text: """
  Generalized Linear Models.
  """
  
  # Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>
  # Fabian Pedregosa <fabian.pedregosa@inria.fr>
  # Olivier Grisel <olivier.grisel@ensta.org>
  #         Vincent Michel <vincent.michel@inria.fr>
  #         Peter Prettenhofer <peter.prettenhofer@gmail.com>
  #         Mathieu Blondel <mathieu@mblondel.org>
  #         Lars Buitinck
  #         Maryan Morel <maryan.morel@polytechnique.edu>
  #         Giorgio Patrini <giorgio.patrini@anu.edu.au>
  #         Maria Telenczuk <https://github.com/maikia>
  # License: BSD 3 clause
  
  from abc import ABCMeta, abstractmethod
  import numbers
  import warnings
  
  import numpy as np
  import scipy.sparse as sp
  from scipy import linalg
  from scipy import optimize
  from scipy import sparse
  from scipy.special import expit
  from joblib import Parallel
  
  from ..base import (BaseEstimator, ClassifierMixin, RegressorMixin,
                      MultiOutputMixin)
  from ..preprocessing._data import _is_constant_feature
  from ..utils import check_array
  from ..utils.validation import FLOAT_DTYPES
  from ..utils.validation import _deprecate_positional_args
  from ..utils import check_random_state
  from ..utils.extmath import safe_sparse_dot
  from ..utils.extmath import _incremental_mean_and_var
  from ..utils.sparsefuncs import mean_variance_axis, inplace_column_scale
  from ..utils.fixes import sparse_lsqr
  from ..utils._seq_dataset import ArrayDataset32, CSRDataset32
  from ..utils._seq_dataset import ArrayDataset64, CSRDataset64
  from ..utils.validation import check_is_fitted, _check_sample_weight
  from ..utils.fixes import delayed
  
  # TODO: bayesian_ridge_regression and bayesian_regression_ard
  # should be squashed into its respective objects.
  
  SPARSE_INTERCEPT_DECAY = 0.01
  # For sparse data intercept updates are scaled by this decay factor to avoid
  # intercept oscillation.
  
  
  # FIXME in 1.2: parameter 'normalize' should be removed from linear models
  # in cases where now normalize=False. The default value of 'normalize' should
  # be changed to False in linear models where now normalize=True
  def _deprecate_normalize(normalize, default, estimator_name):
      """ Normalize is to be deprecated from linear models and a use of
      a pipeline with a StandardScaler is to be recommended instead.
      Here the appropriate message is selected to be displayed to the user
      depending on the default normalize value (as it varies between the linear
      models and normalize value selected by the user).
  
      Parameters
      ----------
      normalize : bool,
          normalize value passed by the user
  
      default : bool,
          default normalize value used by the estimator
  
      estimator_name : string,
          name of the linear estimator which calls this function.
          The name will be used for writing the deprecation warnings
  
      Returns
      -------
      normalize : bool,
          normalize value which should further be used by the estimator at this
          stage of the depreciation process
  
      Notes
      -----
      This function should be updated in 1.2 depending on the value of
      `normalize`:
      - True, warning: `normalize` was deprecated in 1.2 and will be removed in
        1.4. Suggest to use pipeline instead.
      - False, `normalize` was deprecated in 1.2 and it will be removed in 1.4.
        Leave normalize to its default value.
      - `deprecated` - this should only be possible with default == False as from
        1.2 `normalize` in all the linear models should be either removed or the
        default should be set to False.
      This function should be completely removed in 1.4.
      """
  
      if normalize not in [True, False, 'deprecated']:
          raise ValueError("Leave 'normalize' to its default value or set it "
                           "to True or False")
  
      if normalize == 'deprecated':
          _normalize = default
      else:
          _normalize = normalize
  
      pipeline_msg = (
          "If you wish to scale the data, use Pipeline with a StandardScaler "
          "in a preprocessing stage. To reproduce the previous behavior:\n\n"
          "from sklearn.pipeline import make_pipeline\n\n"
          "model = make_pipeline(StandardScaler(with_mean=False), "
          f"{estimator_name}())\n\n"
          "If you wish to pass a sample_weight parameter, you need to pass it "
          "as a fit parameter to each step of the pipeline as follows:\n\n"
          "kwargs = {s[0] + '__sample_weight': sample_weight for s "
          "in model.steps}\n"
          "model.fit(X, y, **kwargs)\n\n"
      )
  
      if estimator_name == 'Ridge' or estimator_name == 'RidgeClassifier':
          alpha_msg = 'Set parameter alpha to: original_alpha * n_samples. '
      elif 'Lasso' in estimator_name:
          alpha_msg = (
              'Set parameter alpha to: original_alpha * np.sqrt(n_samples). '
          )
      elif 'ElasticNet' in estimator_name:
          alpha_msg = (
              'Set parameter alpha to original_alpha * np.sqrt(n_samples) if '
              'l1_ratio is 1, and to original_alpha * n_samples if l1_ratio is '
              '0. For other values of l1_ratio, no analytic formula is '
              'available.'
          )
      elif estimator_name == 'RidgeCV' or estimator_name == 'RidgeClassifierCV':
          alpha_msg = 'Set parameter alphas to: original_alphas * n_samples. '
      else:
          alpha_msg = ""
  
      if default and normalize == 'deprecated':
          warnings.warn(
              "The default of 'normalize' will be set to False in version 1.2 "
              "and deprecated in version 1.4.\n" +
              pipeline_msg + alpha_msg,
              FutureWarning
          )
      elif normalize != 'deprecated' and normalize and not default:
          warnings.warn(
              "'normalize' was deprecated in version 1.0 and will be "
              "removed in 1.2.\n" +
              pipeline_msg + alpha_msg, FutureWarning
          )
      elif not normalize and not default:
          warnings.warn(
              "'normalize' was deprecated in version 1.0 and will be "
              "removed in 1.2. "
              "Please leave the normalize parameter to its default value to "
              "silence this warning. The default behavior of this estimator "
              "is to not do any normalization. If normalization is needed "
              "please use sklearn.preprocessing.StandardScaler instead.",
              FutureWarning
          )
  
      return _normalize
  
  
  def make_dataset(X, y, sample_weight, random_state=None):
      """Create ``Dataset`` abstraction for sparse and dense inputs.
  
      This also returns the ``intercept_decay`` which is different
      for sparse datasets.
  
      Parameters
      ----------
      X : array-like, shape (n_samples, n_features)
          Training data
  
      y : array-like, shape (n_samples, )
          Target values.
  
      sample_weight : numpy array of shape (n_samples,)
          The weight of each sample
  
      random_state : int, RandomState instance or None (default)
          Determines random number generation for dataset shuffling and noise.
          Pass an int for reproducible output across multiple function calls.
          See :term:`Glossary <random_state>`.
  
      Returns
      -------
      dataset
          The ``Dataset`` abstraction
      intercept_decay
          The intercept decay
      """
  
      rng = check_random_state(random_state)
      # seed should never be 0 in SequentialDataset64
      seed = rng.randint(1, np.iinfo(np.int32).max)
  
      if X.dtype == np.float32:
          CSRData = CSRDataset32
          ArrayData = ArrayDataset32
      else:
          CSRData = CSRDataset64
          ArrayData = ArrayDataset64
  
      if sp.issparse(X):
          dataset = CSRData(X.data, X.indptr, X.indices, y, sample_weight,
                            seed=seed)
          intercept_decay = SPARSE_INTERCEPT_DECAY
      else:
          X = np.ascontiguousarray(X)
          dataset = ArrayData(X, y, sample_weight, seed=seed)
          intercept_decay = 1.0
  
      return dataset, intercept_decay
  
  
  def _preprocess_data(X, y, fit_intercept, normalize=False, copy=True,
                       sample_weight=None, return_mean=False, check_input=True):
      """Center and scale data.
  
      Centers data to have mean zero along axis 0. If fit_intercept=False or if
      the X is a sparse matrix, no centering is done, but normalization can still
      be applied. The function returns the statistics necessary to reconstruct
      the input data, which are X_offset, y_offset, X_scale, such that the output
  
          X = (X - X_offset) / X_scale
  
      X_scale is the L2 norm of X - X_offset. If sample_weight is not None,
      then the weighted mean of X and y is zero, and not the mean itself. If
      return_mean=True, the mean, eventually weighted, is returned, independently
      of whether X was centered (option used for optimization with sparse data in
      coordinate_descend).
  
      This is here because nearly all linear models will want their data to be
      centered. This function also systematically makes y consistent with X.dtype
      """
      if isinstance(sample_weight, numbers.Number):
          sample_weight = None
      if sample_weight is not None:
          sample_weight = np.asarray(sample_weight)
  
      if check_input:
          X = check_array(X, copy=copy, accept_sparse=['csr', 'csc'],
                          dtype=FLOAT_DTYPES)
      elif copy:
          if sp.issparse(X):
              X = X.copy()
          else:
              X = X.copy(order='K')
  
      y = np.asarray(y, dtype=X.dtype)
  
      if fit_intercept:
          if sp.issparse(X):
              X_offset, X_var = mean_variance_axis(
                  X, axis=0, weights=sample_weight
              )
              if not return_mean:
                  X_offset[:] = X.dtype.type(0)
          else:
              if normalize:
                  X_offset, X_var, _ = _incremental_mean_and_var(
                      X, last_mean=0., last_variance=0., last_sample_count=0.,
                      sample_weight=sample_weight
                  )
              else:
                  X_offset = np.average(X, axis=0, weights=sample_weight)
  
              X_offset = X_offset.astype(X.dtype, copy=False)
              X -= X_offset
  
          if normalize:
              X_var = X_var.astype(X.dtype, copy=False)
              # Detect constant features on the computed variance, before taking
              # the np.sqrt. Otherwise constant features cannot be detected with
              # sample weights.
              constant_mask = _is_constant_feature(X_var, X_offset, X.shape[0])
              X_var *= X.shape[0]
              X_scale = np.sqrt(X_var, out=X_var)
              X_scale[constant_mask] = 1.
              if sp.issparse(X):
                  inplace_column_scale(X, 1. / X_scale)
              else:
                  X /= X_scale
          else:
              X_scale = np.ones(X.shape[1], dtype=X.dtype)
  
          y_offset = np.average(y, axis=0, weights=sample_weight)
          y = y - y_offset
      else:
          X_offset = np.zeros(X.shape[1], dtype=X.dtype)
          X_scale = np.ones(X.shape[1], dtype=X.dtype)
          if y.ndim == 1:
              y_offset = X.dtype.type(0)
          else:
              y_offset = np.zeros(y.shape[1], dtype=X.dtype)
  
      return X, y, X_offset, y_offset, X_scale
  
  
  # TODO: _rescale_data should be factored into _preprocess_data.
  # Currently, the fact that sag implements its own way to deal with
  # sample_weight makes the refactoring tricky.
  
  def _rescale_data(X, y, sample_weight):
      """Rescale data sample-wise by square root of sample_weight.
  
      For many linear models, this enables easy support for sample_weight.
  
      Returns
      -------
      X_rescaled : {array-like, sparse matrix}
  
      y_rescaled : {array-like, sparse matrix}
      """
      n_samples = X.shape[0]
      sample_weight = np.asarray(sample_weight)
      if sample_weight.ndim == 0:
          sample_weight = np.full(n_samples, sample_weight,
                                  dtype=sample_weight.dtype)
      sample_weight = np.sqrt(sample_weight)
      sw_matrix = sparse.dia_matrix((sample_weight, 0),
                                    shape=(n_samples, n_samples))
      X = safe_sparse_dot(sw_matrix, X)
      y = safe_sparse_dot(sw_matrix, y)
      return X, y
  
  
  class LinearModel(BaseEstimator, metaclass=ABCMeta):
      """Base class for Linear Models"""
  
      @abstractmethod
      def fit(self, X, y):
          """Fit model."""
  
      def _decision_function(self, X):
          check_is_fitted(self)
  
          X = self._validate_data(X, accept_sparse=['csr', 'csc', 'coo'],
                                  reset=False)
          return safe_sparse_dot(X, self.coef_.T,
                                 dense_output=True) + self.intercept_
  
      def predict(self, X):
          """
          Predict using the linear model.
  
          Parameters
          ----------
          X : array-like or sparse matrix, shape (n_samples, n_features)
              Samples.
  
          Returns
          -------
          C : array, shape (n_samples,)
              Returns predicted values.
          """
          return self._decision_function(X)
  
      _preprocess_data = staticmethod(_preprocess_data)
  
      def _set_intercept(self, X_offset, y_offset, X_scale):
          """Set the intercept_
          """
          if self.fit_intercept:
              self.coef_ = self.coef_ / X_scale
              self.intercept_ = y_offset - np.dot(X_offset, self.coef_.T)
          else:
              self.intercept_ = 0.
  
      def _more_tags(self):
          return {'requires_y': True}
  
  
  # XXX Should this derive from LinearModel? It should be a mixin, not an ABC.
  # Maybe the n_features checking can be moved to LinearModel.
  class LinearClassifierMixin(ClassifierMixin):
      """Mixin for linear classifiers.
  
      Handles prediction for sparse and dense X.
      """
  
      def decision_function(self, X):
          """
          Predict confidence scores for samples.
  
          The confidence score for a sample is proportional to the signed
          distance of that sample to the hyperplane.
  
          Parameters
          ----------
          X : array-like or sparse matrix, shape (n_samples, n_features)
              Samples.
  
          Returns
          -------
          array, shape=(n_samples,) if n_classes == 2 else (n_samples, n_classes)
              Confidence scores per (sample, class) combination. In the binary
              case, confidence score for self.classes_[1] where >0 means this
              class would be predicted.
          """
          check_is_fitted(self)
  
          X = self._validate_data(X, accept_sparse='csr', reset=False)
          scores = safe_sparse_dot(X, self.coef_.T,
                                   dense_output=True) + self.intercept_
          return scores.ravel() if scores.shape[1] == 1 else scores
  
      def predict(self, X):
          """
          Predict class labels for samples in X.
  
          Parameters
          ----------
          X : array-like or sparse matrix, shape (n_samples, n_features)
              Samples.
  
          Returns
          -------
          C : array, shape [n_samples]
              Predicted class label per sample.
          """
          scores = self.decision_function(X)
          if len(scores.shape) == 1:
              indices = (scores > 0).astype(int)
          else:
              indices = scores.argmax(axis=1)
          return self.classes_[indices]
  
      def _predict_proba_lr(self, X):
          """Probability estimation for OvR logistic regression.
  
          Positive class probabilities are computed as
          1. / (1. + np.exp(-self.decision_function(X)));
          multiclass is handled by normalizing that over all classes.
          """
          prob = self.decision_function(X)
          expit(prob, out=prob)
          if prob.ndim == 1:
              return np.vstack([1 - prob, prob]).T
          else:
              # OvR normalization, like LibLinear's predict_probability
              prob /= prob.sum(axis=1).reshape((prob.shape[0], -1))
              return prob
  
  
  class SparseCoefMixin:
      """Mixin for converting coef_ to and from CSR format.
  
      L1-regularizing estimators should inherit this.
      """
  
      def densify(self):
          """
          Convert coefficient matrix to dense array format.
  
          Converts the ``coef_`` member (back) to a numpy.ndarray. This is the
          default format of ``coef_`` and is required for fitting, so calling
          this method is only required on models that have previously been
          sparsified; otherwise, it is a no-op.
  
          Returns
          -------
          self
              Fitted estimator.
          """
          msg = "Estimator, %(name)s, must be fitted before densifying."
          check_is_fitted(self, msg=msg)
          if sp.issparse(self.coef_):
              self.coef_ = self.coef_.toarray()
          return self
  
      def sparsify(self):
          """
          Convert coefficient matrix to sparse format.
  
          Converts the ``coef_`` member to a scipy.sparse matrix, which for
          L1-regularized models can be much more memory- and storage-efficient
          than the usual numpy.ndarray representation.
  
          The ``intercept_`` member is not converted.
  
          Returns
          -------
          self
              Fitted estimator.
  
          Notes
          -----
          For non-sparse models, i.e. when there are not many zeros in ``coef_``,
          this may actually *increase* memory usage, so use this method with
          care. A rule of thumb is that the number of zero elements, which can
          be computed with ``(coef_ == 0).sum()``, must be more than 50% for this
          to provide significant benefits.
  
          After calling this method, further fitting with the partial_fit
          method (if any) will not work until you call densify.
          """
          msg = "Estimator, %(name)s, must be fitted before sparsifying."
          check_is_fitted(self, msg=msg)
          self.coef_ = sp.csr_matrix(self.coef_)
          return self
  
  
  class LinearRegression(MultiOutputMixin, RegressorMixin, LinearModel):
      """
      Ordinary least squares Linear Regression.
  
      LinearRegression fits a linear model with coefficients w = (w1, ..., wp)
      to minimize the residual sum of squares between the observed targets in
      the dataset, and the targets predicted by the linear approximation.
  
      Parameters
      ----------
      fit_intercept : bool, default=True
          Whether to calculate the intercept for this model. If set
          to False, no intercept will be used in calculations
          (i.e. data is expected to be centered).
  
      normalize : bool, default=False
          This parameter is ignored when ``fit_intercept`` is set to False.
          If True, the regressors X will be normalized before regression by
          subtracting the mean and dividing by the l2-norm.
          If you wish to standardize, please use
          :class:`~sklearn.preprocessing.StandardScaler` before calling ``fit``
          on an estimator with ``normalize=False``.
  
          .. deprecated:: 1.0
             `normalize` was deprecated in version 1.0 and will be
             removed in 1.2.
  
      copy_X : bool, default=True
          If True, X will be copied; else, it may be overwritten.
  
      n_jobs : int, default=None
          The number of jobs to use for the computation. This will only provide
          speedup for n_targets > 1 and sufficient large problems.
          ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
          ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
          for more details.
  
      positive : bool, default=False
          When set to ``True``, forces the coefficients to be positive. This
          option is only supported for dense arrays.
  
          .. versionadded:: 0.24
  
      Attributes
      ----------
      coef_ : array of shape (n_features, ) or (n_targets, n_features)
          Estimated coefficients for the linear regression problem.
          If multiple targets are passed during the fit (y 2D), this
          is a 2D array of shape (n_targets, n_features), while if only
          one target is passed, this is a 1D array of length n_features.
  
      rank_ : int
          Rank of matrix `X`. Only available when `X` is dense.
  
      singular_ : array of shape (min(X, y),)
          Singular values of `X`. Only available when `X` is dense.
  
      intercept_ : float or array of shape (n_targets,)
          Independent term in the linear model. Set to 0.0 if
          `fit_intercept = False`.
  
      See Also
      --------
      Ridge : Ridge regression addresses some of the
          problems of Ordinary Least Squares by imposing a penalty on the
          size of the coefficients with l2 regularization.
      Lasso : The Lasso is a linear model that estimates
          sparse coefficients with l1 regularization.
      ElasticNet : Elastic-Net is a linear regression
          model trained with both l1 and l2 -norm regularization of the
          coefficients.
  
      Notes
      -----
      From the implementation point of view, this is just plain Ordinary
      Least Squares (scipy.linalg.lstsq) or Non Negative Least Squares
      (scipy.optimize.nnls) wrapped as a predictor object.
  
      Examples
      --------
      >>> import numpy as np
      >>> from sklearn.linear_model import LinearRegression
      >>> X = np.array([[1, 1], [1, 2], [2, 2], [2, 3]])
      >>> # y = 1 * x_0 + 2 * x_1 + 3
      >>> y = np.dot(X, np.array([1, 2])) + 3
      >>> reg = LinearRegression().fit(X, y)
      >>> reg.score(X, y)
      1.0
      >>> reg.coef_
      array([1., 2.])
      >>> reg.intercept_
      3.0000...
      >>> reg.predict(np.array([[3, 5]]))
      array([16.])
      """
      @_deprecate_positional_args
      def __init__(self, *, fit_intercept=True, normalize='deprecated',
                   copy_X=True, n_jobs=None, positive=False):
          self.fit_intercept = fit_intercept
          self.normalize = normalize
          self.copy_X = copy_X
          self.n_jobs = n_jobs
          self.positive = positive
  
      def fit(self, X, y, sample_weight=None):
          """
          Fit linear model.
  
          Parameters
          ----------
          X : {array-like, sparse matrix} of shape (n_samples, n_features)
              Training data
  
          y : array-like of shape (n_samples,) or (n_samples, n_targets)
              Target values. Will be cast to X's dtype if necessary
  
          sample_weight : array-like of shape (n_samples,), default=None
              Individual weights for each sample
  
              .. versionadded:: 0.17
                 parameter *sample_weight* support to LinearRegression.
  
          Returns
          -------
          self : returns an instance of self.
          """
  
          _normalize = _deprecate_normalize(
              self.normalize, default=False,
              estimator_name=self.__class__.__name__
          )
  
          n_jobs_ = self.n_jobs
  
          accept_sparse = False if self.positive else ['csr', 'csc', 'coo']
  
          X, y = self._validate_data(X, y, accept_sparse=accept_sparse,
                                     y_numeric=True, multi_output=True)
  
          if sample_weight is not None:
              sample_weight = _check_sample_weight(sample_weight, X,
                                                   dtype=X.dtype)
  
          X, y, X_offset, y_offset, X_scale = self._preprocess_data(
              X, y, fit_intercept=self.fit_intercept, normalize=_normalize,
              copy=self.copy_X, sample_weight=sample_weight,
              return_mean=True)
  
          if sample_weight is not None:
              # Sample weight can be implemented via a simple rescaling.
              X, y = _rescale_data(X, y, sample_weight)
  
          if self.positive:
              if y.ndim < 2:
                  self.coef_, self._residues = optimize.nnls(X, y)
              else:
                  # scipy.optimize.nnls cannot handle y with shape (M, K)
                  outs = Parallel(n_jobs=n_jobs_)(
                          delayed(optimize.nnls)(X, y[:, j])
                          for j in range(y.shape[1]))
                  self.coef_, self._residues = map(np.vstack, zip(*outs))
          elif sp.issparse(X):
              X_offset_scale = X_offset / X_scale
  
              def matvec(b):
                  return X.dot(b) - b.dot(X_offset_scale)
  
              def rmatvec(b):
                  return X.T.dot(b) - X_offset_scale * np.sum(b)
  
              X_centered = sparse.linalg.LinearOperator(shape=X.shape,
                                                        matvec=matvec,
                                                        rmatvec=rmatvec)
  
              if y.ndim < 2:
                  out = sparse_lsqr(X_centered, y)
                  self.coef_ = out[0]
                  self._residues = out[3]
              else:
                  # sparse_lstsq cannot handle y with shape (M, K)
                  outs = Parallel(n_jobs=n_jobs_)(
                      delayed(sparse_lsqr)(X_centered, y[:, j].ravel())
                      for j in range(y.shape[1]))
                  self.coef_ = np.vstack([out[0] for out in outs])
                  self._residues = np.vstack([out[3] for out in outs])
          else:
              self.coef_, self._residues, self.rank_, self.singular_ = \
                  linalg.lstsq(X, y)
              self.coef_ = self.coef_.T
  
          if y.ndim == 1:
              self.coef_ = np.ravel(self.coef_)
          self._set_intercept(X_offset, y_offset, X_scale)
          return self
  
  
  def _check_precomputed_gram_matrix(X, precompute, X_offset, X_scale,
                                     rtol=1e-7,
                                     atol=1e-5):
      """Computes a single element of the gram matrix and compares it to
      the corresponding element of the user supplied gram matrix.
  
      If the values do not match a ValueError will be thrown.
  
      Parameters
      ----------
      X : ndarray of shape (n_samples, n_features)
          Data array.
  
      precompute : array-like of shape (n_features, n_features)
          User-supplied gram matrix.
  
      X_offset : ndarray of shape (n_features,)
          Array of feature means used to center design matrix.
  
      X_scale : ndarray of shape (n_features,)
          Array of feature scale factors used to normalize design matrix.
  
      rtol : float, default=1e-7
          Relative tolerance; see numpy.allclose.
  
      atol : float, default=1e-5
          absolute tolerance; see :func`numpy.allclose`. Note that the default
          here is more tolerant than the default for
          :func:`numpy.testing.assert_allclose`, where `atol=0`.
  
      Raises
      ------
      ValueError
          Raised when the provided Gram matrix is not consistent.
      """
  
      n_features = X.shape[1]
      f1 = n_features // 2
      f2 = min(f1+1, n_features-1)
  
      v1 = (X[:, f1] - X_offset[f1]) * X_scale[f1]
      v2 = (X[:, f2] - X_offset[f2]) * X_scale[f2]
  
      expected = np.dot(v1, v2)
      actual = precompute[f1, f2]
  
      if not np.isclose(expected, actual, rtol=rtol, atol=atol):
          raise ValueError("Gram matrix passed in via 'precompute' parameter "
                           "did not pass validation when a single element was "
                           "checked - please check that it was computed "
                           f"properly. For element ({f1},{f2}) we computed "
                           f"{expected} but the user-supplied value was "
                           f"{actual}.")
  
  
  def _pre_fit(X, y, Xy, precompute, normalize, fit_intercept, copy,
               check_input=True, sample_weight=None):
      """Aux function used at beginning of fit in linear models
  
      Parameters
      ----------
      order : 'F', 'C' or None, default=None
          Whether X and y will be forced to be fortran or c-style. Only relevant
          if sample_weight is not None.
      """
      n_samples, n_features = X.shape
  
      if sparse.isspmatrix(X):
          # copy is not needed here as X is not modified inplace when X is sparse
          precompute = False
          X, y, X_offset, y_offset, X_scale = _preprocess_data(
              X, y, fit_intercept=fit_intercept, normalize=normalize,
              copy=False, return_mean=True, check_input=check_input)
      else:
          # copy was done in fit if necessary
          X, y, X_offset, y_offset, X_scale = _preprocess_data(
              X, y, fit_intercept=fit_intercept, normalize=normalize, copy=copy,
              check_input=check_input, sample_weight=sample_weight)
      if sample_weight is not None:
          X, y = _rescale_data(X, y, sample_weight=sample_weight)
  
      # FIXME: 'normalize' to be removed in 1.2
      if hasattr(precompute, '__array__'):
          if (fit_intercept and not np.allclose(X_offset, np.zeros(n_features))
                  or normalize and not np.allclose(X_scale, np.ones(n_features)
                                                   )):
              warnings.warn(
                  "Gram matrix was provided but X was centered to fit "
                  "intercept, or X was normalized : recomputing Gram matrix.",
                  UserWarning
              )
              # recompute Gram
              precompute = 'auto'
              Xy = None
          elif check_input:
              # If we're going to use the user's precomputed gram matrix, we
              # do a quick check to make sure its not totally bogus.
              _check_precomputed_gram_matrix(X, precompute, X_offset, X_scale)
  
      # precompute if n_samples > n_features
      if isinstance(precompute, str) and precompute == 'auto':
          precompute = (n_samples > n_features)
  
      if precompute is True:
          # make sure that the 'precompute' array is contiguous.
          precompute = np.empty(shape=(n_features, n_features), dtype=X.dtype,
                                order='C')
          np.dot(X.T, X, out=precompute)
  
      if not hasattr(precompute, '__array__'):
          Xy = None  # cannot use Xy if precompute is not Gram
  
      if hasattr(precompute, '__array__') and Xy is None:
          common_dtype = np.find_common_type([X.dtype, y.dtype], [])
          if y.ndim == 1:
              # Xy is 1d, make sure it is contiguous.
              Xy = np.empty(shape=n_features, dtype=common_dtype, order='C')
              np.dot(X.T, y, out=Xy)
          else:
              # Make sure that Xy is always F contiguous even if X or y are not
              # contiguous: the goal is to make it fast to extract the data for a
              # specific target.
              n_targets = y.shape[1]
              Xy = np.empty(shape=(n_features, n_targets), dtype=common_dtype,
                            order='F')
              np.dot(y.T, X, out=Xy.T)
  
      return X, y, X_offset, y_offset, X_scale, precompute, Xy
  