content: # Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>
  #         Fabian Pedregosa <fabian.pedregosa@inria.fr>
  #
  # License: BSD 3 clause
  
  import numpy as np
  from scipy import sparse
  from scipy import linalg
  from itertools import product
  
  
  from sklearn.utils.testing import assert_array_almost_equal
  from sklearn.utils.testing import assert_almost_equal
  from sklearn.utils.testing import assert_equal
  from sklearn.utils.testing import ignore_warnings
  
  from sklearn.linear_model.base import LinearRegression
  from sklearn.linear_model.base import _preprocess_data
  from sklearn.linear_model.base import sparse_center_data, center_data
  from sklearn.linear_model.base import _rescale_data
  from sklearn.utils import check_random_state
  from sklearn.utils.testing import assert_greater
  from sklearn.datasets.samples_generator import make_sparse_uncorrelated
  from sklearn.datasets.samples_generator import make_regression
  
  rng = np.random.RandomState(0)
  
  
  def test_linear_regression():
      # Test LinearRegression on a simple dataset.
      # a simple dataset
      X = [[1], [2]]
      Y = [1, 2]
  
      reg = LinearRegression()
      reg.fit(X, Y)
  
      assert_array_almost_equal(reg.coef_, [1])
      assert_array_almost_equal(reg.intercept_, [0])
      assert_array_almost_equal(reg.predict(X), [1, 2])
  
      # test it also for degenerate input
      X = [[1]]
      Y = [0]
  
      reg = LinearRegression()
      reg.fit(X, Y)
      assert_array_almost_equal(reg.coef_, [0])
      assert_array_almost_equal(reg.intercept_, [0])
      assert_array_almost_equal(reg.predict(X), [0])
  
  
  def test_linear_regression_sample_weights():
      # TODO: loop over sparse data as well
  
      rng = np.random.RandomState(0)
  
      # It would not work with under-determined systems
      for n_samples, n_features in ((6, 5), ):
  
          y = rng.randn(n_samples)
          X = rng.randn(n_samples, n_features)
          sample_weight = 1.0 + rng.rand(n_samples)
  
          for intercept in (True, False):
  
              # LinearRegression with explicit sample_weight
              reg = LinearRegression(fit_intercept=intercept)
              reg.fit(X, y, sample_weight=sample_weight)
              coefs1 = reg.coef_
              inter1 = reg.intercept_
  
              assert_equal(reg.coef_.shape, (X.shape[1], ))  # sanity checks
              assert_greater(reg.score(X, y), 0.5)
  
              # Closed form of the weighted least square
              # theta = (X^T W X)^(-1) * X^T W y
              W = np.diag(sample_weight)
              if intercept is False:
                  X_aug = X
              else:
                  dummy_column = np.ones(shape=(n_samples, 1))
                  X_aug = np.concatenate((dummy_column, X), axis=1)
  
              coefs2 = linalg.solve(X_aug.T.dot(W).dot(X_aug),
                                    X_aug.T.dot(W).dot(y))
  
              if intercept is False:
                  assert_array_almost_equal(coefs1, coefs2)
              else:
                  assert_array_almost_equal(coefs1, coefs2[1:])
                  assert_almost_equal(inter1, coefs2[0])
  
  
  def test_raises_value_error_if_sample_weights_greater_than_1d():
      # Sample weights must be either scalar or 1D
  
      n_sampless = [2, 3]
      n_featuress = [3, 2]
  
      for n_samples, n_features in zip(n_sampless, n_featuress):
          X = rng.randn(n_samples, n_features)
          y = rng.randn(n_samples)
          sample_weights_OK = rng.randn(n_samples) ** 2 + 1
          sample_weights_OK_1 = 1.
          sample_weights_OK_2 = 2.
  
          reg = LinearRegression()
  
          # make sure the "OK" sample weights actually work
          reg.fit(X, y, sample_weights_OK)
          reg.fit(X, y, sample_weights_OK_1)
          reg.fit(X, y, sample_weights_OK_2)
  
  
  def test_fit_intercept():
      # Test assertions on betas shape.
      X2 = np.array([[0.38349978, 0.61650022],
                     [0.58853682, 0.41146318]])
      X3 = np.array([[0.27677969, 0.70693172, 0.01628859],
                     [0.08385139, 0.20692515, 0.70922346]])
      y = np.array([1, 1])
  
      lr2_without_intercept = LinearRegression(fit_intercept=False).fit(X2, y)
      lr2_with_intercept = LinearRegression(fit_intercept=True).fit(X2, y)
  
      lr3_without_intercept = LinearRegression(fit_intercept=False).fit(X3, y)
      lr3_with_intercept = LinearRegression(fit_intercept=True).fit(X3, y)
  
      assert_equal(lr2_with_intercept.coef_.shape,
                   lr2_without_intercept.coef_.shape)
      assert_equal(lr3_with_intercept.coef_.shape,
                   lr3_without_intercept.coef_.shape)
      assert_equal(lr2_without_intercept.coef_.ndim,
                   lr3_without_intercept.coef_.ndim)
  
  
  def test_linear_regression_sparse(random_state=0):
      # Test that linear regression also works with sparse data
      random_state = check_random_state(random_state)
      for i in range(10):
          n = 100
          X = sparse.eye(n, n)
          beta = random_state.rand(n)
          y = X * beta[:, np.newaxis]
  
          ols = LinearRegression()
          ols.fit(X, y.ravel())
          assert_array_almost_equal(beta, ols.coef_ + ols.intercept_)
  
          assert_array_almost_equal(ols.predict(X) - y.ravel(), 0)
  
  
  def test_linear_regression_multiple_outcome(random_state=0):
      # Test multiple-outcome linear regressions
      X, y = make_regression(random_state=random_state)
  
      Y = np.vstack((y, y)).T
      n_features = X.shape[1]
  
      reg = LinearRegression(fit_intercept=True)
      reg.fit((X), Y)
      assert_equal(reg.coef_.shape, (2, n_features))
      Y_pred = reg.predict(X)
      reg.fit(X, y)
      y_pred = reg.predict(X)
      assert_array_almost_equal(np.vstack((y_pred, y_pred)).T, Y_pred, decimal=3)
  
  
  def test_linear_regression_sparse_multiple_outcome(random_state=0):
      # Test multiple-outcome linear regressions with sparse data
      random_state = check_random_state(random_state)
      X, y = make_sparse_uncorrelated(random_state=random_state)
      X = sparse.coo_matrix(X)
      Y = np.vstack((y, y)).T
      n_features = X.shape[1]
  
      ols = LinearRegression()
      ols.fit(X, Y)
      assert_equal(ols.coef_.shape, (2, n_features))
      Y_pred = ols.predict(X)
      ols.fit(X, y.ravel())
      y_pred = ols.predict(X)
      assert_array_almost_equal(np.vstack((y_pred, y_pred)).T, Y_pred, decimal=3)
  
  
  def test_preprocess_data():
      n_samples = 200
      n_features = 2
      X = rng.rand(n_samples, n_features)
      y = rng.rand(n_samples)
      expected_X_mean = np.mean(X, axis=0)
      expected_X_norm = np.std(X, axis=0) * np.sqrt(X.shape[0])
      expected_y_mean = np.mean(y, axis=0)
  
      Xt, yt, X_mean, y_mean, X_norm = \
          _preprocess_data(X, y, fit_intercept=False, normalize=False)
      assert_array_almost_equal(X_mean, np.zeros(n_features))
      assert_array_almost_equal(y_mean, 0)
      assert_array_almost_equal(X_norm, np.ones(n_features))
      assert_array_almost_equal(Xt, X)
      assert_array_almost_equal(yt, y)
  
      Xt, yt, X_mean, y_mean, X_norm = \
          _preprocess_data(X, y, fit_intercept=True, normalize=False)
      assert_array_almost_equal(X_mean, expected_X_mean)
      assert_array_almost_equal(y_mean, expected_y_mean)
      assert_array_almost_equal(X_norm, np.ones(n_features))
      assert_array_almost_equal(Xt, X - expected_X_mean)
      assert_array_almost_equal(yt, y - expected_y_mean)
  
      Xt, yt, X_mean, y_mean, X_norm = \
          _preprocess_data(X, y, fit_intercept=True, normalize=True)
      assert_array_almost_equal(X_mean, expected_X_mean)
      assert_array_almost_equal(y_mean, expected_y_mean)
      assert_array_almost_equal(X_norm, expected_X_norm)
      assert_array_almost_equal(Xt, (X - expected_X_mean) / expected_X_norm)
      assert_array_almost_equal(yt, y - expected_y_mean)
  
  
  def test_preprocess_data_multioutput():
      n_samples = 200
      n_features = 3
      n_outputs = 2
      X = rng.rand(n_samples, n_features)
      y = rng.rand(n_samples, n_outputs)
      expected_y_mean = np.mean(y, axis=0)
  
      args = [X, sparse.csc_matrix(X)]
      for X in args:
          _, yt, _, y_mean, _ = _preprocess_data(X, y, fit_intercept=False,
                                                 normalize=False)
          assert_array_almost_equal(y_mean, np.zeros(n_outputs))
          assert_array_almost_equal(yt, y)
  
          _, yt, _, y_mean, _ = _preprocess_data(X, y, fit_intercept=True,
                                                 normalize=False)
          assert_array_almost_equal(y_mean, expected_y_mean)
          assert_array_almost_equal(yt, y - y_mean)
  
          _, yt, _, y_mean, _ = _preprocess_data(X, y, fit_intercept=True,
                                                 normalize=True)
          assert_array_almost_equal(y_mean, expected_y_mean)
          assert_array_almost_equal(yt, y - y_mean)
  
  
  def test_preprocess_data_weighted():
      n_samples = 200
      n_features = 2
      X = rng.rand(n_samples, n_features)
      y = rng.rand(n_samples)
      sample_weight = rng.rand(n_samples)
      expected_X_mean = np.average(X, axis=0, weights=sample_weight)
      expected_y_mean = np.average(y, axis=0, weights=sample_weight)
  
      # XXX: if normalize=True, should we expect a weighted standard deviation?
      #      Currently not weighted, but calculated with respect to weighted mean
      expected_X_norm = (np.sqrt(X.shape[0]) *
                         np.mean((X - expected_X_mean) ** 2, axis=0) ** .5)
  
      Xt, yt, X_mean, y_mean, X_norm = \
          _preprocess_data(X, y, fit_intercept=True, normalize=False,
                           sample_weight=sample_weight)
      assert_array_almost_equal(X_mean, expected_X_mean)
      assert_array_almost_equal(y_mean, expected_y_mean)
      assert_array_almost_equal(X_norm, np.ones(n_features))
      assert_array_almost_equal(Xt, X - expected_X_mean)
      assert_array_almost_equal(yt, y - expected_y_mean)
  
      Xt, yt, X_mean, y_mean, X_norm = \
          _preprocess_data(X, y, fit_intercept=True, normalize=True,
                           sample_weight=sample_weight)
      assert_array_almost_equal(X_mean, expected_X_mean)
      assert_array_almost_equal(y_mean, expected_y_mean)
      assert_array_almost_equal(X_norm, expected_X_norm)
      assert_array_almost_equal(Xt, (X - expected_X_mean) / expected_X_norm)
      assert_array_almost_equal(yt, y - expected_y_mean)
  
  
  def test_sparse_preprocess_data_with_return_mean():
      n_samples = 200
      n_features = 2
      # random_state not supported yet in sparse.rand
      X = sparse.rand(n_samples, n_features, density=.5)  # , random_state=rng
      X = X.tolil()
      y = rng.rand(n_samples)
      XA = X.toarray()
      expected_X_norm = np.std(XA, axis=0) * np.sqrt(X.shape[0])
  
      Xt, yt, X_mean, y_mean, X_norm = \
          _preprocess_data(X, y, fit_intercept=False, normalize=False,
                           return_mean=True)
      assert_array_almost_equal(X_mean, np.zeros(n_features))
      assert_array_almost_equal(y_mean, 0)
      assert_array_almost_equal(X_norm, np.ones(n_features))
      assert_array_almost_equal(Xt.A, XA)
      assert_array_almost_equal(yt, y)
  
      Xt, yt, X_mean, y_mean, X_norm = \
          _preprocess_data(X, y, fit_intercept=True, normalize=False,
                           return_mean=True)
      assert_array_almost_equal(X_mean, np.mean(XA, axis=0))
      assert_array_almost_equal(y_mean, np.mean(y, axis=0))
      assert_array_almost_equal(X_norm, np.ones(n_features))
      assert_array_almost_equal(Xt.A, XA)
      assert_array_almost_equal(yt, y - np.mean(y, axis=0))
  
      Xt, yt, X_mean, y_mean, X_norm = \
          _preprocess_data(X, y, fit_intercept=True, normalize=True,
                           return_mean=True)
      assert_array_almost_equal(X_mean, np.mean(XA, axis=0))
      assert_array_almost_equal(y_mean, np.mean(y, axis=0))
      assert_array_almost_equal(X_norm, expected_X_norm)
      assert_array_almost_equal(Xt.A, XA / expected_X_norm)
      assert_array_almost_equal(yt, y - np.mean(y, axis=0))
  
  
  def test_csr_preprocess_data():
      # Test output format of _preprocess_data, when input is csr
      X, y = make_regression()
      X[X < 2.5] = 0.0
      csr = sparse.csr_matrix(X)
      csr_, y, _, _, _ = _preprocess_data(csr, y, True)
      assert_equal(csr_.getformat(), 'csr')
  
  
  def test_rescale_data():
      n_samples = 200
      n_features = 2
  
      sample_weight = 1.0 + rng.rand(n_samples)
      X = rng.rand(n_samples, n_features)
      y = rng.rand(n_samples)
      rescaled_X, rescaled_y = _rescale_data(X, y, sample_weight)
      rescaled_X2 = X * np.sqrt(sample_weight)[:, np.newaxis]
      rescaled_y2 = y * np.sqrt(sample_weight)
      assert_array_almost_equal(rescaled_X, rescaled_X2)
      assert_array_almost_equal(rescaled_y, rescaled_y2)
  
  
  @ignore_warnings  # all deprecation warnings
  def test_deprecation_center_data():
      n_samples = 200
      n_features = 2
  
      w = 1.0 + rng.rand(n_samples)
      X = rng.rand(n_samples, n_features)
      y = rng.rand(n_samples)
  
      param_grid = product([True, False], [True, False], [True, False],
                           [None, w])
  
      for (fit_intercept, normalize, copy, sample_weight) in param_grid:
  
          XX = X.copy()  # such that we can try copy=False as well
  
          X1, y1, X1_mean, X1_var, y1_mean = \
              center_data(XX, y, fit_intercept=fit_intercept,
                          normalize=normalize, copy=copy,
                          sample_weight=sample_weight)
  
          XX = X.copy()
  
          X2, y2, X2_mean, X2_var, y2_mean = \
              _preprocess_data(XX, y, fit_intercept=fit_intercept,
                               normalize=normalize, copy=copy,
                               sample_weight=sample_weight)
  
          assert_array_almost_equal(X1, X2)
          assert_array_almost_equal(y1, y2)
          assert_array_almost_equal(X1_mean, X2_mean)
          assert_array_almost_equal(X1_var, X2_var)
          assert_array_almost_equal(y1_mean, y2_mean)
  
      # Sparse cases
      X = sparse.csr_matrix(X)
  
      for (fit_intercept, normalize, copy, sample_weight) in param_grid:
  
          X1, y1, X1_mean, X1_var, y1_mean = \
              center_data(X, y, fit_intercept=fit_intercept, normalize=normalize,
                          copy=copy, sample_weight=sample_weight)
  
          X2, y2, X2_mean, X2_var, y2_mean = \
              _preprocess_data(X, y, fit_intercept=fit_intercept,
                               normalize=normalize, copy=copy,
                               sample_weight=sample_weight, return_mean=False)
  
          assert_array_almost_equal(X1.toarray(), X2.toarray())
          assert_array_almost_equal(y1, y2)
          assert_array_almost_equal(X1_mean, X2_mean)
          assert_array_almost_equal(X1_var, X2_var)
          assert_array_almost_equal(y1_mean, y2_mean)
  
      for (fit_intercept, normalize) in product([True, False], [True, False]):
  
          X1, y1, X1_mean, X1_var, y1_mean = \
              sparse_center_data(X, y, fit_intercept=fit_intercept,
                                 normalize=normalize)
  
          X2, y2, X2_mean, X2_var, y2_mean = \
              _preprocess_data(X, y, fit_intercept=fit_intercept,
                               normalize=normalize, return_mean=True)
  
          assert_array_almost_equal(X1.toarray(), X2.toarray())
          assert_array_almost_equal(y1, y2)
          assert_array_almost_equal(X1_mean, X2_mean)
          assert_array_almost_equal(X1_var, X2_var)
          assert_array_almost_equal(y1_mean, y2_mean)
  