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_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)
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 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_pca_randomized_solver(setup):
# PCA on dense arrays
X = 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 pytest.raises(ValueError):
pca.fit(X)
pca = PCA(n_components=0, svd_solver='randomized', random_state=0)
with pytest.raises(ValueError):
pca.fit(X)
# Check internal state
assert pca.n_components == PCA(n_components=0,
svd_solver='randomized', random_state=0).n_components
assert pca.svd_solver == PCA(n_components=0,
svd_solver='randomized', random_state=0).svd_solver
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 _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 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)
