1, 2, 3], [2, 4, 6], [3, 6, 9]]) # rank 1
mean = np.array([1, 2, 3])
dist = NormalDistribution(mean, covariance)
# Conditioning X1 on X2 = 1 should give mean = 1/2 and variance = 0
self.assertTrue((dist.conditional([0], [1],
1).mean == np.array([0.5])).all())
self.assertTrue(
(dist.conditional([0], [1],
1).covariance == np.array([[0]])).all())
# Conditioning X1 on X2 = 0 should give mean = 0 and variance = 0
self.assertTrue((dist.conditional([0], [1],
0).mean == np.array([0])).all())
self.assertTrue(
(dist.conditional([0], [1],
0.5).covariance == np.array([[0]])).all())
def test_conditioning_3(self):
covariance = np.array([[1, 0, 1, 1], [0, 1, 1, 1], [1, 1, 3, 3],
[1, 1, 3, 4]])
mean = np.array([0, 0, 0, 0])
dist = NormalDistribution(mean, covariance)
# Conditioning X3 on X1 = 1, X2 = 2 should give mean 3 and variance 1
conditional = dist.conditional([2], [0, 1], np.array([1, 2]))
self.assertTrue((conditional.mean == np.array([3])).all())
self.assertTrue((conditional.covariance == np.array([[1]])).all())
# Conditioning X4 on X1 = 1, X2 = 2 should give mean 3 and variance 2
conditional = dist.conditional([3], [0, 1], np.array([1, 2]))
self.assertTrue((conditional.mean == np.array([3])).all())
self.assertTrue((conditional.covariance == np.array([[2]])).all())
# Conditioning X3, X4 on X1 = 1, X2 = 2
conditional = dist.conditional([2, 3], [0, 1], np.array([1, 2]))
self.assertTrue((conditional.mean == np.array([3, 3])).all())
self.assertTrue((conditional.covariance == np.array([[1, 1],
[1, 2]])).all())
# Conditioning X3 on X1 = 1, X4 = 1 should give mean 1 and variance 3-7/3
conditional = dist.conditional([2], [0, 3], np.array([1, 1]))
self.assertTrue(np.allclose(conditional.mean, np.array([1])))
self.assertTrue(
np.allclose(conditional.covariance, np.array([[3 - 7 / 3]])))
def test_equal(self):
covariance = np.array([[11, 12, 13], [21, 22, 23], [31, 32, 33]])
mean = np.array([1, 2, 3])
dist1 = NormalDistribution(mean, covariance)
dist2 = NormalDistribution(mean, covariance)
self.assertTrue(dist1.equal(dist2))
self.assertTrue(dist2.equal(dist1))
tol = 1e-7
dist3 = NormalDistribution(mean + tol / 2, covariance + tol / 2)
self.assertTrue(dist1.equal(dist3))
self.assertTrue(dist2.equal(dist3))
self.assertTrue(dist3.equal(dist1))
self.assertTrue(dist3.equal(dist2))
def test_conditioning_1(self):
# Test conditioning
covariance = np.eye(3)
mean = np.zeros(3)
distribution = NormalDistribution(mean, covariance)
# Condition X1 on X2=1 should yield the marginal of X1 (as they are independent)
conditional = distribution.conditional([0], [1], np.array([1]))
self.assertTrue((conditional.mean == np.array([0])).all())
self.assertTrue((conditional.covariance == np.array([1])).all())
self.assertTrue(
(conditional.mean == distribution.marginal([0]).mean).all())
self.assertTrue((conditional.covariance == distribution.marginal(
[0]).covariance).all())
# Conditioning X1, X2 on X3 = 1 should yield marginal of X1, X2
conditional = distribution.conditional([0, 1], [2], 0)
self.assertTrue((conditional.mean == np.array([0, 0])).all())
self.assertTrue((conditional.covariance == np.eye(2)).all())
self.assertTrue(
(conditional.mean == distribution.marginal([0, 1]).mean).all())
self.assertTrue((conditional.covariance == distribution.marginal(
[0, 1]).covariance).all())
# Conditioning X1 on X2=2, X3 = 1 should yield marginal of X1, X2
conditional = distribution.conditional([0], [1, 2], np.array([2, 1]))
self.assertTrue((conditional.mean == np.array([0])).all())
self.assertTrue((conditional.covariance == np.eye(1)).all())
self.assertTrue(
(conditional.mean == distribution.marginal([0]).mean).all())
self.assertTrue((conditional.covariance == distribution.marginal(
[0]).covariance).all())