test_pca.py

import numpy as np
import scipy as sp
from itertools import product

import pytest

from sklearn.utils.testing import assert_almost_equal
from sklearn.utils.testing import assert_array_almost_equal
from sklearn.utils.testing import assert_equal
from sklearn.utils.testing import assert_greater
from sklearn.utils.testing import assert_raise_message
from sklearn.utils.testing import assert_raises
from sklearn.utils.testing import assert_raises_regex
from sklearn.utils.testing import assert_no_warnings
from sklearn.utils.testing import ignore_warnings
from sklearn.utils.testing import assert_less

from sklearn import datasets
from sklearn.decomposition import PCA
from sklearn.decomposition.pca import _assess_dimension_
from sklearn.decomposition.pca import _infer_dimension_

iris = datasets.load_iris()
solver_list = ['full', 'arpack', 'randomized', 'auto']


def test_pca():
    # PCA on dense arrays
    X = iris.data

    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)
        np.testing.assert_equal(X_r.shape[1], n_comp)

        X_r2 = pca.fit_transform(X)
        assert_array_almost_equal(X_r, X_r2)

        X_r = pca.transform(X)
        X_r2 = pca.fit_transform(X)
        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(np.dot(cov, precision),
                                  np.eye(X.shape[1]), 12)

    # test explained_variance_ratio_ == 1 with all components
    pca = PCA(svd_solver='full')
    pca.fit(X)
    assert_almost_equal(pca.explained_variance_ratio_.sum(), 1.0, 3)


def test_pca_arpack_solver():
    # PCA on dense arrays
    X = iris.data
    d = X.shape[1]

    # Loop excluding the extremes, invalid inputs for arpack
    for n_comp in np.arange(1, d):
        pca = PCA(n_components=n_comp, svd_solver='arpack', 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, X_r2)

        X_r = pca.transform(X)
        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(np.dot(cov, precision),
                                  np.eye(d), 12)

    pca = PCA(n_components=0, svd_solver='arpack', random_state=0)
    assert_raises(ValueError, pca.fit, X)
    # Check internal state
    assert_equal(pca.n_components,
                 PCA(n_components=0,
                     svd_solver='arpack', random_state=0).n_components)
    assert_equal(pca.svd_solver,
                 PCA(n_components=0,
                     svd_solver='arpack', random_state=0).svd_solver)

    pca = PCA(n_components=d, svd_solver='arpack', random_state=0)
    assert_raises(ValueError, pca.fit, X)
    assert_equal(pca.n_components,
                 PCA(n_components=d,
                     svd_solver='arpack', random_state=0).n_components)
    assert_equal(pca.svd_solver,
                 PCA(n_components=0,
                     svd_solver='arpack', random_state=0).svd_solver)


def test_pca_randomized_solver():
    # PCA on dense arrays
    X = iris.data

    # 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, X_r2)

        X_r = pca.transform(X)
        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(np.dot(cov, precision),
                                  np.eye(X.shape[1]), 12)

    pca = PCA(n_components=0, svd_solver='randomized', random_state=0)
    assert_raises(ValueError, pca.fit, X)

    pca = PCA(n_components=0, svd_solver='randomized', random_state=0)
    assert_raises(ValueError, pca.fit, X)
    # Check internal state
    assert_equal(pca.n_components,
                 PCA(n_components=0,
                     svd_solver='randomized', random_state=0).n_components)
    assert_equal(pca.svd_solver,
                 PCA(n_components=0,
                     svd_solver='randomized', random_state=0).svd_solver)


def test_no_empty_slice_warning():
    # test if we avoid numpy warnings for computing over empty arrays
    n_components = 10
    n_features = n_components + 2  # anything > n_comps triggered it in 0.16
    X = np.random.uniform(-1, 1, size=(n_components, n_features))
    pca = PCA(n_components=n_components)
    assert_no_warnings(pca.fit, X)


def test_whitening():
    # 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 = np.dot(rng.randn(n_samples, rank),
               np.dot(np.diag(np.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

    assert_equal(X.shape, (n_samples, n_features))

    # the component-wise variance is thus highly varying:
    assert_greater(X.std(axis=0).std(), 43.8)

    for solver, copy in product(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())
        assert_equal(X_whitened.shape, (n_samples, n_components))
        X_whitened2 = pca.transform(X_)
        assert_array_almost_equal(X_whitened, X_whitened2)

        assert_almost_equal(X_whitened.std(ddof=1, axis=0),
                            np.ones(n_components),
                            decimal=6)
        assert_almost_equal(X_whitened.mean(axis=0), 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_)
        assert_equal(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(), 74.1, 1)
        # we always center, so no test for non-centering.


# Ignore warnings from switching to more power iterations in randomized_svd
@ignore_warnings
def test_explained_variance():
    # Check that PCA output has unit-variance
    rng = np.random.RandomState(0)
    n_samples = 100
    n_features = 80

    X = rng.randn(n_samples, n_features)

    pca = PCA(n_components=2, svd_solver='full').fit(X)
    apca = PCA(n_components=2, svd_solver='arpack', random_state=0).fit(X)
    assert_array_almost_equal(pca.explained_variance_,
                              apca.explained_variance_, 1)
    assert_array_almost_equal(pca.explained_variance_ratio_,
                              apca.explained_variance_ratio_, 3)

    rpca = PCA(n_components=2, svd_solver='randomized', random_state=42).fit(X)
    assert_array_almost_equal(pca.explained_variance_,
                              rpca.explained_variance_, 1)
    assert_array_almost_equal(pca.explained_variance_ratio_,
                              rpca.explained_variance_ratio_, 1)

    # compare to empirical variances
    expected_result = np.linalg.eig(np.cov(X, rowvar=False))[0]
    expected_result = sorted(expected_result, reverse=True)[:2]

    X_pca = pca.transform(X)
    assert_array_almost_equal(pca.explained_variance_,
                              np.var(X_pca, ddof=1, axis=0))
    assert_array_almost_equal(pca.explained_variance_, expected_result)

    X_pca = apca.transform(X)
    assert_array_almost_equal(apca.explained_variance_,
                              np.var(X_pca, ddof=1, axis=0))
    assert_array_almost_equal(apca.explained_variance_, expected_result)

    X_rpca = rpca.transform(X)
    assert_array_almost_equal(rpca.explained_variance_,
                              np.var(X_rpca, ddof=1, axis=0),
                              decimal=1)
    assert_array_almost_equal(rpca.explained_variance_,
                              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]

    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_,
                              rpca.explained_variance_ratio_, 5)


def test_singular_values():
    # Check that the PCA output has the correct singular values

    rng = np.random.RandomState(0)
    n_samples = 100
    n_features = 80

    X = rng.randn(n_samples, n_features)

    pca = PCA(n_components=2, svd_solver='full',
              random_state=rng).fit(X)
    apca = PCA(n_components=2, svd_solver='arpack',
               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_, apca.singular_values_, 12)
    assert_array_almost_equal(pca.singular_values_, rpca.singular_values_, 1)
    assert_array_almost_equal(apca.singular_values_, rpca.singular_values_, 1)

    # Compare to the Frobenius norm
    X_pca = pca.transform(X)
    X_apca = apca.transform(X)
    X_rpca = rpca.transform(X)
    assert_array_almost_equal(np.sum(pca.singular_values_**2.0),
                              np.linalg.norm(X_pca, "fro")**2.0, 12)
    assert_array_almost_equal(np.sum(apca.singular_values_**2.0),
                              np.linalg.norm(X_apca, "fro")**2.0, 9)
    assert_array_almost_equal(np.sum(rpca.singular_values_**2.0),
                              np.linalg.norm(X_rpca, "fro")**2.0, 0)

    # Compare to the 2-norms of the score vectors
    assert_array_almost_equal(pca.singular_values_,
                              np.sqrt(np.sum(X_pca**2.0, axis=0)), 12)
    assert_array_almost_equal(apca.singular_values_,
                              np.sqrt(np.sum(X_apca**2.0, axis=0)), 12)
    assert_array_almost_equal(rpca.singular_values_,
                              np.sqrt(np.sum(X_rpca**2.0, axis=0)), 2)

    # Set the singular values and see what we get back
    rng = np.random.RandomState(0)
    n_samples = 100
    n_features = 110

    X = rng.randn(n_samples, n_features)

    pca = PCA(n_components=3, svd_solver='full', random_state=rng)
    apca = PCA(n_components=3, svd_solver='arpack', random_state=rng)
    rpca = PCA(n_components=3, svd_solver='randomized', random_state=rng)
    X_pca = pca.fit_transform(X)

    X_pca /= np.sqrt(np.sum(X_pca**2.0, axis=0))
    X_pca[:, 0] *= 3.142
    X_pca[:, 1] *= 2.718

    X_hat = np.dot(X_pca, pca.components_)
    pca.fit(X_hat)
    apca.fit(X_hat)
    rpca.fit(X_hat)
    assert_array_almost_equal(pca.singular_values_, [3.142, 2.718, 1.0], 14)
    assert_array_almost_equal(apca.singular_values_, [3.142, 2.718, 1.0], 14)
    assert_array_almost_equal(rpca.singular_values_, [3.142, 2.718, 1.0], 14)


def test_pca_check_projection():
    # Test that the projection of data is correct
    rng = np.random.RandomState(0)
    n, p = 100, 3
    X = rng.randn(n, p) * .1
    X[:10] += np.array([3, 4, 5])
    Xt = 0.1 * rng.randn(1, p) + np.array([3, 4, 5])

    for solver in solver_list:
        Yt = PCA(n_components=2, svd_solver=solver).fit(X).transform(Xt)
        Yt /= np.sqrt((Yt ** 2).sum())

        assert_almost_equal(np.abs(Yt[0][0]), 1., 1)


def test_pca_inverse():
    # Test that the projection of data can be inverted
    rng = np.random.RandomState(0)
    n, p = 50, 3
    X = 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, Y_inverse, decimal=3)

    # same as above with whitening (approximate reconstruction)
    for solver in 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, Y_inverse, decimal=3)


@pytest.mark.parametrize('solver', solver_list)
def test_pca_validation(solver):
    # Ensures that solver-specific extreme inputs for the n_components
    # parameter raise errors
    X = np.array([[0, 1, 0], [1, 0, 0]])
    smallest_d = 2  # The smallest dimension
    lower_limit = {'randomized': 1, 'arpack': 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)
        if solver == 'arpack':

            n_components = smallest_d

            assert_raises_regex(ValueError,
                                "n_components={}L? must be "
                                "strictly less than "
                                r"min\(n_samples, n_features\)={}L?"
                                " with svd_solver=\'arpack\'"
                                .format(n_components, smallest_d),
                                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)


@pytest.mark.parametrize('solver', solver_list)
def test_n_components_none(solver):
    # Ensures that n_components == None is handled correctly
    X = iris.data
    # 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)
        if solver == 'arpack':
            assert_equal(pca.n_components_, min(data.shape) - 1)
        else:
            assert_equal(pca.n_components_, min(data.shape))


def test_randomized_pca_check_projection():
    # Test that the projection by randomized PCA on dense data is correct
    rng = np.random.RandomState(0)
    n, p = 100, 3
    X = rng.randn(n, p) * .1
    X[:10] += np.array([3, 4, 5])
    Xt = 0.1 * rng.randn(1, p) + np.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(np.abs(Yt[0][0]), 1., 1)


def test_randomized_pca_check_list():
    # Test that the projection by randomized PCA on list data is correct
    X = [[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_equal(X_transformed.shape, (2, 1))
    assert_almost_equal(X_transformed.mean(), 0.00, 2)
    assert_almost_equal(X_transformed.std(), 0.71, 2)


def test_randomized_pca_inverse():
    # Test that randomized PCA is inversible on dense data
    rng = np.random.RandomState(0)
    n, p = 50, 3
    X = 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, Y_inverse, 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 = (np.abs(X - Y_inverse) / np.abs(X).mean()).max()
    assert_less(relative_max_delta, 1e-5)


def test_n_components_mle():
    # 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 = rng.randn(n_samples, n_features)
    n_components_dict = {}
    for solver in 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)
    assert_equal(n_components_dict['auto'], n_components_dict['full'])


def test_pca_dim():
    # Check automated dimensionality setting
    rng = np.random.RandomState(0)
    n, p = 100, 5
    X = rng.randn(n, p) * .1
    X[:10] += np.array([3, 4, 5, 1, 2])
    pca = PCA(n_components='mle', svd_solver='full').fit(X)
    assert_equal(pca.n_components, 'mle')
    assert_equal(pca.n_components_, 1)


def test_infer_dim_1():
    # TODO: explain what this is testing
    # Or at least use explicit variable names...
    n, p = 1000, 5
    rng = np.random.RandomState(0)
    X = (rng.randn(n, p) * .1 + rng.randn(n, 1) * np.array([3, 4, 5, 1, 2]) +
         np.array([1, 0, 7, 4, 6]))
    pca = PCA(n_components=p, svd_solver='full')
    pca.fit(X)
    spect = pca.explained_variance_
    ll = np.array([_assess_dimension_(spect, k, n, p) for k in range(p)])
    assert_greater(ll[1], ll.max() - .01 * n)


def test_infer_dim_2():
    # TODO: explain what this is testing
    # Or at least use explicit variable names...
    n, p = 1000, 5
    rng = np.random.RandomState(0)
    X = rng.randn(n, p) * .1
    X[:10] += np.array([3, 4, 5, 1, 2])
    X[10:20] += np.array([6, 0, 7, 2, -1])
    pca = PCA(n_components=p, svd_solver='full')
    pca.fit(X)
    spect = pca.explained_variance_
    assert_greater(_infer_dimension_(spect, n, p), 1)


def test_infer_dim_3():
    n, p = 100, 5
    rng = np.random.RandomState(0)
    X = rng.randn(n, p) * .1
    X[:10] += np.array([3, 4, 5, 1, 2])
    X[10:20] += np.array([6, 0, 7, 2, -1])
    X[30:40] += 2 * np.array([-1, 1, -1, 1, -1])
    pca = PCA(n_components=p, svd_solver='full')
    pca.fit(X)
    spect = pca.explained_variance_
    assert_greater(_infer_dimension_(spect, n, p), 2)


def test_infer_dim_by_explained_variance():
    X = iris.data
    pca = PCA(n_components=0.95, svd_solver='full')
    pca.fit(X)
    assert_equal(pca.n_components, 0.95)
    assert_equal(pca.n_components_, 2)

    pca = PCA(n_components=0.01, svd_solver='full')
    pca.fit(X)
    assert_equal(pca.n_components, 0.01)
    assert_equal(pca.n_components_, 1)

    rng = np.random.RandomState(0)
    # more features than samples
    X = rng.rand(5, 20)
    pca = PCA(n_components=.5, svd_solver='full').fit(X)
    assert_equal(pca.n_components, 0.5)
    assert_equal(pca.n_components_, 2)


def test_pca_score():
    # Test that probabilistic PCA scoring yields a reasonable score
    n, p = 1000, 3
    rng = np.random.RandomState(0)
    X = rng.randn(n, p) * .1 + np.array([3, 4, 5])
    for solver in solver_list:
        pca = PCA(n_components=2, svd_solver=solver)
        pca.fit(X)
        ll1 = pca.score(X)
        h = -0.5 * np.log(2 * np.pi * np.exp(1) * 0.1 ** 2) * p
        np.testing.assert_almost_equal(ll1 / h, 1, 0)


def test_pca_score2():
    # Test that probabilistic PCA correctly separated different datasets
    n, p = 100, 3
    rng = np.random.RandomState(0)
    X = rng.randn(n, p) * .1 + np.array([3, 4, 5])
    for solver in solver_list:
        pca = PCA(n_components=2, svd_solver=solver)
        pca.fit(X)
        ll1 = pca.score(X)
        ll2 = pca.score(rng.randn(n, p) * .2 + np.array([3, 4, 5]))
        assert_greater(ll1, ll2)

        # 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 > ll2


def test_pca_score3():
    # Check that probabilistic PCA selects the right model
    n, p = 200, 3
    rng = np.random.RandomState(0)
    Xl = (rng.randn(n, p) + rng.randn(n, 1) * np.array([3, 4, 5]) +
          np.array([1, 0, 7]))
    Xt = (rng.randn(n, p) + rng.randn(n, 1) * np.array([3, 4, 5]) +
          np.array([1, 0, 7]))
    ll = np.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() == 1


def test_pca_score_with_different_solvers():
    digits = datasets.load_digits()
    X_digits = digits.data

    pca_dict = {svd_solver: PCA(n_components=30, svd_solver=svd_solver,
                                random_state=0)
                for svd_solver in 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 np.all((pca.explained_variance_ - pca.noise_variance_) >= 0)

    # Compare scores with different svd_solvers
    score_dict = {svd_solver: pca.score(X_digits)
                  for svd_solver, pca in pca_dict.items()}
    assert_almost_equal(score_dict['full'], score_dict['arpack'])
    assert_almost_equal(score_dict['full'], score_dict['randomized'],
                        decimal=3)


def test_pca_zero_noise_variance_edge_cases():
    # 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 = rng.randn(n, p) * .1 + np.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)
        assert pca.noise_variance_ == 0

        pca.fit(X.T)
        assert pca.noise_variance_ == 0


def test_svd_solver_auto():
    rng = np.random.RandomState(0)
    X = 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_, pca_test.components_)

    # 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_, pca_test.components_)

    # 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_, pca_test.components_)

    # 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_, pca_test.components_)


@pytest.mark.parametrize('svd_solver', solver_list)
def test_pca_sparse_input(svd_solver):
    X = np.random.RandomState(0).rand(5, 4)
    X = sp.sparse.csr_matrix(X)
    assert(sp.sparse.issparse(X))

    pca = PCA(n_components=3, svd_solver=svd_solver)

    assert_raises(TypeError, pca.fit, X)


def test_pca_bad_solver():
    X = np.random.RandomState(0).rand(5, 4)
    pca = PCA(n_components=3, svd_solver='bad_argument')
    assert_raises(ValueError, pca.fit, X)


@pytest.mark.parametrize('svd_solver', solver_list)
def test_pca_dtype_preservation(svd_solver):
    check_pca_float_dtype_preservation(svd_solver)
    check_pca_int_dtype_upcast_to_double(svd_solver)


def check_pca_float_dtype_preservation(svd_solver):
    # Ensure that PCA does not upscale the dtype when input is float32
    X_64 = np.random.RandomState(0).rand(1000, 4).astype(np.float64)
    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_, pca_32.components_,
                              decimal=4)


def check_pca_int_dtype_upcast_to_double(svd_solver):
    # Ensure that all int types will be upcast to float64
    X_i64 = np.random.RandomState(0).randint(0, 1000, (1000, 4))
    X_i64 = X_i64.astype(np.int64)
    X_i32 = X_i64.astype(np.int32)

    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_, pca_32.components_,
                              decimal=5)