Python numpy.cov() Examples

def PCA_components(x):
    """
    Principal Component Analysis helper to check out eigenvalues of components.

    **Args:**

    * `x` : input matrix (2d array), every row represents new sample

    **Returns:**
    
    * `components`: sorted array of principal components eigenvalues 
        
    """ 
    # validate inputs
    try:    
        x = np.array(x)
    except:
        raise ValueError('Impossible to convert x to a numpy array.')
    # eigen values and eigen vectors of data covariance matrix
    eigen_values, eigen_vectors = np.linalg.eig(np.cov(x.T))
    # sort eigen vectors according biggest eigen value
    eigen_order = eigen_vectors.T[(-eigen_values).argsort()]
    # form output - order the eigenvalues
    return eigen_values[(-eigen_values).argsort()] 

def test_sample(self):
        with self.fixed_randomness_session(233):
            def test_sample_with(seed):
                mean, cov, cov_chol = self._gen_test_params(seed)
                dst = MultivariateNormalCholesky(
                    tf.constant(mean), tf.constant(cov_chol))
                n_exp = 20000
                samples = dst.sample(n_exp)
                sample_shape = (n_exp, 10, 11, 3)
                self.assertEqual(samples.shape.as_list(), list(sample_shape))
                samples = dst.sample(n_exp).eval()
                self.assertEqual(samples.shape, sample_shape)
                self.assertAllClose(
                    np.mean(samples, axis=0), mean, rtol=5e-2, atol=5e-2)
                for i in range(10):
                    for j in range(11):
                        self.assertAllClose(
                            np.cov(samples[:, i, j, :].T), cov[i, j],
                            rtol=1e-1, atol=1e-1)

            for seed in [23, 233, 2333]:
                test_sample_with(seed) 

def test_expanding_cov_diff_index(self):
        # GH 7512
        s1 = Series([1, 2, 3], index=[0, 1, 2])
        s2 = Series([1, 3], index=[0, 2])
        result = s1.expanding().cov(s2)
        expected = Series([None, None, 2.0])
        tm.assert_series_equal(result, expected)

        s2a = Series([1, None, 3], index=[0, 1, 2])
        result = s1.expanding().cov(s2a)
        tm.assert_series_equal(result, expected)

        s1 = Series([7, 8, 10], index=[0, 1, 3])
        s2 = Series([7, 9, 10], index=[0, 2, 3])
        result = s1.expanding().cov(s2)
        expected = Series([None, None, None, 4.5])
        tm.assert_series_equal(result, expected) 

def test_rolling_functions_window_non_shrinkage_binary(self):

        # corr/cov return a MI DataFrame
        df = DataFrame([[1, 5], [3, 2], [3, 9], [-1, 0]],
                       columns=Index(['A', 'B'], name='foo'),
                       index=Index(range(4), name='bar'))
        df_expected = DataFrame(
            columns=Index(['A', 'B'], name='foo'),
            index=pd.MultiIndex.from_product([df.index, df.columns],
                                             names=['bar', 'foo']),
            dtype='float64')
        functions = [lambda x: (x.rolling(window=10, min_periods=5)
                                .cov(x, pairwise=True)),
                     lambda x: (x.rolling(window=10, min_periods=5)
                                .corr(x, pairwise=True))]
        for f in functions:
            df_result = f(df)
            tm.assert_frame_equal(df_result, df_expected) 

def test_2d_with_missing(self):
        # Test cov on 2D variable w/ missing value
        x = self.data
        x[-1] = masked
        x = x.reshape(3, 4)
        valid = np.logical_not(getmaskarray(x)).astype(int)
        frac = np.dot(valid, valid.T)
        xf = (x - x.mean(1)[:, None]).filled(0)
        assert_almost_equal(cov(x),
                            np.cov(xf) * (x.shape[1] - 1) / (frac - 1.))
        assert_almost_equal(cov(x, bias=True),
                            np.cov(xf, bias=True) * x.shape[1] / frac)
        frac = np.dot(valid.T, valid)
        xf = (x - x.mean(0)).filled(0)
        assert_almost_equal(cov(x, rowvar=False),
                            (np.cov(xf, rowvar=False) *
                             (x.shape[0] - 1) / (frac - 1.)))
        assert_almost_equal(cov(x, rowvar=False, bias=True),
                            (np.cov(xf, rowvar=False, bias=True) *
                             x.shape[0] / frac)) 

def test_1d_with_missing(self):
        # Test cov 1 1D variable w/missing values
        x = self.data
        x[-1] = masked
        x -= x.mean()
        nx = x.compressed()
        assert_almost_equal(np.cov(nx), cov(x))
        assert_almost_equal(np.cov(nx, rowvar=False), cov(x, rowvar=False))
        assert_almost_equal(np.cov(nx, rowvar=False, bias=True),
                            cov(x, rowvar=False, bias=True))
        #
        try:
            cov(x, allow_masked=False)
        except ValueError:
            pass
        #
        # 2 1D variables w/ missing values
        nx = x[1:-1]
        assert_almost_equal(np.cov(nx, nx[::-1]), cov(x, x[::-1]))
        assert_almost_equal(np.cov(nx, nx[::-1], rowvar=False),
                            cov(x, x[::-1], rowvar=False))
        assert_almost_equal(np.cov(nx, nx[::-1], rowvar=False, bias=True),
                            cov(x, x[::-1], rowvar=False, bias=True)) 

def testCovExecution(self):
        data = np.array([[0, 2], [1, 1], [2, 0]]).T
        x = tensor(data, chunk_size=1)

        t = cov(x)

        res = self.executor.execute_tensor(t, concat=True)[0]
        expected = np.cov(data)
        np.testing.assert_equal(res, expected)

        data_x = [-2.1, -1, 4.3]
        data_y = [3, 1.1, 0.12]
        x = tensor(data_x, chunk_size=1)
        y = tensor(data_y, chunk_size=1)

        X = stack((x, y), axis=0)
        t = cov(x, y)
        r = tall(t == cov(X))
        self.assertTrue(self.executor.execute_tensor(r)[0]) 

def test_expanding_cov_pairwise_diff_length(self):
        # GH 7512
        df1 = DataFrame([[1, 5], [3, 2], [3, 9]],
                        columns=Index(['A', 'B'], name='foo'))
        df1a = DataFrame([[1, 5], [3, 9]],
                         index=[0, 2],
                         columns=Index(['A', 'B'], name='foo'))
        df2 = DataFrame([[5, 6], [None, None], [2, 1]],
                        columns=Index(['X', 'Y'], name='foo'))
        df2a = DataFrame([[5, 6], [2, 1]],
                         index=[0, 2],
                         columns=Index(['X', 'Y'], name='foo'))
        # TODO: xref gh-15826
        # .loc is not preserving the names
        result1 = df1.expanding().cov(df2a, pairwise=True).loc[2]
        result2 = df1.expanding().cov(df2a, pairwise=True).loc[2]
        result3 = df1a.expanding().cov(df2, pairwise=True).loc[2]
        result4 = df1a.expanding().cov(df2a, pairwise=True).loc[2]
        expected = DataFrame([[-3.0, -6.0], [-5.0, -10.0]],
                             columns=Index(['A', 'B'], name='foo'),
                             index=Index(['X', 'Y'], name='foo'))
        tm.assert_frame_equal(result1, expected)
        tm.assert_frame_equal(result2, expected)
        tm.assert_frame_equal(result3, expected)
        tm.assert_frame_equal(result4, expected) 

def test_1d_with_missing(self):
        # Test cov 1 1D variable w/missing values
        x = self.data
        x[-1] = masked
        x -= x.mean()
        nx = x.compressed()
        assert_almost_equal(np.cov(nx), cov(x))
        assert_almost_equal(np.cov(nx, rowvar=False), cov(x, rowvar=False))
        assert_almost_equal(np.cov(nx, rowvar=False, bias=True),
                            cov(x, rowvar=False, bias=True))
        #
        try:
            cov(x, allow_masked=False)
        except ValueError:
            pass
        #
        # 2 1D variables w/ missing values
        nx = x[1:-1]
        assert_almost_equal(np.cov(nx, nx[::-1]), cov(x, x[::-1]))
        assert_almost_equal(np.cov(nx, nx[::-1], rowvar=False),
                            cov(x, x[::-1], rowvar=False))
        assert_almost_equal(np.cov(nx, nx[::-1], rowvar=False, bias=True),
                            cov(x, x[::-1], rowvar=False, bias=True)) 

def test_2d_with_missing(self):
        # Test cov on 2D variable w/ missing value
        x = self.data
        x[-1] = masked
        x = x.reshape(3, 4)
        valid = np.logical_not(getmaskarray(x)).astype(int)
        frac = np.dot(valid, valid.T)
        xf = (x - x.mean(1)[:, None]).filled(0)
        assert_almost_equal(cov(x),
                            np.cov(xf) * (x.shape[1] - 1) / (frac - 1.))
        assert_almost_equal(cov(x, bias=True),
                            np.cov(xf, bias=True) * x.shape[1] / frac)
        frac = np.dot(valid.T, valid)
        xf = (x - x.mean(0)).filled(0)
        assert_almost_equal(cov(x, rowvar=False),
                            (np.cov(xf, rowvar=False) *
                             (x.shape[0] - 1) / (frac - 1.)))
        assert_almost_equal(cov(x, rowvar=False, bias=True),
                            (np.cov(xf, rowvar=False, bias=True) *
                             x.shape[0] / frac)) 

def test_expanding_corr_cov(self):
        g = self.frame.groupby('A')
        r = g.expanding()

        for f in ['corr', 'cov']:
            result = getattr(r, f)(self.frame)

            def func(x):
                return getattr(x.expanding(), f)(self.frame)
            expected = g.apply(func)
            tm.assert_frame_equal(result, expected)

            result = getattr(r.B, f)(pairwise=True)

            def func(x):
                return getattr(x.B.expanding(), f)(pairwise=True)
            expected = g.apply(func)
            tm.assert_series_equal(result, expected) 

def calculate_activation_statistics(
    dataloader, device="cuda", num_images=500, real_dataset=False
):
    """Calculate the activation statistics for a dataset.

    Args:
        dataloader: Dataloader of data from which to obtain activations.
        device: to perform the evaluation (e.g. 'cuda' for GPU).
        num_images: number of images to evaluate.
        real_dataset: bool (whether the dataset is real or generated).

    Returns:
        Mean activations (np.array), std of activations (np.array).
    """
    act = get_activations(
        dataloader,
        device,
        dims=2048,
        num_images=num_images,
        real_dataset=real_dataset,
    )
    mu = np.mean(act, axis=0)
    sigma = np.cov(act, rowvar=False)
    return mu, sigma 

def calculate_activation_statistics(images, model, batch_size=50,#
                                    dims=2048, cuda=False, verbose=False):
    """Calculation of the statistics used by the FID.
    Params:
    -- images      : Numpy array of dimension (n_images, 3, hi, wi). The values
                     must lie between 0 and 1.
    -- model       : Instance of inception model
    -- batch_size  : The images numpy array is split into batches with
                     batch size batch_size. A reasonable batch size
                     depends on the hardware.
    -- dims        : Dimensionality of features returned by Inception
    -- cuda        : If set to True, use GPU
    -- verbose     : If set to True and parameter out_step is given, the
                     number of calculated batches is reported.
    Returns:
    -- mu    : The mean over samples of the activations of the pool_3 layer of
               the inception model.
    -- sigma : The covariance matrix of the activations of the pool_3 layer of
               the inception model.
    """
    act = get_activations(images, model, batch_size, dims, cuda, verbose)
    mu = np.mean(act, axis=0)
    sigma = np.cov(act, rowvar=False)
    return mu, sigma 

def calculate_activation_statistics(pointclouds, model, batch_size=100,
                                    dims=1808, device=None, verbose=False):
    """Calculation of the statistics used by the FID.
    Params:
    -- pointcloud       : pytorch Tensor of pointclouds.
    -- model       : Instance of inception model
    -- batch_size  : The images numpy array is split into batches with
                     batch size batch_size. A reasonable batch size
                     depends on the hardware.
    -- dims        : Dimensionality of features returned by Inception
    -- device      : If set to device, use GPU
    -- verbose     : If set to True and parameter out_step is given, the
                     number of calculated batches is reported.
    Returns:
    -- mu    : The mean over samples of the activations of the pool_3 layer of
               the inception model.
    -- sigma : The covariance matrix of the activations of the pool_3 layer of
               the inception model.
    """
    act = get_activations(pointclouds, model, batch_size, dims, device, verbose)
    mu = np.mean(act, axis=0)
    sigma = np.cov(act, rowvar=False)
    return mu, sigma 

def _internal_tell_candidate(self, candidate: p.Parameter, value: float) -> None:
        self.children.append(candidate)
        if self._POPSIZE_ADAPTATION:
            self.popsize.add_value(value)
        if len(self.children) >= self.popsize.llambda:
            self.children = sorted(self.children, key=lambda c: c.loss)
            population_data = [c.get_standardized_data(reference=self.parametrization) for c in self.children]
            mu = self.popsize.mu
            arrays = population_data[:mu]
            # covariance
            # TODO: check actual covariance that should be used
            centered_arrays = np.array([x - self.current_center for x in arrays])
            cov = centered_arrays.T.dot(centered_arrays)
            # cov = np.cov(np.array(population_data).T)
            mem_factor = 0.9 if self._COVARIANCE_MEMORY else 0
            self.covariance *= mem_factor
            self.covariance += (1 - mem_factor) * cov
            # Computing the new parent
            self.current_center = sum(arrays) / mu  # type: ignore
            self.sigma = np.exp(sum([np.log(c._meta["sigma"]) for c in self.children[:mu]]) / mu)
            self.parents = self.children[:mu]
            self.children = [] 

def test_rolling_cov_offset(self):
        # GH16058

        idx = pd.date_range('2017-01-01', periods=24, freq='1h')
        ss = Series(np.arange(len(idx)), index=idx)

        result = ss.rolling('2h').cov()
        expected = Series([np.nan] + [0.5] * (len(idx) - 1), index=idx)
        tm.assert_series_equal(result, expected)

        expected2 = ss.rolling(2, min_periods=1).cov()
        tm.assert_series_equal(result, expected2)

        result = ss.rolling('3h').cov()
        expected = Series([np.nan, 0.5] + [1.0] * (len(idx) - 2), index=idx)
        tm.assert_series_equal(result, expected)

        expected2 = ss.rolling(3, min_periods=1).cov()
        tm.assert_series_equal(result, expected2) 

def calculate_activation_statistics(images, model, batch_size=64, dims=2048, device=None):
    """Calculation of the statistics used by the FID.
    Params:
    -- images      : Numpy array of dimension (n_images, 3, hi, wi). The values
                     must lie between 0 and 1.
    -- model       : Instance of inception model
    -- batch_size  : The images numpy array is split into batches with
                     batch size batch_size. A reasonable batch size
                     depends on the hardware.
    -- dims        : Dimensionality of features returned by Inception
    -- device      : If set to True, use GPU
    -- verbose     : If set to True and parameter out_step is given, the
                     number of calculated batches is reported.
    Returns:
    -- mu    : The mean over samples of the activations of the pool_3 layer of
               the inception model.
    -- sigma : The covariance matrix of the activations of the pool_3 layer of
               the inception model.
    """
    act = get_activations(images, model, batch_size, dims, device)
    mu = np.mean(act, axis=0)
    sigma = np.cov(act, rowvar=False)
    return mu, sigma 

def test_1d_w_missing(self):
        # Test cov 1 1D variable w/missing values
        x = self.data
        x[-1] = masked
        x -= x.mean()
        nx = x.compressed()
        assert_almost_equal(np.cov(nx), cov(x))
        assert_almost_equal(np.cov(nx, rowvar=False), cov(x, rowvar=False))
        assert_almost_equal(np.cov(nx, rowvar=False, bias=True),
                            cov(x, rowvar=False, bias=True))
        #
        try:
            cov(x, allow_masked=False)
        except ValueError:
            pass
        #
        # 2 1D variables w/ missing values
        nx = x[1:-1]
        assert_almost_equal(np.cov(nx, nx[::-1]), cov(x, x[::-1]))
        assert_almost_equal(np.cov(nx, nx[::-1], rowvar=False),
                            cov(x, x[::-1], rowvar=False))
        assert_almost_equal(np.cov(nx, nx[::-1], rowvar=False, bias=True),
                            cov(x, x[::-1], rowvar=False, bias=True)) 

def test_2d_w_missing(self):
        # Test cov on 2D variable w/ missing value
        x = self.data
        x[-1] = masked
        x = x.reshape(3, 4)
        valid = np.logical_not(getmaskarray(x)).astype(int)
        frac = np.dot(valid, valid.T)
        xf = (x - x.mean(1)[:, None]).filled(0)
        assert_almost_equal(cov(x),
                            np.cov(xf) * (x.shape[1] - 1) / (frac - 1.))
        assert_almost_equal(cov(x, bias=True),
                            np.cov(xf, bias=True) * x.shape[1] / frac)
        frac = np.dot(valid.T, valid)
        xf = (x - x.mean(0)).filled(0)
        assert_almost_equal(cov(x, rowvar=False),
                            (np.cov(xf, rowvar=False) *
                             (x.shape[0] - 1) / (frac - 1.)))
        assert_almost_equal(cov(x, rowvar=False, bias=True),
                            (np.cov(xf, rowvar=False, bias=True) *
                             x.shape[0] / frac)) 

def get_ma_dist(A, B):
    Y = A.copy()
    X = B.copy()
    
    S = np.cov(X.T)
    try:
        SI = np.linalg.inv(S)
    except:
        print("Singular Matrix: using np.linalg.pinv")
        SI = np.linalg.pinv(S)
    mu = np.mean(X, axis=0)
    
    diff = Y - mu
    Dct_c = np.diag(diff @ SI @ diff.T)
    
    return Dct_c 

def test_prob(self):
        with self.fixed_randomness_session(233):
            def test_prob_with(seed):
                mean, cov, cov_chol = self._gen_test_params(seed)
                dst = MultivariateNormalCholesky(
                    tf.constant(mean), tf.constant(cov_chol),
                    check_numerics=True)
                n_exp = 200
                samples = dst.sample(n_exp).eval()
                log_pdf = dst.log_prob(tf.constant(samples))
                pdf_shape = (n_exp, 10, 11)
                self.assertEqual(log_pdf.shape.as_list(), list(pdf_shape))
                log_pdf = log_pdf.eval()
                self.assertEqual(log_pdf.shape, pdf_shape)
                for i in range(10):
                    for j in range(11):
                        log_pdf_exact = stats.multivariate_normal.logpdf(
                                samples[:, i, j, :], mean[i, j], cov[i, j])
                        self.assertAllClose(
                            log_pdf_exact, log_pdf[:, i, j])
                self.assertAllClose(
                    np.exp(log_pdf), dst.prob(tf.constant(samples)).eval())

            for seed in [23, 233, 2333]:
                test_prob_with(seed) 

def nancov(a, b, min_periods=None):
    if len(a) != len(b):
        raise AssertionError('Operands to nancov must have same size')

    if min_periods is None:
        min_periods = 1

    valid = notna(a) & notna(b)
    if not valid.all():
        a = a[valid]
        b = b[valid]

    if len(a) < min_periods:
        return np.nan

    return np.cov(a, b)[0, 1] 

def test_shape_inference(self):
        with self.session(use_gpu=True):
            # Static
            mean = 10 * np.random.normal(size=(10, 11, 2)).astype('d')
            cov = np.zeros((10, 11, 2, 2))
            dst = MultivariateNormalCholesky(
                tf.constant(mean), tf.constant(cov))
            self.assertEqual(dst.get_batch_shape().as_list(), [10, 11])
            self.assertEqual(dst.get_value_shape().as_list(), [2])
            # Dynamic
            unk_mean = tf.placeholder(tf.float32, None)
            unk_cov = tf.placeholder(tf.float32, None)
            dst = MultivariateNormalCholesky(unk_mean, unk_cov)
            self.assertEqual(dst.get_value_shape().as_list(), [None])
            feed_dict = {unk_mean: np.ones(2), unk_cov: np.eye(2)}
            self.assertEqual(list(dst.batch_shape.eval(feed_dict)), [])
            self.assertEqual(list(dst.value_shape.eval(feed_dict)), [2]) 

def test_cholesky_and_cholesky_grad_shape():
    if not imported_scipy:
        raise SkipTest("Scipy needed for the Cholesky op.")

    rng = numpy.random.RandomState(utt.fetch_seed())
    x = tensor.matrix()
    for l in (cholesky(x), Cholesky(lower=True)(x), Cholesky(lower=False)(x)):
        f_chol = theano.function([x], l.shape)
        g = tensor.grad(l.sum(), x)
        f_cholgrad = theano.function([x], g.shape)
        topo_chol = f_chol.maker.fgraph.toposort()
        topo_cholgrad = f_cholgrad.maker.fgraph.toposort()
        if config.mode != 'FAST_COMPILE':
            assert sum([node.op.__class__ == Cholesky
                        for node in topo_chol]) == 0
            assert sum([node.op.__class__ == CholeskyGrad
                        for node in topo_cholgrad]) == 0
        for shp in [2, 3, 5]:
            m = numpy.cov(rng.randn(shp, shp + 10)).astype(config.floatX)
            yield numpy.testing.assert_equal, f_chol(m), (shp, shp)
            yield numpy.testing.assert_equal, f_cholgrad(m), (shp, shp) 

def test_sample_reparameterized(self):
        mean, cov, cov_chol = self._gen_test_params(23)
        mean = tf.constant(mean)
        cov_chol = tf.constant(cov_chol)
        mvn_rep = MultivariateNormalCholesky(mean, cov_chol)
        samples = mvn_rep.sample(tf.placeholder(tf.int32, shape=[]))
        mean_grads, cov_grads = tf.gradients(samples, [mean, cov_chol])
        self.assertTrue(mean_grads is not None)
        self.assertTrue(cov_grads is not None)
        mvn_no_rep = MultivariateNormalCholesky(
                mean, cov_chol, is_reparameterized=False)
        samples = mvn_no_rep.sample(tf.placeholder(tf.int32, shape=[]))
        mean_grads, cov_grads = tf.gradients(samples, [mean, cov_chol])
        self.assertEqual(mean_grads, None)
        self.assertEqual(cov_grads, None) 

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_cov_parameters(self, level=rlevel):
        # Ticket #91
        x = np.random.random((3, 3))
        y = x.copy()
        np.cov(x, rowvar=1)
        np.cov(y, rowvar=0)
        assert_array_equal(x, y) 

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_2d_wo_missing(self):
        # Test cov on 1 2D variable w/o missing values
        x = self.data.reshape(3, 4)
        assert_almost_equal(np.cov(x), cov(x))
        assert_almost_equal(np.cov(x, rowvar=False), cov(x, rowvar=False))
        assert_almost_equal(np.cov(x, rowvar=False, bias=True),
                            cov(x, rowvar=False, bias=True)) 

def test_naive_covariance(self):
        assert_allclose(covar(self.d[:, 1:], method='naive'),
                                   np.cov(self.d[:, 1:], rowvar=False))

        assert_allclose(covar(self.d[:, 1:3], self.d[:, 3:], 'naive'),
                                   np.cov(self.d[:, 1:], rowvar=False))