content: """
  Least Angle Regression algorithm. See the documentation on the
  Generalized Linear Model for a complete discussion.
  """
  # Author: Fabian Pedregosa <fabian.pedregosa@inria.fr>
  #         Alexandre Gramfort <alexandre.gramfort@inria.fr>
  #         Gael Varoquaux
  #
  # License: BSD 3 clause
  
  from math import log
  import sys
  import warnings
  
  import numpy as np
  from scipy import linalg, interpolate
  from scipy.linalg.lapack import get_lapack_funcs
  from joblib import Parallel
  
  from ._base import LinearModel
  from ..base import RegressorMixin, MultiOutputMixin
  # mypy error: Module 'sklearn.utils' has no attribute 'arrayfuncs'
  from ..utils import arrayfuncs, as_float_array  # type: ignore
  from ..utils import check_random_state
  from ..model_selection import check_cv
  from ..exceptions import ConvergenceWarning
  from ..utils.validation import _deprecate_positional_args
  from ..utils.fixes import delayed
  
  SOLVE_TRIANGULAR_ARGS = {'check_finite': False}
  
  
  @_deprecate_positional_args
  def lars_path(
      X,
      y,
      Xy=None,
      *,
      Gram=None,
      max_iter=500,
      alpha_min=0,
      method="lar",
      copy_X=True,
      eps=np.finfo(float).eps,
      copy_Gram=True,
      verbose=0,
      return_path=True,
      return_n_iter=False,
      positive=False
  ):
      """Compute Least Angle Regression or Lasso path using LARS algorithm [1]
  
      The optimization objective for the case method='lasso' is::
  
      (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1
  
      in the case of method='lars', the objective function is only known in
      the form of an implicit equation (see discussion in [1])
  
      Read more in the :ref:`User Guide <least_angle_regression>`.
  
      Parameters
      ----------
      X : None or array-like of shape (n_samples, n_features)
          Input data. Note that if X is None then the Gram matrix must be
          specified, i.e., cannot be None or False.
  
      y : None or array-like of shape (n_samples,)
          Input targets.
  
      Xy : array-like of shape (n_samples,) or (n_samples, n_targets), \
              default=None
          Xy = np.dot(X.T, y) that can be precomputed. It is useful
          only when the Gram matrix is precomputed.
  
      Gram : None, 'auto', array-like of shape (n_features, n_features), \
              default=None
          Precomputed Gram matrix (X' * X), if ``'auto'``, the Gram
          matrix is precomputed from the given X, if there are more samples
          than features.
  
      max_iter : int, default=500
          Maximum number of iterations to perform, set to infinity for no limit.
  
      alpha_min : float, default=0
          Minimum correlation along the path. It corresponds to the
          regularization parameter alpha parameter in the Lasso.
  
      method : {'lar', 'lasso'}, default='lar'
          Specifies the returned model. Select ``'lar'`` for Least Angle
          Regression, ``'lasso'`` for the Lasso.
  
      copy_X : bool, default=True
          If ``False``, ``X`` is overwritten.
  
      eps : float, default=np.finfo(float).eps
          The machine-precision regularization in the computation of the
          Cholesky diagonal factors. Increase this for very ill-conditioned
          systems. Unlike the ``tol`` parameter in some iterative
          optimization-based algorithms, this parameter does not control
          the tolerance of the optimization.
  
      copy_Gram : bool, default=True
          If ``False``, ``Gram`` is overwritten.
  
      verbose : int, default=0
          Controls output verbosity.
  
      return_path : bool, default=True
          If ``return_path==True`` returns the entire path, else returns only the
          last point of the path.
  
      return_n_iter : bool, default=False
          Whether to return the number of iterations.
  
      positive : bool, default=False
          Restrict coefficients to be >= 0.
          This option is only allowed with method 'lasso'. Note that the model
          coefficients will not converge to the ordinary-least-squares solution
          for small values of alpha. Only coefficients up to the smallest alpha
          value (``alphas_[alphas_ > 0.].min()`` when fit_path=True) reached by
          the stepwise Lars-Lasso algorithm are typically in congruence with the
          solution of the coordinate descent lasso_path function.
  
      Returns
      -------
      alphas : array-like of shape (n_alphas + 1,)
          Maximum of covariances (in absolute value) at each iteration.
          ``n_alphas`` is either ``max_iter``, ``n_features`` or the
          number of nodes in the path with ``alpha >= alpha_min``, whichever
          is smaller.
  
      active : array-like of shape (n_alphas,)
          Indices of active variables at the end of the path.
  
      coefs : array-like of shape (n_features, n_alphas + 1)
          Coefficients along the path
  
      n_iter : int
          Number of iterations run. Returned only if return_n_iter is set
          to True.
  
      See Also
      --------
      lars_path_gram
      lasso_path
      lasso_path_gram
      LassoLars
      Lars
      LassoLarsCV
      LarsCV
      sklearn.decomposition.sparse_encode
  
      References
      ----------
      .. [1] "Least Angle Regression", Efron et al.
             http://statweb.stanford.edu/~tibs/ftp/lars.pdf
  
      .. [2] `Wikipedia entry on the Least-angle regression
             <https://en.wikipedia.org/wiki/Least-angle_regression>`_
  
      .. [3] `Wikipedia entry on the Lasso
             <https://en.wikipedia.org/wiki/Lasso_(statistics)>`_
  
      """
      if X is None and Gram is not None:
          raise ValueError(
              'X cannot be None if Gram is not None'
              'Use lars_path_gram to avoid passing X and y.'
          )
      return _lars_path_solver(
          X=X, y=y, Xy=Xy, Gram=Gram, n_samples=None, max_iter=max_iter,
          alpha_min=alpha_min, method=method, copy_X=copy_X,
          eps=eps, copy_Gram=copy_Gram, verbose=verbose, return_path=return_path,
          return_n_iter=return_n_iter, positive=positive)
  
  
  @_deprecate_positional_args
  def lars_path_gram(
      Xy,
      Gram,
      *,
      n_samples,
      max_iter=500,
      alpha_min=0,
      method="lar",
      copy_X=True,
      eps=np.finfo(float).eps,
      copy_Gram=True,
      verbose=0,
      return_path=True,
      return_n_iter=False,
      positive=False
  ):
      """lars_path in the sufficient stats mode [1]
  
      The optimization objective for the case method='lasso' is::
  
      (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1
  
      in the case of method='lars', the objective function is only known in
      the form of an implicit equation (see discussion in [1])
  
      Read more in the :ref:`User Guide <least_angle_regression>`.
  
      Parameters
      ----------
      Xy : array-like of shape (n_samples,) or (n_samples, n_targets)
          Xy = np.dot(X.T, y).
  
      Gram : array-like of shape (n_features, n_features)
          Gram = np.dot(X.T * X).
  
      n_samples : int or float
          Equivalent size of sample.
  
      max_iter : int, default=500
          Maximum number of iterations to perform, set to infinity for no limit.
  
      alpha_min : float, default=0
          Minimum correlation along the path. It corresponds to the
          regularization parameter alpha parameter in the Lasso.
  
      method : {'lar', 'lasso'}, default='lar'
          Specifies the returned model. Select ``'lar'`` for Least Angle
          Regression, ``'lasso'`` for the Lasso.
  
      copy_X : bool, default=True
          If ``False``, ``X`` is overwritten.
  
      eps : float, default=np.finfo(float).eps
          The machine-precision regularization in the computation of the
          Cholesky diagonal factors. Increase this for very ill-conditioned
          systems. Unlike the ``tol`` parameter in some iterative
          optimization-based algorithms, this parameter does not control
          the tolerance of the optimization.
  
      copy_Gram : bool, default=True
          If ``False``, ``Gram`` is overwritten.
  
      verbose : int, default=0
          Controls output verbosity.
  
      return_path : bool, default=True
          If ``return_path==True`` returns the entire path, else returns only the
          last point of the path.
  
      return_n_iter : bool, default=False
          Whether to return the number of iterations.
  
      positive : bool, default=False
          Restrict coefficients to be >= 0.
          This option is only allowed with method 'lasso'. Note that the model
          coefficients will not converge to the ordinary-least-squares solution
          for small values of alpha. Only coefficients up to the smallest alpha
          value (``alphas_[alphas_ > 0.].min()`` when fit_path=True) reached by
          the stepwise Lars-Lasso algorithm are typically in congruence with the
          solution of the coordinate descent lasso_path function.
  
      Returns
      -------
      alphas : array-like of shape (n_alphas + 1,)
          Maximum of covariances (in absolute value) at each iteration.
          ``n_alphas`` is either ``max_iter``, ``n_features`` or the
          number of nodes in the path with ``alpha >= alpha_min``, whichever
          is smaller.
  
      active : array-like of shape (n_alphas,)
          Indices of active variables at the end of the path.
  
      coefs : array-like of shape (n_features, n_alphas + 1)
          Coefficients along the path
  
      n_iter : int
          Number of iterations run. Returned only if return_n_iter is set
          to True.
  
      See Also
      --------
      lars_path
      lasso_path
      lasso_path_gram
      LassoLars
      Lars
      LassoLarsCV
      LarsCV
      sklearn.decomposition.sparse_encode
  
      References
      ----------
      .. [1] "Least Angle Regression", Efron et al.
             http://statweb.stanford.edu/~tibs/ftp/lars.pdf
  
      .. [2] `Wikipedia entry on the Least-angle regression
             <https://en.wikipedia.org/wiki/Least-angle_regression>`_
  
      .. [3] `Wikipedia entry on the Lasso
             <https://en.wikipedia.org/wiki/Lasso_(statistics)>`_
  
      """
      return _lars_path_solver(
          X=None, y=None, Xy=Xy, Gram=Gram, n_samples=n_samples,
          max_iter=max_iter, alpha_min=alpha_min, method=method,
          copy_X=copy_X, eps=eps, copy_Gram=copy_Gram,
          verbose=verbose, return_path=return_path,
          return_n_iter=return_n_iter, positive=positive)
  
  
  def _lars_path_solver(
      X,
      y,
      Xy=None,
      Gram=None,
      n_samples=None,
      max_iter=500,
      alpha_min=0,
      method="lar",
      copy_X=True,
      eps=np.finfo(float).eps,
      copy_Gram=True,
      verbose=0,
      return_path=True,
      return_n_iter=False,
      positive=False,
  ):
      """Compute Least Angle Regression or Lasso path using LARS algorithm [1]
  
      The optimization objective for the case method='lasso' is::
  
      (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1
  
      in the case of method='lars', the objective function is only known in
      the form of an implicit equation (see discussion in [1])
  
      Read more in the :ref:`User Guide <least_angle_regression>`.
  
      Parameters
      ----------
      X : None or ndarray of shape (n_samples, n_features)
          Input data. Note that if X is None then Gram must be specified,
          i.e., cannot be None or False.
  
      y : None or ndarray of shape (n_samples,)
          Input targets.
  
      Xy : array-like of shape (n_samples,) or (n_samples, n_targets), \
              default=None
          `Xy = np.dot(X.T, y)` that can be precomputed. It is useful
          only when the Gram matrix is precomputed.
  
      Gram : None, 'auto' or array-like of shape (n_features, n_features), \
              default=None
          Precomputed Gram matrix `(X' * X)`, if ``'auto'``, the Gram
          matrix is precomputed from the given X, if there are more samples
          than features.
  
      n_samples : int or float, default=None
          Equivalent size of sample. If `None`, it will be `n_samples`.
  
      max_iter : int, default=500
          Maximum number of iterations to perform, set to infinity for no limit.
  
      alpha_min : float, default=0
          Minimum correlation along the path. It corresponds to the
          regularization parameter alpha parameter in the Lasso.
  
      method : {'lar', 'lasso'}, default='lar'
          Specifies the returned model. Select ``'lar'`` for Least Angle
          Regression, ``'lasso'`` for the Lasso.
  
      copy_X : bool, default=True
          If ``False``, ``X`` is overwritten.
  
      eps : float, default=np.finfo(float).eps
          The machine-precision regularization in the computation of the
          Cholesky diagonal factors. Increase this for very ill-conditioned
          systems. Unlike the ``tol`` parameter in some iterative
          optimization-based algorithms, this parameter does not control
          the tolerance of the optimization.
  
      copy_Gram : bool, default=True
          If ``False``, ``Gram`` is overwritten.
  
      verbose : int, default=0
          Controls output verbosity.
  
      return_path : bool, default=True
          If ``return_path==True`` returns the entire path, else returns only the
          last point of the path.
  
      return_n_iter : bool, default=False
          Whether to return the number of iterations.
  
      positive : bool, default=False
          Restrict coefficients to be >= 0.
          This option is only allowed with method 'lasso'. Note that the model
          coefficients will not converge to the ordinary-least-squares solution
          for small values of alpha. Only coefficients up to the smallest alpha
          value (``alphas_[alphas_ > 0.].min()`` when fit_path=True) reached by
          the stepwise Lars-Lasso algorithm are typically in congruence with the
          solution of the coordinate descent lasso_path function.
  
      Returns
      -------
      alphas : array-like of shape (n_alphas + 1,)
          Maximum of covariances (in absolute value) at each iteration.
          ``n_alphas`` is either ``max_iter``, ``n_features`` or the
          number of nodes in the path with ``alpha >= alpha_min``, whichever
          is smaller.
  
      active : array-like of shape (n_alphas,)
          Indices of active variables at the end of the path.
  
      coefs : array-like of shape (n_features, n_alphas + 1)
          Coefficients along the path
  
      n_iter : int
          Number of iterations run. Returned only if return_n_iter is set
          to True.
  
      See Also
      --------
      lasso_path
      LassoLars
      Lars
      LassoLarsCV
      LarsCV
      sklearn.decomposition.sparse_encode
  
      References
      ----------
      .. [1] "Least Angle Regression", Efron et al.
             http://statweb.stanford.edu/~tibs/ftp/lars.pdf
  
      .. [2] `Wikipedia entry on the Least-angle regression
             <https://en.wikipedia.org/wiki/Least-angle_regression>`_
  
      .. [3] `Wikipedia entry on the Lasso
             <https://en.wikipedia.org/wiki/Lasso_(statistics)>`_
  
      """
      if method == "lar" and positive:
          raise ValueError(
              "Positive constraint not supported for 'lar' " "coding method."
          )
  
      n_samples = n_samples if n_samples is not None else y.size
  
      if Xy is None:
          Cov = np.dot(X.T, y)
      else:
          Cov = Xy.copy()
  
      if Gram is None or Gram is False:
          Gram = None
          if X is None:
              raise ValueError('X and Gram cannot both be unspecified.')
      elif isinstance(Gram, str) and Gram == 'auto' or Gram is True:
          if Gram is True or X.shape[0] > X.shape[1]:
              Gram = np.dot(X.T, X)
          else:
              Gram = None
      elif copy_Gram:
          Gram = Gram.copy()
  
      if Gram is None:
          n_features = X.shape[1]
      else:
          n_features = Cov.shape[0]
          if Gram.shape != (n_features, n_features):
              raise ValueError('The shapes of the inputs Gram and Xy'
                               ' do not match.')
  
      if copy_X and X is not None and Gram is None:
          # force copy. setting the array to be fortran-ordered
          # speeds up the calculation of the (partial) Gram matrix
          # and allows to easily swap columns
          X = X.copy('F')
  
      max_features = min(max_iter, n_features)
  
      if return_path:
          coefs = np.zeros((max_features + 1, n_features))
          alphas = np.zeros(max_features + 1)
      else:
          coef, prev_coef = np.zeros(n_features), np.zeros(n_features)
          alpha, prev_alpha = np.array([0.]), np.array([0.])  # better ideas?
  
      n_iter, n_active = 0, 0
      active, indices = list(), np.arange(n_features)
      # holds the sign of covariance
      sign_active = np.empty(max_features, dtype=np.int8)
      drop = False
  
      # will hold the cholesky factorization. Only lower part is
      # referenced.
      if Gram is None:
          L = np.empty((max_features, max_features), dtype=X.dtype)
          swap, nrm2 = linalg.get_blas_funcs(('swap', 'nrm2'), (X,))
      else:
          L = np.empty((max_features, max_features), dtype=Gram.dtype)
          swap, nrm2 = linalg.get_blas_funcs(('swap', 'nrm2'), (Cov,))
      solve_cholesky, = get_lapack_funcs(('potrs',), (L,))
  
      if verbose:
          if verbose > 1:
              print("Step\t\tAdded\t\tDropped\t\tActive set size\t\tC")
          else:
              sys.stdout.write('.')
              sys.stdout.flush()
  
      tiny32 = np.finfo(np.float32).tiny  # to avoid division by 0 warning
      equality_tolerance = np.finfo(np.float32).eps
  
      if Gram is not None:
          Gram_copy = Gram.copy()
          Cov_copy = Cov.copy()
  
      while True:
          if Cov.size:
              if positive:
                  C_idx = np.argmax(Cov)
              else:
                  C_idx = np.argmax(np.abs(Cov))
  
              C_ = Cov[C_idx]
  
              if positive:
                  C = C_
              else:
                  C = np.fabs(C_)
          else:
              C = 0.
  
          if return_path:
              alpha = alphas[n_iter, np.newaxis]
              coef = coefs[n_iter]
              prev_alpha = alphas[n_iter - 1, np.newaxis]
              prev_coef = coefs[n_iter - 1]
  
          alpha[0] = C / n_samples
          if alpha[0] <= alpha_min + equality_tolerance:  # early stopping
              if abs(alpha[0] - alpha_min) > equality_tolerance:
                  # interpolation factor 0 <= ss < 1
                  if n_iter > 0:
                      # In the first iteration, all alphas are zero, the formula
                      # below would make ss a NaN
                      ss = ((prev_alpha[0] - alpha_min) /
                            (prev_alpha[0] - alpha[0]))
                      coef[:] = prev_coef + ss * (coef - prev_coef)
                  alpha[0] = alpha_min
              if return_path:
                  coefs[n_iter] = coef
              break
  
          if n_iter >= max_iter or n_active >= n_features:
              break
          if not drop:
  
              ##########################################################
              # Append x_j to the Cholesky factorization of (Xa * Xa') #
              #                                                        #
              #            ( L   0 )                                   #
              #     L  ->  (       )  , where L * w = Xa' x_j          #
              #            ( w   z )    and z = ||x_j||                #
              #                                                        #
              ##########################################################
  
              if positive:
                  sign_active[n_active] = np.ones_like(C_)
              else:
                  sign_active[n_active] = np.sign(C_)
              m, n = n_active, C_idx + n_active
  
              Cov[C_idx], Cov[0] = swap(Cov[C_idx], Cov[0])
              indices[n], indices[m] = indices[m], indices[n]
              Cov_not_shortened = Cov
              Cov = Cov[1:]  # remove Cov[0]
  
              if Gram is None:
                  X.T[n], X.T[m] = swap(X.T[n], X.T[m])
                  c = nrm2(X.T[n_active]) ** 2
                  L[n_active, :n_active] = \
                      np.dot(X.T[n_active], X.T[:n_active].T)
              else:
                  # swap does only work inplace if matrix is fortran
                  # contiguous ...
                  Gram[m], Gram[n] = swap(Gram[m], Gram[n])
                  Gram[:, m], Gram[:, n] = swap(Gram[:, m], Gram[:, n])
                  c = Gram[n_active, n_active]
                  L[n_active, :n_active] = Gram[n_active, :n_active]
  
              # Update the cholesky decomposition for the Gram matrix
              if n_active:
                  linalg.solve_triangular(L[:n_active, :n_active],
                                          L[n_active, :n_active],
                                          trans=0, lower=1,
                                          overwrite_b=True,
                                          **SOLVE_TRIANGULAR_ARGS)
  
              v = np.dot(L[n_active, :n_active], L[n_active, :n_active])
              diag = max(np.sqrt(np.abs(c - v)), eps)
              L[n_active, n_active] = diag
  
              if diag < 1e-7:
                  # The system is becoming too ill-conditioned.
                  # We have degenerate vectors in our active set.
                  # We'll 'drop for good' the last regressor added.
  
                  # Note: this case is very rare. It is no longer triggered by
                  # the test suite. The `equality_tolerance` margin added in 0.16
                  # to get early stopping to work consistently on all versions of
                  # Python including 32 bit Python under Windows seems to make it
                  # very difficult to trigger the 'drop for good' strategy.
                  warnings.warn('Regressors in active set degenerate. '
                                'Dropping a regressor, after %i iterations, '
                                'i.e. alpha=%.3e, '
                                'with an active set of %i regressors, and '
                                'the smallest cholesky pivot element being %.3e.'
                                ' Reduce max_iter or increase eps parameters.'
                                % (n_iter, alpha, n_active, diag),
                                ConvergenceWarning)
  
                  # XXX: need to figure a 'drop for good' way
                  Cov = Cov_not_shortened
                  Cov[0] = 0
                  Cov[C_idx], Cov[0] = swap(Cov[C_idx], Cov[0])
                  continue
  
              active.append(indices[n_active])
              n_active += 1
  
              if verbose > 1:
                  print("%s\t\t%s\t\t%s\t\t%s\t\t%s" % (n_iter, active[-1], '',
                                                        n_active, C))
  
          if method == 'lasso' and n_iter > 0 and prev_alpha[0] < alpha[0]:
              # alpha is increasing. This is because the updates of Cov are
              # bringing in too much numerical error that is greater than
              # than the remaining correlation with the
              # regressors. Time to bail out
              warnings.warn('Early stopping the lars path, as the residues '
                            'are small and the current value of alpha is no '
                            'longer well controlled. %i iterations, alpha=%.3e, '
                            'previous alpha=%.3e, with an active set of %i '
                            'regressors.'
                            % (n_iter, alpha, prev_alpha, n_active),
                            ConvergenceWarning)
              break
  
          # least squares solution
          least_squares, _ = solve_cholesky(L[:n_active, :n_active],
                                            sign_active[:n_active],
                                            lower=True)
  
          if least_squares.size == 1 and least_squares == 0:
              # This happens because sign_active[:n_active] = 0
              least_squares[...] = 1
              AA = 1.
          else:
              # is this really needed ?
              AA = 1. / np.sqrt(np.sum(least_squares * sign_active[:n_active]))
  
              if not np.isfinite(AA):
                  # L is too ill-conditioned
                  i = 0
                  L_ = L[:n_active, :n_active].copy()
                  while not np.isfinite(AA):
                      L_.flat[::n_active + 1] += (2 ** i) * eps
                      least_squares, _ = solve_cholesky(
                          L_, sign_active[:n_active], lower=True)
                      tmp = max(np.sum(least_squares * sign_active[:n_active]),
                                eps)
                      AA = 1. / np.sqrt(tmp)
                      i += 1
              least_squares *= AA
  
          if Gram is None:
              # equiangular direction of variables in the active set
              eq_dir = np.dot(X.T[:n_active].T, least_squares)
              # correlation between each unactive variables and
              # eqiangular vector
              corr_eq_dir = np.dot(X.T[n_active:], eq_dir)
          else:
              # if huge number of features, this takes 50% of time, I
              # think could be avoided if we just update it using an
              # orthogonal (QR) decomposition of X
              corr_eq_dir = np.dot(Gram[:n_active, n_active:].T,
                                   least_squares)
  
          g1 = arrayfuncs.min_pos((C - Cov) / (AA - corr_eq_dir + tiny32))
          if positive:
              gamma_ = min(g1, C / AA)
          else:
              g2 = arrayfuncs.min_pos((C + Cov) / (AA + corr_eq_dir + tiny32))
              gamma_ = min(g1, g2, C / AA)
  
          # TODO: better names for these variables: z
          drop = False
          z = -coef[active] / (least_squares + tiny32)
          z_pos = arrayfuncs.min_pos(z)
          if z_pos < gamma_:
              # some coefficients have changed sign
              idx = np.where(z == z_pos)[0][::-1]
  
              # update the sign, important for LAR
              sign_active[idx] = -sign_active[idx]
  
              if method == 'lasso':
                  gamma_ = z_pos
              drop = True
  
          n_iter += 1
  
          if return_path:
              if n_iter >= coefs.shape[0]:
                  del coef, alpha, prev_alpha, prev_coef
                  # resize the coefs and alphas array
                  add_features = 2 * max(1, (max_features - n_active))
                  coefs = np.resize(coefs, (n_iter + add_features, n_features))
                  coefs[-add_features:] = 0
                  alphas = np.resize(alphas, n_iter + add_features)
                  alphas[-add_features:] = 0
              coef = coefs[n_iter]
              prev_coef = coefs[n_iter - 1]
          else:
              # mimic the effect of incrementing n_iter on the array references
              prev_coef = coef
              prev_alpha[0] = alpha[0]
              coef = np.zeros_like(coef)
  
          coef[active] = prev_coef[active] + gamma_ * least_squares
  
          # update correlations
          Cov -= gamma_ * corr_eq_dir
  
          # See if any coefficient has changed sign
          if drop and method == 'lasso':
  
              # handle the case when idx is not length of 1
              for ii in idx:
                  arrayfuncs.cholesky_delete(L[:n_active, :n_active], ii)
  
              n_active -= 1
              # handle the case when idx is not length of 1
              drop_idx = [active.pop(ii) for ii in idx]
  
              if Gram is None:
                  # propagate dropped variable
                  for ii in idx:
                      for i in range(ii, n_active):
                          X.T[i], X.T[i + 1] = swap(X.T[i], X.T[i + 1])
                          # yeah this is stupid
                          indices[i], indices[i + 1] = indices[i + 1], indices[i]
  
                  # TODO: this could be updated
                  residual = y - np.dot(X[:, :n_active], coef[active])
                  temp = np.dot(X.T[n_active], residual)
  
                  Cov = np.r_[temp, Cov]
              else:
                  for ii in idx:
                      for i in range(ii, n_active):
                          indices[i], indices[i + 1] = indices[i + 1], indices[i]
                          Gram[i], Gram[i + 1] = swap(Gram[i], Gram[i + 1])
                          Gram[:, i], Gram[:, i + 1] = swap(Gram[:, i],
                                                            Gram[:, i + 1])
  
                  # Cov_n = Cov_j + x_j * X + increment(betas) TODO:
                  # will this still work with multiple drops ?
  
                  # recompute covariance. Probably could be done better
                  # wrong as Xy is not swapped with the rest of variables
  
                  # TODO: this could be updated
                  temp = Cov_copy[drop_idx] - np.dot(Gram_copy[drop_idx], coef)
                  Cov = np.r_[temp, Cov]
  
              sign_active = np.delete(sign_active, idx)
              sign_active = np.append(sign_active, 0.)  # just to maintain size
              if verbose > 1:
                  print("%s\t\t%s\t\t%s\t\t%s\t\t%s" % (n_iter, '', drop_idx,
                                                        n_active, abs(temp)))
  
      if return_path:
          # resize coefs in case of early stop
          alphas = alphas[:n_iter + 1]
          coefs = coefs[:n_iter + 1]
  
          if return_n_iter:
              return alphas, active, coefs.T, n_iter
          else:
              return alphas, active, coefs.T
      else:
          if return_n_iter:
              return alpha, active, coef, n_iter
          else:
              return alpha, active, coef
  
  
  ###############################################################################
  # Estimator classes
  
  class Lars(MultiOutputMixin, RegressorMixin, LinearModel):
      """Least Angle Regression model a.k.a. LAR
  
      Read more in the :ref:`User Guide <least_angle_regression>`.
  
      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).
  
      verbose : bool or int, default=False
          Sets the verbosity amount.
  
      normalize : bool, default=True
          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``.
  
      precompute : bool, 'auto' or array-like , default='auto'
          Whether to use a precomputed Gram matrix to speed up
          calculations. If set to ``'auto'`` let us decide. The Gram
          matrix can also be passed as argument.
  
      n_nonzero_coefs : int, default=500
          Target number of non-zero coefficients. Use ``np.inf`` for no limit.
  
      eps : float, default=np.finfo(float).eps
          The machine-precision regularization in the computation of the
          Cholesky diagonal factors. Increase this for very ill-conditioned
          systems. Unlike the ``tol`` parameter in some iterative
          optimization-based algorithms, this parameter does not control
          the tolerance of the optimization.
  
      copy_X : bool, default=True
          If ``True``, X will be copied; else, it may be overwritten.
  
      fit_path : bool, default=True
          If True the full path is stored in the ``coef_path_`` attribute.
          If you compute the solution for a large problem or many targets,
          setting ``fit_path`` to ``False`` will lead to a speedup, especially
          with a small alpha.
  
      jitter : float, default=None
          Upper bound on a uniform noise parameter to be added to the
          `y` values, to satisfy the model's assumption of
          one-at-a-time computations. Might help with stability.
  
          .. versionadded:: 0.23
  
      random_state : int, RandomState instance or None, default=None
          Determines random number generation for jittering. Pass an int
          for reproducible output across multiple function calls.
          See :term:`Glossary <random_state>`. Ignored if `jitter` is None.
  
          .. versionadded:: 0.23
  
      Attributes
      ----------
      alphas_ : array-like of shape (n_alphas + 1,) or list of such arrays
          Maximum of covariances (in absolute value) at each iteration.
          ``n_alphas`` is either ``max_iter``, ``n_features`` or the
          number of nodes in the path with ``alpha >= alpha_min``, whichever
          is smaller. If this is a list of array-like, the length of the outer
          list is `n_targets`.
  
      active_ : list of shape (n_alphas,) or list of such lists
          Indices of active variables at the end of the path.
          If this is a list of list, the length of the outer list is `n_targets`.
  
      coef_path_ : array-like of shape (n_features, n_alphas + 1) or list \
              of such arrays
          The varying values of the coefficients along the path. It is not
          present if the ``fit_path`` parameter is ``False``. If this is a list
          of array-like, the length of the outer list is `n_targets`.
  
      coef_ : array-like of shape (n_features,) or (n_targets, n_features)
          Parameter vector (w in the formulation formula).
  
      intercept_ : float or array-like of shape (n_targets,)
          Independent term in decision function.
  
      n_iter_ : array-like or int
          The number of iterations taken by lars_path to find the
          grid of alphas for each target.
  
      Examples
      --------
      >>> from sklearn import linear_model
      >>> reg = linear_model.Lars(n_nonzero_coefs=1)
      >>> reg.fit([[-1, 1], [0, 0], [1, 1]], [-1.1111, 0, -1.1111])
      Lars(n_nonzero_coefs=1)
      >>> print(reg.coef_)
      [ 0. -1.11...]
  
      See Also
      --------
      lars_path, LarsCV
      sklearn.decomposition.sparse_encode
  
      """
  
      method = "lar"
      positive = False
  
      @_deprecate_positional_args
      def __init__(self, *, fit_intercept=True, verbose=False, normalize=True,
                   precompute='auto', n_nonzero_coefs=500,
                   eps=np.finfo(float).eps, copy_X=True, fit_path=True,
                   jitter=None, random_state=None):
          self.fit_intercept = fit_intercept
          self.verbose = verbose
          self.normalize = normalize
          self.precompute = precompute
          self.n_nonzero_coefs = n_nonzero_coefs
          self.eps = eps
          self.copy_X = copy_X
          self.fit_path = fit_path
          self.jitter = jitter
          self.random_state = random_state
  
      @staticmethod
      def _get_gram(precompute, X, y):
          if (not hasattr(precompute, '__array__')) and (
                  (precompute is True) or
                  (precompute == 'auto' and X.shape[0] > X.shape[1]) or
                  (precompute == 'auto' and y.shape[1] > 1)):
              precompute = np.dot(X.T, X)
  
          return precompute
  
      def _fit(self, X, y, max_iter, alpha, fit_path, Xy=None):
          """Auxiliary method to fit the model using X, y as training data"""
          n_features = X.shape[1]
  
          X, y, X_offset, y_offset, X_scale = self._preprocess_data(
              X, y, self.fit_intercept, self.normalize, self.copy_X)
  
          if y.ndim == 1:
              y = y[:, np.newaxis]
  
          n_targets = y.shape[1]
  
          Gram = self._get_gram(self.precompute, X, y)
  
          self.alphas_ = []
          self.n_iter_ = []
          self.coef_ = np.empty((n_targets, n_features))
  
          if fit_path:
              self.active_ = []
              self.coef_path_ = []
              for k in range(n_targets):
                  this_Xy = None if Xy is None else Xy[:, k]
                  alphas, active, coef_path, n_iter_ = lars_path(
                      X, y[:, k], Gram=Gram, Xy=this_Xy, copy_X=self.copy_X,
                      copy_Gram=True, alpha_min=alpha, method=self.method,
                      verbose=max(0, self.verbose - 1), max_iter=max_iter,
                      eps=self.eps, return_path=True,
                      return_n_iter=True, positive=self.positive)
                  self.alphas_.append(alphas)
                  self.active_.append(active)
                  self.n_iter_.append(n_iter_)
                  self.coef_path_.append(coef_path)
                  self.coef_[k] = coef_path[:, -1]
  
              if n_targets == 1:
                  self.alphas_, self.active_, self.coef_path_, self.coef_ = [
                      a[0] for a in (self.alphas_, self.active_, self.coef_path_,
                                     self.coef_)]
                  self.n_iter_ = self.n_iter_[0]
          else:
              for k in range(n_targets):
                  this_Xy = None if Xy is None else Xy[:, k]
                  alphas, _, self.coef_[k], n_iter_ = lars_path(
                      X, y[:, k], Gram=Gram, Xy=this_Xy, copy_X=self.copy_X,
                      copy_Gram=True, alpha_min=alpha, method=self.method,
                      verbose=max(0, self.verbose - 1), max_iter=max_iter,
                      eps=self.eps, return_path=False, return_n_iter=True,
                      positive=self.positive)
                  self.alphas_.append(alphas)
                  self.n_iter_.append(n_iter_)
              if n_targets == 1:
                  self.alphas_ = self.alphas_[0]
                  self.n_iter_ = self.n_iter_[0]
  
          self._set_intercept(X_offset, y_offset, X_scale)
          return self
  
      def fit(self, X, y, Xy=None):
          """Fit the model using X, y as training data.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Training data.
  
          y : array-like of shape (n_samples,) or (n_samples, n_targets)
              Target values.
  
          Xy : array-like of shape (n_samples,) or (n_samples, n_targets), \
                  default=None
              Xy = np.dot(X.T, y) that can be precomputed. It is useful
              only when the Gram matrix is precomputed.
  
          Returns
          -------
          self : object
              returns an instance of self.
          """
          X, y = self._validate_data(X, y, y_numeric=True, multi_output=True)
  
          alpha = getattr(self, 'alpha', 0.)
          if hasattr(self, 'n_nonzero_coefs'):
              alpha = 0.  # n_nonzero_coefs parametrization takes priority
              max_iter = self.n_nonzero_coefs
          else:
              max_iter = self.max_iter
  
          if self.jitter is not None:
              rng = check_random_state(self.random_state)
  
              noise = rng.uniform(high=self.jitter, size=len(y))
              y = y + noise
  
          self._fit(X, y, max_iter=max_iter, alpha=alpha, fit_path=self.fit_path,
                    Xy=Xy)
  
          return self
  
  
  class LassoLars(Lars):
      """Lasso model fit with Least Angle Regression a.k.a. Lars
  
      It is a Linear Model trained with an L1 prior as regularizer.
  
      The optimization objective for Lasso is::
  
      (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1
  
      Read more in the :ref:`User Guide <least_angle_regression>`.
  
      Parameters
      ----------
      alpha : float, default=1.0
          Constant that multiplies the penalty term. Defaults to 1.0.
          ``alpha = 0`` is equivalent to an ordinary least square, solved
          by :class:`LinearRegression`. For numerical reasons, using
          ``alpha = 0`` with the LassoLars object is not advised and you
          should prefer the LinearRegression object.
  
      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).
  
      verbose : bool or int, default=False
          Sets the verbosity amount.
  
      normalize : bool, default=True
          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``.
  
      precompute : bool, 'auto' or array-like, default='auto'
          Whether to use a precomputed Gram matrix to speed up
          calculations. If set to ``'auto'`` let us decide. The Gram
          matrix can also be passed as argument.
  
      max_iter : int, default=500
          Maximum number of iterations to perform.
  
      eps : float, default=np.finfo(float).eps
          The machine-precision regularization in the computation of the
          Cholesky diagonal factors. Increase this for very ill-conditioned
          systems. Unlike the ``tol`` parameter in some iterative
          optimization-based algorithms, this parameter does not control
          the tolerance of the optimization.
  
      copy_X : bool, default=True
          If True, X will be copied; else, it may be overwritten.
  
      fit_path : bool, default=True
          If ``True`` the full path is stored in the ``coef_path_`` attribute.
          If you compute the solution for a large problem or many targets,
          setting ``fit_path`` to ``False`` will lead to a speedup, especially
          with a small alpha.
  
      positive : bool, default=False
          Restrict coefficients to be >= 0. Be aware that you might want to
          remove fit_intercept which is set True by default.
          Under the positive restriction the model coefficients will not converge
          to the ordinary-least-squares solution for small values of alpha.
          Only coefficients up to the smallest alpha value (``alphas_[alphas_ >
          0.].min()`` when fit_path=True) reached by the stepwise Lars-Lasso
          algorithm are typically in congruence with the solution of the
          coordinate descent Lasso estimator.
  
      jitter : float, default=None
          Upper bound on a uniform noise parameter to be added to the
          `y` values, to satisfy the model's assumption of
          one-at-a-time computations. Might help with stability.
  
          .. versionadded:: 0.23
  
      random_state : int, RandomState instance or None, default=None
          Determines random number generation for jittering. Pass an int
          for reproducible output across multiple function calls.
          See :term:`Glossary <random_state>`. Ignored if `jitter` is None.
  
          .. versionadded:: 0.23
  
      Attributes
      ----------
      alphas_ : array-like of shape (n_alphas + 1,) or list of such arrays
          Maximum of covariances (in absolute value) at each iteration.
          ``n_alphas`` is either ``max_iter``, ``n_features`` or the
          number of nodes in the path with ``alpha >= alpha_min``, whichever
          is smaller. If this is a list of array-like, the length of the outer
          list is `n_targets`.
  
      active_ : list of length n_alphas or list of such lists
          Indices of active variables at the end of the path.
          If this is a list of list, the length of the outer list is `n_targets`.
  
      coef_path_ : array-like of shape (n_features, n_alphas + 1) or list \
              of such arrays
          If a list is passed it's expected to be one of n_targets such arrays.
          The varying values of the coefficients along the path. It is not
          present if the ``fit_path`` parameter is ``False``. If this is a list
          of array-like, the length of the outer list is `n_targets`.
  
      coef_ : array-like of shape (n_features,) or (n_targets, n_features)
          Parameter vector (w in the formulation formula).
  
      intercept_ : float or array-like of shape (n_targets,)
          Independent term in decision function.
  
      n_iter_ : array-like or int
          The number of iterations taken by lars_path to find the
          grid of alphas for each target.
  
      Examples
      --------
      >>> from sklearn import linear_model
      >>> reg = linear_model.LassoLars(alpha=0.01)
      >>> reg.fit([[-1, 1], [0, 0], [1, 1]], [-1, 0, -1])
      LassoLars(alpha=0.01)
      >>> print(reg.coef_)
      [ 0.         -0.963257...]
  
      See Also
      --------
      lars_path
      lasso_path
      Lasso
      LassoCV
      LassoLarsCV
      LassoLarsIC
      sklearn.decomposition.sparse_encode
  
      """
      method = 'lasso'
  
      @_deprecate_positional_args
      def __init__(self, alpha=1.0, *, fit_intercept=True, verbose=False,
                   normalize=True, precompute='auto', max_iter=500,
                   eps=np.finfo(float).eps, copy_X=True, fit_path=True,
                   positive=False, jitter=None, random_state=None):
          self.alpha = alpha
          self.fit_intercept = fit_intercept
          self.max_iter = max_iter
          self.verbose = verbose
          self.normalize = normalize
          self.positive = positive
          self.precompute = precompute
          self.copy_X = copy_X
          self.eps = eps
          self.fit_path = fit_path
          self.jitter = jitter
          self.random_state = random_state
  
  
  ###############################################################################
  # Cross-validated estimator classes
  
  def _check_copy_and_writeable(array, copy=False):
      if copy or not array.flags.writeable:
          return array.copy()
      return array
  
  
  def _lars_path_residues(X_train, y_train, X_test, y_test, Gram=None,
                          copy=True, method='lars', verbose=False,
                          fit_intercept=True, normalize=True, max_iter=500,
                          eps=np.finfo(float).eps, positive=False):
      """Compute the residues on left-out data for a full LARS path
  
      Parameters
      -----------
      X_train : array-like of shape (n_samples, n_features)
          The data to fit the LARS on
  
      y_train : array-like of shape (n_samples,)
          The target variable to fit LARS on
  
      X_test : array-like of shape (n_samples, n_features)
          The data to compute the residues on
  
      y_test : array-like of shape (n_samples,)
          The target variable to compute the residues on
  
      Gram : None, 'auto' or array-like of shape (n_features, n_features), \
              default=None
          Precomputed Gram matrix (X' * X), if ``'auto'``, the Gram
          matrix is precomputed from the given X, if there are more samples
          than features
  
      copy : bool, default=True
          Whether X_train, X_test, y_train and y_test should be copied;
          if False, they may be overwritten.
  
      method : {'lar' , 'lasso'}, default='lar'
          Specifies the returned model. Select ``'lar'`` for Least Angle
          Regression, ``'lasso'`` for the Lasso.
  
      verbose : bool or int, default=False
          Sets the amount of verbosity
  
      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).
  
      positive : bool, default=False
          Restrict coefficients to be >= 0. Be aware that you might want to
          remove fit_intercept which is set True by default.
          See reservations for using this option in combination with method
          'lasso' for expected small values of alpha in the doc of LassoLarsCV
          and LassoLarsIC.
  
      normalize : bool, default=True
          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``.
  
      max_iter : int, default=500
          Maximum number of iterations to perform.
  
      eps : float, default=np.finfo(float).eps
          The machine-precision regularization in the computation of the
          Cholesky diagonal factors. Increase this for very ill-conditioned
          systems. Unlike the ``tol`` parameter in some iterative
          optimization-based algorithms, this parameter does not control
          the tolerance of the optimization.
  
      Returns
      --------
      alphas : array-like of shape (n_alphas,)
          Maximum of covariances (in absolute value) at each iteration.
          ``n_alphas`` is either ``max_iter`` or ``n_features``, whichever
          is smaller.
  
      active : list
          Indices of active variables at the end of the path.
  
      coefs : array-like of shape (n_features, n_alphas)
          Coefficients along the path
  
      residues : array-like of shape (n_alphas, n_samples)
          Residues of the prediction on the test data
      """
      X_train = _check_copy_and_writeable(X_train, copy)
      y_train = _check_copy_and_writeable(y_train, copy)
      X_test = _check_copy_and_writeable(X_test, copy)
      y_test = _check_copy_and_writeable(y_test, copy)
  
      if fit_intercept:
          X_mean = X_train.mean(axis=0)
          X_train -= X_mean
          X_test -= X_mean
          y_mean = y_train.mean(axis=0)
          y_train = as_float_array(y_train, copy=False)
          y_train -= y_mean
          y_test = as_float_array(y_test, copy=False)
          y_test -= y_mean
  
      if normalize:
          norms = np.sqrt(np.sum(X_train ** 2, axis=0))
          nonzeros = np.flatnonzero(norms)
          X_train[:, nonzeros] /= norms[nonzeros]
  
      alphas, active, coefs = lars_path(
          X_train, y_train, Gram=Gram, copy_X=False, copy_Gram=False,
          method=method, verbose=max(0, verbose - 1), max_iter=max_iter, eps=eps,
          positive=positive)
      if normalize:
          coefs[nonzeros] /= norms[nonzeros][:, np.newaxis]
      residues = np.dot(X_test, coefs) - y_test[:, np.newaxis]
      return alphas, active, coefs, residues.T
  
  
  class LarsCV(Lars):
      """Cross-validated Least Angle Regression model.
  
      See glossary entry for :term:`cross-validation estimator`.
  
      Read more in the :ref:`User Guide <least_angle_regression>`.
  
      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).
  
      verbose : bool or int, default=False
          Sets the verbosity amount.
  
      max_iter : int, default=500
          Maximum number of iterations to perform.
  
      normalize : bool, default=True
          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``.
  
      precompute : bool, 'auto' or array-like , default='auto'
          Whether to use a precomputed Gram matrix to speed up
          calculations. If set to ``'auto'`` let us decide. The Gram matrix
          cannot be passed as argument since we will use only subsets of X.
  
      cv : int, cross-validation generator or an iterable, default=None
          Determines the cross-validation splitting strategy.
          Possible inputs for cv are:
  
          - None, to use the default 5-fold cross-validation,
          - integer, to specify the number of folds.
          - :term:`CV splitter`,
          - An iterable yielding (train, test) splits as arrays of indices.
  
          For integer/None inputs, :class:`KFold` is used.
  
          Refer :ref:`User Guide <cross_validation>` for the various
          cross-validation strategies that can be used here.
  
          .. versionchanged:: 0.22
              ``cv`` default value if None changed from 3-fold to 5-fold.
  
      max_n_alphas : int, default=1000
          The maximum number of points on the path used to compute the
          residuals in the cross-validation
  
      n_jobs : int or None, default=None
          Number of CPUs to use during the cross validation.
          ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
          ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
          for more details.
  
      eps : float, default=np.finfo(float).eps
          The machine-precision regularization in the computation of the
          Cholesky diagonal factors. Increase this for very ill-conditioned
          systems. Unlike the ``tol`` parameter in some iterative
          optimization-based algorithms, this parameter does not control
          the tolerance of the optimization.
  
      copy_X : bool, default=True
          If ``True``, X will be copied; else, it may be overwritten.
  
      Attributes
      ----------
      active_ : list of length n_alphas or list of such lists
          Indices of active variables at the end of the path.
          If this is a list of lists, the outer list length is `n_targets`.
  
      coef_ : array-like of shape (n_features,)
          parameter vector (w in the formulation formula)
  
      intercept_ : float
          independent term in decision function
  
      coef_path_ : array-like of shape (n_features, n_alphas)
          the varying values of the coefficients along the path
  
      alpha_ : float
          the estimated regularization parameter alpha
  
      alphas_ : array-like of shape (n_alphas,)
          the different values of alpha along the path
  
      cv_alphas_ : array-like of shape (n_cv_alphas,)
          all the values of alpha along the path for the different folds
  
      mse_path_ : array-like of shape (n_folds, n_cv_alphas)
          the mean square error on left-out for each fold along the path
          (alpha values given by ``cv_alphas``)
  
      n_iter_ : array-like or int
          the number of iterations run by Lars with the optimal alpha.
  
      Examples
      --------
      >>> from sklearn.linear_model import LarsCV
      >>> from sklearn.datasets import make_regression
      >>> X, y = make_regression(n_samples=200, noise=4.0, random_state=0)
      >>> reg = LarsCV(cv=5).fit(X, y)
      >>> reg.score(X, y)
      0.9996...
      >>> reg.alpha_
      0.0254...
      >>> reg.predict(X[:1,])
      array([154.0842...])
  
      See Also
      --------
      lars_path, LassoLars, LassoLarsCV
      """
  
      method = "lar"
  
      @_deprecate_positional_args
      def __init__(self, *, fit_intercept=True, verbose=False, max_iter=500,
                   normalize=True, precompute='auto', cv=None,
                   max_n_alphas=1000, n_jobs=None, eps=np.finfo(float).eps,
                   copy_X=True):
          self.max_iter = max_iter
          self.cv = cv
          self.max_n_alphas = max_n_alphas
          self.n_jobs = n_jobs
          super().__init__(fit_intercept=fit_intercept,
                           verbose=verbose, normalize=normalize,
                           precompute=precompute,
                           n_nonzero_coefs=500,
                           eps=eps, copy_X=copy_X, fit_path=True)
  
      def _more_tags(self):
          return {'multioutput': False}
  
      def fit(self, X, y):
          """Fit the model using X, y as training data.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Training data.
  
          y : array-like of shape (n_samples,)
              Target values.
  
          Returns
          -------
          self : object
              returns an instance of self.
          """
          X, y = self._validate_data(X, y, y_numeric=True)
          X = as_float_array(X, copy=self.copy_X)
          y = as_float_array(y, copy=self.copy_X)
  
          # init cross-validation generator
          cv = check_cv(self.cv, classifier=False)
  
          # As we use cross-validation, the Gram matrix is not precomputed here
          Gram = self.precompute
          if hasattr(Gram, '__array__'):
              warnings.warn('Parameter "precompute" cannot be an array in '
                            '%s. Automatically switch to "auto" instead.'
                            % self.__class__.__name__)
              Gram = 'auto'
  
          cv_paths = Parallel(n_jobs=self.n_jobs, verbose=self.verbose)(
              delayed(_lars_path_residues)(
                  X[train], y[train], X[test], y[test], Gram=Gram, copy=False,
                  method=self.method, verbose=max(0, self.verbose - 1),
                  normalize=self.normalize, fit_intercept=self.fit_intercept,
                  max_iter=self.max_iter, eps=self.eps, positive=self.positive)
              for train, test in cv.split(X, y))
          all_alphas = np.concatenate(list(zip(*cv_paths))[0])
          # Unique also sorts
          all_alphas = np.unique(all_alphas)
          # Take at most max_n_alphas values
          stride = int(max(1, int(len(all_alphas) / float(self.max_n_alphas))))
          all_alphas = all_alphas[::stride]
  
          mse_path = np.empty((len(all_alphas), len(cv_paths)))
          for index, (alphas, _, _, residues) in enumerate(cv_paths):
              alphas = alphas[::-1]
              residues = residues[::-1]
              if alphas[0] != 0:
                  alphas = np.r_[0, alphas]
                  residues = np.r_[residues[0, np.newaxis], residues]
              if alphas[-1] != all_alphas[-1]:
                  alphas = np.r_[alphas, all_alphas[-1]]
                  residues = np.r_[residues, residues[-1, np.newaxis]]
              this_residues = interpolate.interp1d(alphas,
                                                   residues,
                                                   axis=0)(all_alphas)
              this_residues **= 2
              mse_path[:, index] = np.mean(this_residues, axis=-1)
  
          mask = np.all(np.isfinite(mse_path), axis=-1)
          all_alphas = all_alphas[mask]
          mse_path = mse_path[mask]
          # Select the alpha that minimizes left-out error
          i_best_alpha = np.argmin(mse_path.mean(axis=-1))
          best_alpha = all_alphas[i_best_alpha]
  
          # Store our parameters
          self.alpha_ = best_alpha
          self.cv_alphas_ = all_alphas
          self.mse_path_ = mse_path
  
          # Now compute the full model
          # it will call a lasso internally when self if LassoLarsCV
          # as self.method == 'lasso'
          self._fit(X, y, max_iter=self.max_iter, alpha=best_alpha,
                    Xy=None, fit_path=True)
          return self
  
  
  class LassoLarsCV(LarsCV):
      """Cross-validated Lasso, using the LARS algorithm.
  
      See glossary entry for :term:`cross-validation estimator`.
  
      The optimization objective for Lasso is::
  
      (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1
  
      Read more in the :ref:`User Guide <least_angle_regression>`.
  
      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).
  
      verbose : bool or int, default=False
          Sets the verbosity amount.
  
      max_iter : int, default=500
          Maximum number of iterations to perform.
  
      normalize : bool, default=True
          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``.
  
      precompute : bool or 'auto' , default='auto'
          Whether to use a precomputed Gram matrix to speed up
          calculations. If set to ``'auto'`` let us decide. The Gram matrix
          cannot be passed as argument since we will use only subsets of X.
  
      cv : int, cross-validation generator or an iterable, default=None
          Determines the cross-validation splitting strategy.
          Possible inputs for cv are:
  
          - None, to use the default 5-fold cross-validation,
          - integer, to specify the number of folds.
          - :term:`CV splitter`,
          - An iterable yielding (train, test) splits as arrays of indices.
  
          For integer/None inputs, :class:`KFold` is used.
  
          Refer :ref:`User Guide <cross_validation>` for the various
          cross-validation strategies that can be used here.
  
          .. versionchanged:: 0.22
              ``cv`` default value if None changed from 3-fold to 5-fold.
  
      max_n_alphas : int, default=1000
          The maximum number of points on the path used to compute the
          residuals in the cross-validation
  
      n_jobs : int or None, default=None
          Number of CPUs to use during the cross validation.
          ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
          ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
          for more details.
  
      eps : float, default=np.finfo(float).eps
          The machine-precision regularization in the computation of the
          Cholesky diagonal factors. Increase this for very ill-conditioned
          systems. Unlike the ``tol`` parameter in some iterative
          optimization-based algorithms, this parameter does not control
          the tolerance of the optimization.
  
      copy_X : bool, default=True
          If True, X will be copied; else, it may be overwritten.
  
      positive : bool, default=False
          Restrict coefficients to be >= 0. Be aware that you might want to
          remove fit_intercept which is set True by default.
          Under the positive restriction the model coefficients do not converge
          to the ordinary-least-squares solution for small values of alpha.
          Only coefficients up to the smallest alpha value (``alphas_[alphas_ >
          0.].min()`` when fit_path=True) reached by the stepwise Lars-Lasso
          algorithm are typically in congruence with the solution of the
          coordinate descent Lasso estimator.
          As a consequence using LassoLarsCV only makes sense for problems where
          a sparse solution is expected and/or reached.
  
      Attributes
      ----------
      coef_ : array-like of shape (n_features,)
          parameter vector (w in the formulation formula)
  
      intercept_ : float
          independent term in decision function.
  
      coef_path_ : array-like of shape (n_features, n_alphas)
          the varying values of the coefficients along the path
  
      alpha_ : float
          the estimated regularization parameter alpha
  
      alphas_ : array-like of shape (n_alphas,)
          the different values of alpha along the path
  
      cv_alphas_ : array-like of shape (n_cv_alphas,)
          all the values of alpha along the path for the different folds
  
      mse_path_ : array-like of shape (n_folds, n_cv_alphas)
          the mean square error on left-out for each fold along the path
          (alpha values given by ``cv_alphas``)
  
      n_iter_ : array-like or int
          the number of iterations run by Lars with the optimal alpha.
  
      active_ : list of int
          Indices of active variables at the end of the path.
  
      Examples
      --------
      >>> from sklearn.linear_model import LassoLarsCV
      >>> from sklearn.datasets import make_regression
      >>> X, y = make_regression(noise=4.0, random_state=0)
      >>> reg = LassoLarsCV(cv=5).fit(X, y)
      >>> reg.score(X, y)
      0.9992...
      >>> reg.alpha_
      0.0484...
      >>> reg.predict(X[:1,])
      array([-77.8723...])
  
      Notes
      -----
  
      The object solves the same problem as the LassoCV object. However,
      unlike the LassoCV, it find the relevant alphas values by itself.
      In general, because of this property, it will be more stable.
      However, it is more fragile to heavily multicollinear datasets.
  
      It is more efficient than the LassoCV if only a small number of
      features are selected compared to the total number, for instance if
      there are very few samples compared to the number of features.
  
      See Also
      --------
      lars_path, LassoLars, LarsCV, LassoCV
      """
  
      method = 'lasso'
  
      @_deprecate_positional_args
      def __init__(self, *, fit_intercept=True, verbose=False, max_iter=500,
                   normalize=True, precompute='auto', cv=None,
                   max_n_alphas=1000, n_jobs=None, eps=np.finfo(float).eps,
                   copy_X=True, positive=False):
          self.fit_intercept = fit_intercept
          self.verbose = verbose
          self.max_iter = max_iter
          self.normalize = normalize
          self.precompute = precompute
          self.cv = cv
          self.max_n_alphas = max_n_alphas
          self.n_jobs = n_jobs
          self.eps = eps
          self.copy_X = copy_X
          self.positive = positive
          # XXX : we don't use super().__init__
          # to avoid setting n_nonzero_coefs
  
  
  class LassoLarsIC(LassoLars):
      """Lasso model fit with Lars using BIC or AIC for model selection
  
      The optimization objective for Lasso is::
  
      (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1
  
      AIC is the Akaike information criterion and BIC is the Bayes
      Information criterion. Such criteria are useful to select the value
      of the regularization parameter by making a trade-off between the
      goodness of fit and the complexity of the model. A good model should
      explain well the data while being simple.
  
      Read more in the :ref:`User Guide <least_angle_regression>`.
  
      Parameters
      ----------
      criterion : {'bic' , 'aic'}, default='aic'
          The type of criterion to use.
  
      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).
  
      verbose : bool or int, default=False
          Sets the verbosity amount.
  
      normalize : bool, default=True
          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``.
  
      precompute : bool, 'auto' or array-like, default='auto'
          Whether to use a precomputed Gram matrix to speed up
          calculations. If set to ``'auto'`` let us decide. The Gram
          matrix can also be passed as argument.
  
      max_iter : int, default=500
          Maximum number of iterations to perform. Can be used for
          early stopping.
  
      eps : float, default=np.finfo(float).eps
          The machine-precision regularization in the computation of the
          Cholesky diagonal factors. Increase this for very ill-conditioned
          systems. Unlike the ``tol`` parameter in some iterative
          optimization-based algorithms, this parameter does not control
          the tolerance of the optimization.
  
      copy_X : bool, default=True
          If True, X will be copied; else, it may be overwritten.
  
      positive : bool, default=False
          Restrict coefficients to be >= 0. Be aware that you might want to
          remove fit_intercept which is set True by default.
          Under the positive restriction the model coefficients do not converge
          to the ordinary-least-squares solution for small values of alpha.
          Only coefficients up to the smallest alpha value (``alphas_[alphas_ >
          0.].min()`` when fit_path=True) reached by the stepwise Lars-Lasso
          algorithm are typically in congruence with the solution of the
          coordinate descent Lasso estimator.
          As a consequence using LassoLarsIC only makes sense for problems where
          a sparse solution is expected and/or reached.
  
      Attributes
      ----------
      coef_ : array-like of shape (n_features,)
          parameter vector (w in the formulation formula)
  
      intercept_ : float
          independent term in decision function.
  
      alpha_ : float
          the alpha parameter chosen by the information criterion
  
      alphas_ : array-like of shape (n_alphas + 1,) or list of such arrays
          Maximum of covariances (in absolute value) at each iteration.
          ``n_alphas`` is either ``max_iter``, ``n_features`` or the
          number of nodes in the path with ``alpha >= alpha_min``, whichever
          is smaller. If a list, it will be of length `n_targets`.
  
      n_iter_ : int
          number of iterations run by lars_path to find the grid of
          alphas.
  
      criterion_ : array-like of shape (n_alphas,)
          The value of the information criteria ('aic', 'bic') across all
          alphas. The alpha which has the smallest information criterion is
          chosen. This value is larger by a factor of ``n_samples`` compared to
          Eqns. 2.15 and 2.16 in (Zou et al, 2007).
  
  
      Examples
      --------
      >>> from sklearn import linear_model
      >>> reg = linear_model.LassoLarsIC(criterion='bic')
      >>> reg.fit([[-1, 1], [0, 0], [1, 1]], [-1.1111, 0, -1.1111])
      LassoLarsIC(criterion='bic')
      >>> print(reg.coef_)
      [ 0.  -1.11...]
  
      Notes
      -----
      The estimation of the number of degrees of freedom is given by:
  
      "On the degrees of freedom of the lasso"
      Hui Zou, Trevor Hastie, and Robert Tibshirani
      Ann. Statist. Volume 35, Number 5 (2007), 2173-2192.
  
      https://en.wikipedia.org/wiki/Akaike_information_criterion
      https://en.wikipedia.org/wiki/Bayesian_information_criterion
  
      See Also
      --------
      lars_path, LassoLars, LassoLarsCV
      """
      @_deprecate_positional_args
      def __init__(self, criterion='aic', *, fit_intercept=True, verbose=False,
                   normalize=True, precompute='auto', max_iter=500,
                   eps=np.finfo(float).eps, copy_X=True, positive=False):
          self.criterion = criterion
          self.fit_intercept = fit_intercept
          self.positive = positive
          self.max_iter = max_iter
          self.verbose = verbose
          self.normalize = normalize
          self.copy_X = copy_X
          self.precompute = precompute
          self.eps = eps
          self.fit_path = True
  
      def _more_tags(self):
          return {'multioutput': False}
  
      def fit(self, X, y, copy_X=None):
          """Fit the model using X, y as training data.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              training data.
  
          y : array-like of shape (n_samples,)
              target values. Will be cast to X's dtype if necessary
  
          copy_X : bool, default=None
              If provided, this parameter will override the choice
              of copy_X made at instance creation.
              If ``True``, X will be copied; else, it may be overwritten.
  
          Returns
          -------
          self : object
              returns an instance of self.
          """
          if copy_X is None:
              copy_X = self.copy_X
          X, y = self._validate_data(X, y, y_numeric=True)
  
          X, y, Xmean, ymean, Xstd = LinearModel._preprocess_data(
              X, y, self.fit_intercept, self.normalize, copy_X)
  
          Gram = self.precompute
  
          alphas_, _, coef_path_, self.n_iter_ = lars_path(
              X, y, Gram=Gram, copy_X=copy_X, copy_Gram=True, alpha_min=0.0,
              method='lasso', verbose=self.verbose, max_iter=self.max_iter,
              eps=self.eps, return_n_iter=True, positive=self.positive)
  
          n_samples = X.shape[0]
  
          if self.criterion == 'aic':
              K = 2  # AIC
          elif self.criterion == 'bic':
              K = log(n_samples)  # BIC
          else:
              raise ValueError('criterion should be either bic or aic')
  
          R = y[:, np.newaxis] - np.dot(X, coef_path_)  # residuals
          mean_squared_error = np.mean(R ** 2, axis=0)
          sigma2 = np.var(y)
  
          df = np.zeros(coef_path_.shape[1], dtype=int)  # Degrees of freedom
          for k, coef in enumerate(coef_path_.T):
              mask = np.abs(coef) > np.finfo(coef.dtype).eps
              if not np.any(mask):
                  continue
              # get the number of degrees of freedom equal to:
              # Xc = X[:, mask]
              # Trace(Xc * inv(Xc.T, Xc) * Xc.T) ie the number of non-zero coefs
              df[k] = np.sum(mask)
  
          self.alphas_ = alphas_
          eps64 = np.finfo('float64').eps
          self.criterion_ = (n_samples * mean_squared_error / (sigma2 + eps64) +
                             K * df)  # Eqns. 2.15--16 in (Zou et al, 2007)
          n_best = np.argmin(self.criterion_)
  
          self.alpha_ = alphas_[n_best]
          self.coef_ = coef_path_[:, n_best]
          self._set_intercept(Xmean, ymean, Xstd)
          return self
  