def test_pca_validation(self):
for solver in self.solver_list:
# Ensures that solver-specific extreme inputs for the n_components
# parameter raise errors
X = mt.array([[0, 1, 0], [1, 0, 0]])
smallest_d = 2 # The smallest dimension
lower_limit = {'randomized': 1, 'full': 0, 'auto': 0}
# We conduct the same test on X.T so that it is invariant to axis.
for data in [X, X.T]:
for n_components in [-1, 3]:
if solver == 'auto':
solver_reported = 'full'
else:
solver_reported = solver
assert_raises_regex(
ValueError, "n_components={}L? must be between "
r"{}L? and min\(n_samples, n_features\)="
"{}L? with svd_solver=\'{}\'".format(
n_components, lower_limit[solver], smallest_d,
solver_reported),
PCA(n_components, svd_solver=solver).fit, data)
n_components = 1.0
type_ncom = type(n_components)
assert_raise_message(
ValueError, "n_components={} must be of type int "
"when greater than or equal to 1, was of type={}".format(
n_components, type_ncom),
PCA(n_components, svd_solver=solver).fit, data)
def test_randomized_pca_check_list(setup):
# Test that the projection by randomized PCA on list data is correct
X = mt.tensor([[1.0, 0.0], [0.0, 1.0]])
X_transformed = PCA(n_components=1, svd_solver='randomized',
random_state=0).fit(X).transform(X)
assert X_transformed.shape == (2, 1)
assert_almost_equal(X_transformed.mean().to_numpy(), 0.00, 2)
assert_almost_equal(X_transformed.std().to_numpy(), 0.71, 2)
def test_n_components_none(self):
for solver in self.solver_list:
# Ensures that n_components == None is handled correctly
X = self.iris
# We conduct the same test on X.T so that it is invariant to axis.
for data in [X, X.T]:
pca = PCA(svd_solver=solver)
pca.fit(data)
self.assertEqual(pca.n_components_, min(data.shape))
def test_pca_score(self):
# Test that probabilistic PCA scoring yields a reasonable score
n, p = 1000, 3
rng = np.random.RandomState(0)
X = mt.tensor(rng.randn(n, p) * .1) + mt.array([3, 4, 5])
for solver in self.solver_list:
pca = PCA(n_components=2, svd_solver=solver)
pca.fit(X)
ll1 = pca.score(X)
h = -0.5 * mt.log(2 * mt.pi * mt.exp(1) * 0.1**2) * p
np.testing.assert_almost_equal((ll1 / h).to_numpy(), 1, 0)
def test_infer_dim_3(self):
n, p = 100, 5
rng = np.random.RandomState(0)
X = mt.tensor(rng.randn(n, p) * .1)
X[:10] += mt.array([3, 4, 5, 1, 2])
X[10:20] += mt.array([6, 0, 7, 2, -1])
X[30:40] += 2 * mt.array([-1, 1, -1, 1, -1])
pca = PCA(n_components=p, svd_solver='full')
pca.fit(X)
spect = pca.explained_variance_.fetch()
self.assertGreater(_infer_dimension(spect, n), 2)
def test_pca_deterministic_output(setup):
rng = np.random.RandomState(0)
X = mt.tensor(rng.rand(10, 10))
for solver in solver_list:
transformed_X = np.zeros((20, 2))
for i in range(20):
pca = PCA(n_components=2, svd_solver=solver, random_state=rng)
transformed_X[i, :] = pca.fit_transform(X)[0].fetch()
np.testing.assert_allclose(
transformed_X, np.tile(transformed_X[0, :], 20).reshape(20, 2))
def test_infer_dim_2(self):
# TODO: explain what this is testing
# Or at least use explicit variable names...
n, p = 1000, 5
rng = np.random.RandomState(0)
X = mt.tensor(rng.randn(n, p) * .1)
X[:10] += mt.array([3, 4, 5, 1, 2])
X[10:20] += mt.array([6, 0, 7, 2, -1])
pca = PCA(n_components=p, svd_solver='full')
pca.fit(X)
spect = pca.explained_variance_.fetch()
self.assertGreater(_infer_dimension(spect, n), 1)
def test_infer_dim_1(setup):
# TODO: explain what this is testing
# Or at least use explicit variable names...
n, p = 1000, 5
rng = np.random.RandomState(0)
X = (mt.tensor(rng.randn(n, p)) * .1 + mt.tensor(rng.randn(n, 1)) * mt.array([3, 4, 5, 1, 2]) +
mt.array([1, 0, 7, 4, 6]))
pca = PCA(n_components=p, svd_solver='full')
pca.fit(X)
spect = pca.explained_variance_.to_numpy()
ll = np.array([_assess_dimension(spect, k, n) for k in range(1, p)])
assert ll[1] > ll.max() - .01 * n
def test_randomized_pca_inverse(self):
# Test that randomized PCA is inversible on dense data
rng = np.random.RandomState(0)
n, p = 50, 3
X = mt.tensor(rng.randn(n, p)) # spherical data
X[:, 1] *= .00001 # make middle component relatively small
X += [5, 4, 3] # make a large mean
# same check that we can find the original data from the transformed signal
# (since the data is almost of rank n_components)
pca = PCA(n_components=2, svd_solver='randomized',
random_state=0).fit(X)
Y = pca.transform(X)
Y_inverse = pca.inverse_transform(Y)
assert_almost_equal(X.to_numpy(), Y_inverse.to_numpy(), decimal=2)
# same as above with whitening (approximate reconstruction)
pca = PCA(n_components=2,
whiten=True,
svd_solver='randomized',
random_state=0).fit(X)
Y = pca.transform(X)
Y_inverse = pca.inverse_transform(Y)
relative_max_delta = (mt.abs(X - Y_inverse) / mt.abs(X).mean()).max()
self.assertLess(relative_max_delta.to_numpy(), 1e-5)
def test_pca_score3(setup):
# Check that probabilistic PCA selects the right model
n, p = 200, 3
rng = np.random.RandomState(0)
Xl = mt.tensor(rng.randn(n, p) + rng.randn(n, 1) * np.array([3, 4, 5]) +
np.array([1, 0, 7]))
Xt = mt.tensor(rng.randn(n, p) + rng.randn(n, 1) * np.array([3, 4, 5]) +
np.array([1, 0, 7]))
ll = mt.zeros(p)
for k in range(p):
pca = PCA(n_components=k, svd_solver='full')
pca.fit(Xl)
ll[k] = pca.score(Xt)
assert ll.argmax().to_numpy() == 1
def test_pca_score_with_different_solvers(self):
digits = datasets.load_digits()
X_digits = mt.tensor(digits.data)
pca_dict = {
svd_solver: PCA(n_components=30,
svd_solver=svd_solver,
random_state=0)
for svd_solver in self.solver_list
}
for pca in pca_dict.values():
pca.fit(X_digits)
# Sanity check for the noise_variance_. For more details see
# https://github.com/scikit-learn/scikit-learn/issues/7568
# https://github.com/scikit-learn/scikit-learn/issues/8541
# https://github.com/scikit-learn/scikit-learn/issues/8544
assert mt.all((pca.explained_variance_ -
pca.noise_variance_) >= 0).to_numpy()
# Compare scores with different svd_solvers
score_dict = {
svd_solver: pca.score(X_digits).to_numpy()
for svd_solver, pca in pca_dict.items()
}
assert_almost_equal(score_dict['full'],
score_dict['randomized'],
decimal=3)
def testExplainedVariance(self):
# Check that PCA output has unit-variance
rng = np.random.RandomState(0)
n_samples = 100
n_features = 80
X = mt.tensor(rng.randn(n_samples, n_features))
pca = PCA(n_components=2, svd_solver='full').fit(X)
rpca = PCA(n_components=2, svd_solver='randomized',
random_state=42).fit(X)
assert_array_almost_equal(pca.explained_variance_.to_numpy(),
rpca.explained_variance_.to_numpy(), 1)
assert_array_almost_equal(pca.explained_variance_ratio_.to_numpy(),
rpca.explained_variance_ratio_.to_numpy(), 1)
# compare to empirical variances
expected_result = np.linalg.eig(np.cov(X.to_numpy(), rowvar=False))[0]
expected_result = sorted(expected_result, reverse=True)[:2]
X_pca = pca.transform(X)
assert_array_almost_equal(pca.explained_variance_.to_numpy(),
mt.var(X_pca, ddof=1, axis=0).to_numpy())
assert_array_almost_equal(pca.explained_variance_.to_numpy(),
expected_result)
X_rpca = rpca.transform(X)
assert_array_almost_equal(rpca.explained_variance_.to_numpy(),
mt.var(X_rpca, ddof=1, axis=0).to_numpy(),
decimal=1)
assert_array_almost_equal(rpca.explained_variance_.to_numpy(),
expected_result,
decimal=1)
# Same with correlated data
X = datasets.make_classification(n_samples,
n_features,
n_informative=n_features - 2,
random_state=rng)[0]
X = mt.tensor(X)
pca = PCA(n_components=2).fit(X)
rpca = PCA(n_components=2, svd_solver='randomized',
random_state=rng).fit(X)
assert_array_almost_equal(pca.explained_variance_ratio_.to_numpy(),
rpca.explained_variance_ratio_.to_numpy(), 5)
def test_pca_dim(self):
# Check automated dimensionality setting
rng = np.random.RandomState(0)
n, p = 100, 5
X = mt.tensor(rng.randn(n, p) * .1)
X[:10] += mt.array([3, 4, 5, 1, 2])
pca = PCA(n_components='mle', svd_solver='full').fit(X)
self.assertEqual(pca.n_components, 'mle')
self.assertEqual(pca.n_components_, 1)
def test_n_components_mle(self):
# Ensure that n_components == 'mle' doesn't raise error for auto/full
# svd_solver and raises error for arpack/randomized svd_solver
rng = np.random.RandomState(0)
n_samples = 600
n_features = 10
X = mt.tensor(rng.randn(n_samples, n_features))
n_components_dict = {}
for solver in self.solver_list:
pca = PCA(n_components='mle', svd_solver=solver)
if solver in ['auto', 'full']:
pca.fit(X)
n_components_dict[solver] = pca.n_components_
else: # arpack/randomized solver
error_message = ("n_components='mle' cannot be a string with "
"svd_solver='{}'".format(solver))
assert_raise_message(ValueError, error_message, pca.fit, X)
self.assertEqual(n_components_dict['auto'], n_components_dict['full'])
def test_infer_dim_by_explained_variance(setup):
X = iris
pca = PCA(n_components=0.95, svd_solver='full')
pca.fit(X)
assert pca.n_components == 0.95
assert pca.n_components_ == 2
pca = PCA(n_components=0.01, svd_solver='full')
pca.fit(X)
assert pca.n_components == 0.01
assert pca.n_components_ == 1
rng = np.random.RandomState(0)
# more features than samples
X = mt.tensor(rng.rand(5, 20))
pca = PCA(n_components=.5, svd_solver='full').fit(X)
assert pca.n_components == 0.5
assert pca.n_components_ == 2
def test_infer_dim_by_explained_variance(self):
X = self.iris
pca = PCA(n_components=0.95, svd_solver='full')
pca.fit(X)
self.assertEqual(pca.n_components, 0.95)
self.assertEqual(pca.n_components_, 2)
pca = PCA(n_components=0.01, svd_solver='full')
pca.fit(X)
self.assertEqual(pca.n_components, 0.01)
self.assertEqual(pca.n_components_, 1)
rng = np.random.RandomState(0)
# more features than samples
X = mt.tensor(rng.rand(5, 20))
pca = PCA(n_components=.5, svd_solver='full').fit(X)
self.assertEqual(pca.n_components, 0.5)
self.assertEqual(pca.n_components_, 2)
def test_pca_sparse_input(self):
for svd_solver in self.solver_list:
X = np.random.RandomState(0).rand(5, 4)
X = mt.tensor(sp.sparse.csr_matrix(X))
self.assertTrue(X.issparse())
pca = PCA(n_components=3, svd_solver=svd_solver)
assert_raises(TypeError, pca.fit, X)
def test_randomized_pca_check_projection(self):
# Test that the projection by randomized PCA on dense data is correct
rng = np.random.RandomState(0)
n, p = 100, 3
X = mt.tensor(rng.randn(n, p) * .1)
X[:10] += mt.array([3, 4, 5])
Xt = 0.1 * mt.tensor(rng.randn(1, p)) + mt.array([3, 4, 5])
Yt = PCA(n_components=2, svd_solver='randomized',
random_state=0).fit(X).transform(Xt)
Yt /= np.sqrt((Yt**2).sum())
assert_almost_equal(mt.abs(Yt[0][0]).to_numpy(), 1., 1)
def test_pca_inverse(self):
# Test that the projection of data can be inverted
rng = np.random.RandomState(0)
n, p = 50, 3
X = mt.tensor(rng.randn(n, p)) # spherical data
X[:, 1] *= .00001 # make middle component relatively small
X += [5, 4, 3] # make a large mean
# same check that we can find the original data from the transformed
# signal (since the data is almost of rank n_components)
pca = PCA(n_components=2, svd_solver='full').fit(X)
Y = pca.transform(X)
Y_inverse = pca.inverse_transform(Y)
assert_almost_equal(X.to_numpy(), Y_inverse.to_numpy(), decimal=3)
# same as above with whitening (approximate reconstruction)
for solver in self.solver_list:
pca = PCA(n_components=2, whiten=True, svd_solver=solver)
pca.fit(X)
Y = pca.transform(X)
Y_inverse = pca.inverse_transform(Y)
assert_almost_equal(X.to_numpy(), Y_inverse.to_numpy(), decimal=3)
def testWhitening(self):
# Check that PCA output has unit-variance
rng = np.random.RandomState(0)
n_samples = 100
n_features = 80
n_components = 30
rank = 50
# some low rank data with correlated features
X = mt.dot(
rng.randn(n_samples, rank),
mt.dot(mt.diag(mt.linspace(10.0, 1.0, rank)),
rng.randn(rank, n_features)))
# the component-wise variance of the first 50 features is 3 times the
# mean component-wise variance of the remaining 30 features
X[:, :50] *= 3
self.assertEqual(X.shape, (n_samples, n_features))
# the component-wise variance is thus highly varying:
self.assertGreater(X.std(axis=0).std().to_numpy(), 43.8)
for solver, copy in product(self.solver_list, (True, False)):
# whiten the data while projecting to the lower dim subspace
X_ = X.copy() # make sure we keep an original across iterations.
pca = PCA(n_components=n_components,
whiten=True,
copy=copy,
svd_solver=solver,
random_state=0,
iterated_power=7)
# test fit_transform
X_whitened = pca.fit_transform(X_.copy())
self.assertEqual(X_whitened.shape, (n_samples, n_components))
X_whitened2 = pca.transform(X_)
assert_array_almost_equal(X_whitened.fetch(), X_whitened2.fetch())
assert_almost_equal(X_whitened.std(ddof=1, axis=0).to_numpy(),
np.ones(n_components),
decimal=6)
assert_almost_equal(
X_whitened.mean(axis=0).to_numpy(), np.zeros(n_components))
X_ = X.copy()
pca = PCA(n_components=n_components,
whiten=False,
copy=copy,
svd_solver=solver).fit(X_)
X_unwhitened = pca.transform(X_)
self.assertEqual(X_unwhitened.shape, (n_samples, n_components))
# in that case the output components still have varying variances
assert_almost_equal(
X_unwhitened.std(axis=0).std().to_numpy(), 74.1, 1)
def test_pca_check_projection(setup):
# Test that the projection of data is correct
rng = np.random.RandomState(0)
n, p = 100, 3
X = mt.tensor(rng.randn(n, p) * .1)
X[:10] += mt.array([3, 4, 5])
Xt = 0.1 * mt.tensor(rng.randn(1, p)) + mt.array([3, 4, 5])
for solver in solver_list:
Yt = PCA(n_components=2, svd_solver=solver).fit(X).transform(Xt)
Yt /= mt.sqrt((Yt ** 2).sum())
assert_almost_equal(mt.abs(Yt[0][0]).to_numpy(), 1., 1)
def test_pca_zero_noise_variance_edge_cases(self):
# ensure that noise_variance_ is 0 in edge cases
# when n_components == min(n_samples, n_features)
n, p = 100, 3
rng = np.random.RandomState(0)
X = mt.tensor(rng.randn(n, p) * .1) + mt.array([3, 4, 5])
# arpack raises ValueError for n_components == min(n_samples,
# n_features)
svd_solvers = ['full', 'randomized']
for svd_solver in svd_solvers:
pca = PCA(svd_solver=svd_solver, n_components=p)
pca.fit(X)
self.assertEqual(pca.noise_variance_, 0)
pca.fit(X.T)
self.assertEqual(pca.noise_variance_, 0)
def _check_pca_int_dtype_upcast_to_double(svd_solver):
# Ensure that all int types will be upcast to float64
X_i64 = mt.tensor(np.random.RandomState(0).randint(0, 1000, (1000, 4)))
X_i64 = X_i64.astype(np.int64, copy=False)
X_i32 = X_i64.astype(np.int32, copy=False)
pca_64 = PCA(n_components=3, svd_solver=svd_solver,
random_state=0).fit(X_i64)
pca_32 = PCA(n_components=3, svd_solver=svd_solver,
random_state=0).fit(X_i32)
assert pca_64.components_.dtype == np.float64
assert pca_32.components_.dtype == np.float64
assert pca_64.transform(X_i64).dtype == np.float64
assert pca_32.transform(X_i32).dtype == np.float64
assert_array_almost_equal(pca_64.components_.to_numpy(), pca_32.components_.to_numpy(),
decimal=5)
def _check_pca_float_dtype_preservation(svd_solver):
# Ensure that PCA does not upscale the dtype when input is float32
X_64 = mt.tensor(np.random.RandomState(0).rand(1000, 4).astype(np.float64,
copy=False))
X_32 = X_64.astype(np.float32)
pca_64 = PCA(n_components=3, svd_solver=svd_solver,
random_state=0).fit(X_64)
pca_32 = PCA(n_components=3, svd_solver=svd_solver,
random_state=0).fit(X_32)
assert pca_64.components_.dtype == np.float64
assert pca_32.components_.dtype == np.float32
assert pca_64.transform(X_64).dtype == np.float64
assert pca_32.transform(X_32).dtype == np.float32
# decimal=5 fails on mac with scipy = 1.1.0
assert_array_almost_equal(pca_64.components_.to_numpy(), pca_32.components_.to_numpy(),
decimal=4)
def testPCARandomizedSolver(self):
# PCA on dense arrays
X = self.iris
# Loop excluding the 0, invalid for randomized
for n_comp in np.arange(1, X.shape[1]):
pca = PCA(n_components=n_comp,
svd_solver='randomized',
random_state=0)
X_r = pca.fit(X).transform(X)
np.testing.assert_equal(X_r.shape[1], n_comp)
X_r2 = pca.fit_transform(X)
assert_array_almost_equal(X_r.fetch(), X_r2.fetch())
X_r = pca.transform(X)
assert_array_almost_equal(X_r.fetch(), X_r2.fetch())
# Test get_covariance and get_precision
cov = pca.get_covariance()
precision = pca.get_precision()
assert_array_almost_equal(
mt.dot(cov, precision).to_numpy(),
mt.eye(X.shape[1]).to_numpy(), 12)
pca = PCA(n_components=0, svd_solver='randomized', random_state=0)
with self.assertRaises(ValueError):
pca.fit(X)
pca = PCA(n_components=0, svd_solver='randomized', random_state=0)
with self.assertRaises(ValueError):
pca.fit(X)
# Check internal state
self.assertEqual(
pca.n_components,
PCA(n_components=0, svd_solver='randomized',
random_state=0).n_components)
self.assertEqual(
pca.svd_solver,
PCA(n_components=0, svd_solver='randomized',
random_state=0).svd_solver)
def test_pca_bad_solver(self):
X = mt.tensor(np.random.RandomState(0).rand(5, 4))
pca = PCA(n_components=3, svd_solver='bad_argument')
with self.assertRaises(ValueError):
pca.fit(X)
def test_svd_solver_auto(self):
rng = np.random.RandomState(0)
X = mt.tensor(rng.uniform(size=(1000, 50)))
# case: n_components in (0,1) => 'full'
pca = PCA(n_components=.5)
pca.fit(X)
pca_test = PCA(n_components=.5, svd_solver='full')
pca_test.fit(X)
assert_array_almost_equal(pca.components_.to_numpy(),
pca_test.components_.to_numpy())
# case: max(X.shape) <= 500 => 'full'
pca = PCA(n_components=5, random_state=0)
Y = X[:10, :]
pca.fit(Y)
pca_test = PCA(n_components=5, svd_solver='full', random_state=0)
pca_test.fit(Y)
assert_array_almost_equal(pca.components_.to_numpy(),
pca_test.components_.to_numpy())
# case: n_components >= .8 * min(X.shape) => 'full'
pca = PCA(n_components=50)
pca.fit(X)
pca_test = PCA(n_components=50, svd_solver='full')
pca_test.fit(X)
assert_array_almost_equal(pca.components_.to_numpy(),
pca_test.components_.to_numpy())
# n_components >= 1 and n_components < .8 * min(X.shape) => 'randomized'
pca = PCA(n_components=10, random_state=0)
pca.fit(X)
pca_test = PCA(n_components=10,
svd_solver='randomized',
random_state=0)
pca_test.fit(X)
assert_array_almost_equal(pca.components_.to_numpy(),
pca_test.components_.to_numpy())
def test_pca_score2(self):
# Test that probabilistic PCA correctly separated different datasets
n, p = 100, 3
rng = np.random.RandomState(0)
X = mt.tensor(rng.randn(n, p) * .1) + mt.array([3, 4, 5])
for solver in self.solver_list:
pca = PCA(n_components=2, svd_solver=solver)
pca.fit(X)
ll1 = pca.score(X)
ll2 = pca.score(
mt.tensor(rng.randn(n, p) * .2) + mt.array([3, 4, 5]))
self.assertGreater(ll1.fetch(), ll2.fetch())
# Test that it gives different scores if whiten=True
pca = PCA(n_components=2, whiten=True, svd_solver=solver)
pca.fit(X)
ll2 = pca.score(X)
assert ll1.fetch() > ll2.fetch()
def testPCA(self):
X = self.iris
for n_comp in np.arange(X.shape[1]):
pca = PCA(n_components=n_comp, svd_solver='full')
X_r = pca.fit(X).transform(X).fetch()
np.testing.assert_equal(X_r.shape[1], n_comp)
X_r2 = pca.fit_transform(X).fetch()
assert_array_almost_equal(X_r, X_r2)
X_r = pca.transform(X).fetch()
X_r2 = pca.fit_transform(X).fetch()
assert_array_almost_equal(X_r, X_r2)
# Test get_covariance and get_precision
cov = pca.get_covariance()
precision = pca.get_precision()
assert_array_almost_equal(
mt.dot(cov, precision).to_numpy(), np.eye(X.shape[1]), 12)
# test explained_variance_ratio_ == 1 with all components
pca = PCA(svd_solver='full')
pca.fit(X)
np.testing.assert_allclose(
pca.explained_variance_ratio_.sum().to_numpy(), 1.0, 3)
def test_singular_values(self):
# Check that the PCA output has the correct singular values
rng = np.random.RandomState(0)
n_samples = 100
n_features = 80
X = mt.tensor(rng.randn(n_samples, n_features))
pca = PCA(n_components=2, svd_solver='full', random_state=rng).fit(X)
rpca = PCA(n_components=2, svd_solver='randomized',
random_state=rng).fit(X)
assert_array_almost_equal(pca.singular_values_.fetch(),
rpca.singular_values_.fetch(), 1)
# Compare to the Frobenius norm
X_pca = pca.transform(X)
X_rpca = rpca.transform(X)
assert_array_almost_equal(
mt.sum(pca.singular_values_**2.0).to_numpy(),
(mt.linalg.norm(X_pca, "fro")**2.0).to_numpy(), 12)
assert_array_almost_equal(
mt.sum(rpca.singular_values_**2.0).to_numpy(),
(mt.linalg.norm(X_rpca, "fro")**2.0).to_numpy(), 0)
# Compare to the 2-norms of the score vectors
assert_array_almost_equal(
pca.singular_values_.fetch(),
mt.sqrt(mt.sum(X_pca**2.0, axis=0)).to_numpy(), 12)
assert_array_almost_equal(
rpca.singular_values_.fetch(),
mt.sqrt(mt.sum(X_rpca**2.0, axis=0)).to_numpy(), 2)
# Set the singular values and see what we get back
rng = np.random.RandomState(0)
n_samples = 100
n_features = 110
X = mt.tensor(rng.randn(n_samples, n_features))
pca = PCA(n_components=3, svd_solver='full', random_state=rng)
rpca = PCA(n_components=3, svd_solver='randomized', random_state=rng)
X_pca = pca.fit_transform(X)
X_pca /= mt.sqrt(mt.sum(X_pca**2.0, axis=0))
X_pca[:, 0] *= 3.142
X_pca[:, 1] *= 2.718
X_hat = mt.dot(X_pca, pca.components_)
pca.fit(X_hat)
rpca.fit(X_hat)
assert_array_almost_equal(pca.singular_values_.fetch(),
[3.142, 2.718, 1.0], 14)
assert_array_almost_equal(rpca.singular_values_.fetch(),
[3.142, 2.718, 1.0], 14)
