text: """
  The :mod:`sklearn.model_selection._validation` module includes classes and
  functions to validate the model.
  """
  
  # Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>
  #         Gael Varoquaux <gael.varoquaux@normalesup.org>
  #         Olivier Grisel <olivier.grisel@ensta.org>
  #         Raghav RV <rvraghav93@gmail.com>
  # License: BSD 3 clause
  
  
  import warnings
  import numbers
  import time
  from traceback import format_exc
  from contextlib import suppress
  
  import numpy as np
  import scipy.sparse as sp
  from joblib import Parallel, logger
  
  from ..base import is_classifier, clone
  from ..utils import indexable, check_random_state, _safe_indexing
  from ..utils.validation import _check_fit_params
  from ..utils.validation import _num_samples
  from ..utils.validation import _deprecate_positional_args
  from ..utils.fixes import delayed
  from ..utils.metaestimators import _safe_split
  from ..metrics import check_scoring
  from ..metrics._scorer import _check_multimetric_scoring, _MultimetricScorer
  from ..exceptions import FitFailedWarning, NotFittedError
  from ._split import check_cv
  from ..preprocessing import LabelEncoder
  
  
  __all__ = ['cross_validate', 'cross_val_score', 'cross_val_predict',
             'permutation_test_score', 'learning_curve', 'validation_curve']
  
  
  @_deprecate_positional_args
  def cross_validate(estimator, X, y=None, *, groups=None, scoring=None, cv=None,
                     n_jobs=None, verbose=0, fit_params=None,
                     pre_dispatch='2*n_jobs', return_train_score=False,
                     return_estimator=False, error_score=np.nan):
      """Evaluate metric(s) by cross-validation and also record fit/score times.
  
      Read more in the :ref:`User Guide <multimetric_cross_validation>`.
  
      Parameters
      ----------
      estimator : estimator object implementing 'fit'
          The object to use to fit the data.
  
      X : array-like of shape (n_samples, n_features)
          The data to fit. Can be for example a list, or an array.
  
      y : array-like of shape (n_samples,) or (n_samples, n_outputs), \
              default=None
          The target variable to try to predict in the case of
          supervised learning.
  
      groups : array-like of shape (n_samples,), default=None
          Group labels for the samples used while splitting the dataset into
          train/test set. Only used in conjunction with a "Group" :term:`cv`
          instance (e.g., :class:`GroupKFold`).
  
      scoring : str, callable, list/tuple, or dict, default=None
          A single str (see :ref:`scoring_parameter`) or a callable
          (see :ref:`scoring`) to evaluate the predictions on the test set.
  
          For evaluating multiple metrics, either give a list of (unique) strings
          or a dict with names as keys and callables as values.
  
          NOTE that when using custom scorers, each scorer should return a single
          value. Metric functions returning a list/array of values can be wrapped
          into multiple scorers that return one value each.
  
          See :ref:`multimetric_grid_search` for an example.
  
          If None, the estimator's score method is used.
  
      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,
          - int, to specify the number of folds in a `(Stratified)KFold`,
          - :term:`CV splitter`,
          - An iterable yielding (train, test) splits as arrays of indices.
  
          For int/None inputs, if the estimator is a classifier and ``y`` is
          either binary or multiclass, :class:`StratifiedKFold` is used. In all
          other cases, :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.
  
      n_jobs : int, default=None
          Number of jobs to run in parallel. Training the estimator and computing
          the score are parallelized over the cross-validation splits.
          ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
          ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
          for more details.
  
      verbose : int, default=0
          The verbosity level.
  
      fit_params : dict, default=None
          Parameters to pass to the fit method of the estimator.
  
      pre_dispatch : int or str, default='2*n_jobs'
          Controls the number of jobs that get dispatched during parallel
          execution. Reducing this number can be useful to avoid an
          explosion of memory consumption when more jobs get dispatched
          than CPUs can process. This parameter can be:
  
              - None, in which case all the jobs are immediately
                created and spawned. Use this for lightweight and
                fast-running jobs, to avoid delays due to on-demand
                spawning of the jobs
  
              - An int, giving the exact number of total jobs that are
                spawned
  
              - A str, giving an expression as a function of n_jobs,
                as in '2*n_jobs'
  
      return_train_score : bool, default=False
          Whether to include train scores.
          Computing training scores is used to get insights on how different
          parameter settings impact the overfitting/underfitting trade-off.
          However computing the scores on the training set can be computationally
          expensive and is not strictly required to select the parameters that
          yield the best generalization performance.
  
          .. versionadded:: 0.19
  
          .. versionchanged:: 0.21
              Default value was changed from ``True`` to ``False``
  
      return_estimator : bool, default=False
          Whether to return the estimators fitted on each split.
  
          .. versionadded:: 0.20
  
      error_score : 'raise' or numeric, default=np.nan
          Value to assign to the score if an error occurs in estimator fitting.
          If set to 'raise', the error is raised.
          If a numeric value is given, FitFailedWarning is raised. This parameter
          does not affect the refit step, which will always raise the error.
  
          .. versionadded:: 0.20
  
      Returns
      -------
      scores : dict of float arrays of shape (n_splits,)
          Array of scores of the estimator for each run of the cross validation.
  
          A dict of arrays containing the score/time arrays for each scorer is
          returned. The possible keys for this ``dict`` are:
  
              ``test_score``
                  The score array for test scores on each cv split.
                  Suffix ``_score`` in ``test_score`` changes to a specific
                  metric like ``test_r2`` or ``test_auc`` if there are
                  multiple scoring metrics in the scoring parameter.
              ``train_score``
                  The score array for train scores on each cv split.
                  Suffix ``_score`` in ``train_score`` changes to a specific
                  metric like ``train_r2`` or ``train_auc`` if there are
                  multiple scoring metrics in the scoring parameter.
                  This is available only if ``return_train_score`` parameter
                  is ``True``.
              ``fit_time``
                  The time for fitting the estimator on the train
                  set for each cv split.
              ``score_time``
                  The time for scoring the estimator on the test set for each
                  cv split. (Note time for scoring on the train set is not
                  included even if ``return_train_score`` is set to ``True``
              ``estimator``
                  The estimator objects for each cv split.
                  This is available only if ``return_estimator`` parameter
                  is set to ``True``.
  
      Examples
      --------
      >>> from sklearn import datasets, linear_model
      >>> from sklearn.model_selection import cross_validate
      >>> from sklearn.metrics import make_scorer
      >>> from sklearn.metrics import confusion_matrix
      >>> from sklearn.svm import LinearSVC
      >>> diabetes = datasets.load_diabetes()
      >>> X = diabetes.data[:150]
      >>> y = diabetes.target[:150]
      >>> lasso = linear_model.Lasso()
  
      Single metric evaluation using ``cross_validate``
  
      >>> cv_results = cross_validate(lasso, X, y, cv=3)
      >>> sorted(cv_results.keys())
      ['fit_time', 'score_time', 'test_score']
      >>> cv_results['test_score']
      array([0.33150734, 0.08022311, 0.03531764])
  
      Multiple metric evaluation using ``cross_validate``
      (please refer the ``scoring`` parameter doc for more information)
  
      >>> scores = cross_validate(lasso, X, y, cv=3,
      ...                         scoring=('r2', 'neg_mean_squared_error'),
      ...                         return_train_score=True)
      >>> print(scores['test_neg_mean_squared_error'])
      [-3635.5... -3573.3... -6114.7...]
      >>> print(scores['train_r2'])
      [0.28010158 0.39088426 0.22784852]
  
      See Also
      ---------
      cross_val_score : Run cross-validation for single metric evaluation.
  
      cross_val_predict : Get predictions from each split of cross-validation for
          diagnostic purposes.
  
      sklearn.metrics.make_scorer : Make a scorer from a performance metric or
          loss function.
  
      """
      X, y, groups = indexable(X, y, groups)
  
      cv = check_cv(cv, y, classifier=is_classifier(estimator))
  
      if callable(scoring):
          scorers = scoring
      elif scoring is None or isinstance(scoring, str):
          scorers = check_scoring(estimator, scoring)
      else:
          scorers = _check_multimetric_scoring(estimator, scoring)
  
      # We clone the estimator to make sure that all the folds are
      # independent, and that it is pickle-able.
      parallel = Parallel(n_jobs=n_jobs, verbose=verbose,
                          pre_dispatch=pre_dispatch)
      results = parallel(
          delayed(_fit_and_score)(
              clone(estimator), X, y, scorers, train, test, verbose, None,
              fit_params, return_train_score=return_train_score,
              return_times=True, return_estimator=return_estimator,
              error_score=error_score)
          for train, test in cv.split(X, y, groups))
  
      # For callabe scoring, the return type is only know after calling. If the
      # return type is a dictionary, the error scores can now be inserted with
      # the correct key.
      if callable(scoring):
          _insert_error_scores(results, error_score)
  
      results = _aggregate_score_dicts(results)
  
      ret = {}
      ret['fit_time'] = results["fit_time"]
      ret['score_time'] = results["score_time"]
  
      if return_estimator:
          ret['estimator'] = results["estimator"]
  
      test_scores_dict = _normalize_score_results(results["test_scores"])
      if return_train_score:
          train_scores_dict = _normalize_score_results(results["train_scores"])
  
      for name in test_scores_dict:
          ret['test_%s' % name] = test_scores_dict[name]
          if return_train_score:
              key = 'train_%s' % name
              ret[key] = train_scores_dict[name]
  
      return ret
  
  
  def _insert_error_scores(results, error_score):
      """Insert error in `results` by replacing them inplace with `error_score`.
  
      This only applies to multimetric scores because `_fit_and_score` will
      handle the single metric case.
      """
      successful_score = None
      failed_indices = []
      for i, result in enumerate(results):
          if result["fit_failed"]:
              failed_indices.append(i)
          elif successful_score is None:
              successful_score = result["test_scores"]
  
      if successful_score is None:
          raise NotFittedError("All estimators failed to fit")
  
      if isinstance(successful_score, dict):
          formatted_error = {name: error_score for name in successful_score}
          for i in failed_indices:
              results[i]["test_scores"] = formatted_error.copy()
              if "train_scores" in results[i]:
                  results[i]["train_scores"] = formatted_error.copy()
  
  
  def _normalize_score_results(scores, scaler_score_key='score'):
      """Creates a scoring dictionary based on the type of `scores`"""
      if isinstance(scores[0], dict):
          # multimetric scoring
          return _aggregate_score_dicts(scores)
      # scaler
      return {scaler_score_key: scores}
  
  
  @_deprecate_positional_args
  def cross_val_score(estimator, X, y=None, *, groups=None, scoring=None,
                      cv=None, n_jobs=None, verbose=0, fit_params=None,
                      pre_dispatch='2*n_jobs', error_score=np.nan):
      """Evaluate a score by cross-validation
  
      Read more in the :ref:`User Guide <cross_validation>`.
  
      Parameters
      ----------
      estimator : estimator object implementing 'fit'
          The object to use to fit the data.
  
      X : array-like of shape (n_samples, n_features)
          The data to fit. Can be for example a list, or an array.
  
      y : array-like of shape (n_samples,) or (n_samples, n_outputs), \
              default=None
          The target variable to try to predict in the case of
          supervised learning.
  
      groups : array-like of shape (n_samples,), default=None
          Group labels for the samples used while splitting the dataset into
          train/test set. Only used in conjunction with a "Group" :term:`cv`
          instance (e.g., :class:`GroupKFold`).
  
      scoring : str or callable, default=None
          A str (see model evaluation documentation) or
          a scorer callable object / function with signature
          ``scorer(estimator, X, y)`` which should return only
          a single value.
  
          Similar to :func:`cross_validate`
          but only a single metric is permitted.
  
          If None, the estimator's default scorer (if available) is used.
  
      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,
          - int, to specify the number of folds in a `(Stratified)KFold`,
          - :term:`CV splitter`,
          - An iterable yielding (train, test) splits as arrays of indices.
  
          For int/None inputs, if the estimator is a classifier and ``y`` is
          either binary or multiclass, :class:`StratifiedKFold` is used. In all
          other cases, :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.
  
      n_jobs : int, default=None
          Number of jobs to run in parallel. Training the estimator and computing
          the score are parallelized over the cross-validation splits.
          ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
          ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
          for more details.
  
      verbose : int, default=0
          The verbosity level.
  
      fit_params : dict, default=None
          Parameters to pass to the fit method of the estimator.
  
      pre_dispatch : int or str, default='2*n_jobs'
          Controls the number of jobs that get dispatched during parallel
          execution. Reducing this number can be useful to avoid an
          explosion of memory consumption when more jobs get dispatched
          than CPUs can process. This parameter can be:
  
              - None, in which case all the jobs are immediately
                created and spawned. Use this for lightweight and
                fast-running jobs, to avoid delays due to on-demand
                spawning of the jobs
  
              - An int, giving the exact number of total jobs that are
                spawned
  
              - A str, giving an expression as a function of n_jobs,
                as in '2*n_jobs'
  
      error_score : 'raise' or numeric, default=np.nan
          Value to assign to the score if an error occurs in estimator fitting.
          If set to 'raise', the error is raised.
          If a numeric value is given, FitFailedWarning is raised. This parameter
          does not affect the refit step, which will always raise the error.
  
          .. versionadded:: 0.20
  
      Returns
      -------
      scores : ndarray of float of shape=(len(list(cv)),)
          Array of scores of the estimator for each run of the cross validation.
  
      Examples
      --------
      >>> from sklearn import datasets, linear_model
      >>> from sklearn.model_selection import cross_val_score
      >>> diabetes = datasets.load_diabetes()
      >>> X = diabetes.data[:150]
      >>> y = diabetes.target[:150]
      >>> lasso = linear_model.Lasso()
      >>> print(cross_val_score(lasso, X, y, cv=3))
      [0.33150734 0.08022311 0.03531764]
  
      See Also
      ---------
      cross_validate : To run cross-validation on multiple metrics and also to
          return train scores, fit times and score times.
  
      cross_val_predict : Get predictions from each split of cross-validation for
          diagnostic purposes.
  
      sklearn.metrics.make_scorer : Make a scorer from a performance metric or
          loss function.
  
      """
      # To ensure multimetric format is not supported
      scorer = check_scoring(estimator, scoring=scoring)
  
      cv_results = cross_validate(estimator=estimator, X=X, y=y, groups=groups,
                                  scoring={'score': scorer}, cv=cv,
                                  n_jobs=n_jobs, verbose=verbose,
                                  fit_params=fit_params,
                                  pre_dispatch=pre_dispatch,
                                  error_score=error_score)
      return cv_results['test_score']
  
  
  def _fit_and_score(estimator, X, y, scorer, train, test, verbose,
                     parameters, fit_params, return_train_score=False,
                     return_parameters=False, return_n_test_samples=False,
                     return_times=False, return_estimator=False,
                     split_progress=None, candidate_progress=None,
                     error_score=np.nan):
  
      """Fit estimator and compute scores for a given dataset split.
  
      Parameters
      ----------
      estimator : estimator object implementing 'fit'
          The object to use to fit the data.
  
      X : array-like of shape (n_samples, n_features)
          The data to fit.
  
      y : array-like of shape (n_samples,) or (n_samples, n_outputs) or None
          The target variable to try to predict in the case of
          supervised learning.
  
      scorer : A single callable or dict mapping scorer name to the callable
          If it is a single callable, the return value for ``train_scores`` and
          ``test_scores`` is a single float.
  
          For a dict, it should be one mapping the scorer name to the scorer
          callable object / function.
  
          The callable object / fn should have signature
          ``scorer(estimator, X, y)``.
  
      train : array-like of shape (n_train_samples,)
          Indices of training samples.
  
      test : array-like of shape (n_test_samples,)
          Indices of test samples.
  
      verbose : int
          The verbosity level.
  
      error_score : 'raise' or numeric, default=np.nan
          Value to assign to the score if an error occurs in estimator fitting.
          If set to 'raise', the error is raised.
          If a numeric value is given, FitFailedWarning is raised. This parameter
          does not affect the refit step, which will always raise the error.
  
      parameters : dict or None
          Parameters to be set on the estimator.
  
      fit_params : dict or None
          Parameters that will be passed to ``estimator.fit``.
  
      return_train_score : bool, default=False
          Compute and return score on training set.
  
      return_parameters : bool, default=False
          Return parameters that has been used for the estimator.
  
      split_progress : list or tuple, optional, default: None
          A list or tuple of format (<current_split_id>, <total_num_of_splits>)
  
      candidate_progress : list or tuple, optional, default: None
          A list or tuple of format
          (<current_candidate_id>, <total_number_of_candidates>)
  
      return_n_test_samples : bool, default=False
          Whether to return the ``n_test_samples``
  
      return_times : bool, default=False
          Whether to return the fit/score times.
  
      return_estimator : bool, default=False
          Whether to return the fitted estimator.
  
      Returns
      -------
      result : dict with the following attributes
          train_scores : dict of scorer name -> float
              Score on training set (for all the scorers),
              returned only if `return_train_score` is `True`.
          test_scores : dict of scorer name -> float
              Score on testing set (for all the scorers).
          n_test_samples : int
              Number of test samples.
          fit_time : float
              Time spent for fitting in seconds.
          score_time : float
              Time spent for scoring in seconds.
          parameters : dict or None
              The parameters that have been evaluated.
          estimator : estimator object
              The fitted estimator.
          fit_failed : bool
              The estimator failed to fit.
      """
      progress_msg = ""
      if verbose > 2:
          if split_progress is not None:
              progress_msg = f" {split_progress[0]+1}/{split_progress[1]}"
          if candidate_progress and verbose > 9:
              progress_msg += (f"; {candidate_progress[0]+1}/"
                               f"{candidate_progress[1]}")
  
      if verbose > 1:
          if parameters is None:
              params_msg = ''
          else:
              sorted_keys = sorted(parameters)  # Ensure deterministic o/p
              params_msg = (', '.join(f'{k}={parameters[k]}'
                                      for k in sorted_keys))
      if verbose > 9:
          start_msg = f"[CV{progress_msg}] START {params_msg}"
          print(f"{start_msg}{(80 - len(start_msg)) * '.'}")
  
      # Adjust length of sample weights
      fit_params = fit_params if fit_params is not None else {}
      fit_params = _check_fit_params(X, fit_params, train)
  
      if parameters is not None:
          # clone after setting parameters in case any parameters
          # are estimators (like pipeline steps)
          # because pipeline doesn't clone steps in fit
          cloned_parameters = {}
          for k, v in parameters.items():
              cloned_parameters[k] = clone(v, safe=False)
  
          estimator = estimator.set_params(**cloned_parameters)
  
      start_time = time.time()
  
      X_train, y_train = _safe_split(estimator, X, y, train)
      X_test, y_test = _safe_split(estimator, X, y, test, train)
  
      result = {}
      try:
          if y_train is None:
              estimator.fit(X_train, **fit_params)
          else:
              estimator.fit(X_train, y_train, **fit_params)
  
      except Exception as e:
          # Note fit time as time until error
          fit_time = time.time() - start_time
          score_time = 0.0
          if error_score == 'raise':
              raise
          elif isinstance(error_score, numbers.Number):
              if isinstance(scorer, dict):
                  test_scores = {name: error_score for name in scorer}
                  if return_train_score:
                      train_scores = test_scores.copy()
              else:
                  test_scores = error_score
                  if return_train_score:
                      train_scores = error_score
              warnings.warn("Estimator fit failed. The score on this train-test"
                            " partition for these parameters will be set to %f. "
                            "Details: \n%s" %
                            (error_score, format_exc()),
                            FitFailedWarning)
          else:
              raise ValueError("error_score must be the string 'raise' or a"
                               " numeric value. (Hint: if using 'raise', please"
                               " make sure that it has been spelled correctly.)")
          result["fit_failed"] = True
      else:
          result["fit_failed"] = False
  
          fit_time = time.time() - start_time
          test_scores = _score(estimator, X_test, y_test, scorer)
          score_time = time.time() - start_time - fit_time
          if return_train_score:
              train_scores = _score(estimator, X_train, y_train, scorer)
  
      if verbose > 1:
          total_time = score_time + fit_time
          end_msg = f"[CV{progress_msg}] END "
          result_msg = params_msg + (";" if params_msg else "")
          if verbose > 2 and isinstance(test_scores, dict):
              for scorer_name in sorted(test_scores):
                  result_msg += f" {scorer_name}: ("
                  if return_train_score:
                      scorer_scores = train_scores[scorer_name]
                      result_msg += f"train={scorer_scores:.3f}, "
                  result_msg += f"test={test_scores[scorer_name]:.3f})"
          result_msg += f" total time={logger.short_format_time(total_time)}"
  
          # Right align the result_msg
          end_msg += "." * (80 - len(end_msg) - len(result_msg))
          end_msg += result_msg
          print(end_msg)
  
      result["test_scores"] = test_scores
      if return_train_score:
          result["train_scores"] = train_scores
      if return_n_test_samples:
          result["n_test_samples"] = _num_samples(X_test)
      if return_times:
          result["fit_time"] = fit_time
          result["score_time"] = score_time
      if return_parameters:
          result["parameters"] = parameters
      if return_estimator:
          result["estimator"] = estimator
      return result
  
  
  def _score(estimator, X_test, y_test, scorer):
      """Compute the score(s) of an estimator on a given test set.
  
      Will return a dict of floats if `scorer` is a dict, otherwise a single
      float is returned.
      """
      if isinstance(scorer, dict):
          # will cache method calls if needed. scorer() returns a dict
          scorer = _MultimetricScorer(**scorer)
      if y_test is None:
          scores = scorer(estimator, X_test)
      else:
          scores = scorer(estimator, X_test, y_test)
  
      error_msg = ("scoring must return a number, got %s (%s) "
                   "instead. (scorer=%s)")
      if isinstance(scores, dict):
          for name, score in scores.items():
              if hasattr(score, 'item'):
                  with suppress(ValueError):
                      # e.g. unwrap memmapped scalars
                      score = score.item()
              if not isinstance(score, numbers.Number):
                  raise ValueError(error_msg % (score, type(score), name))
              scores[name] = score
      else:  # scalar
          if hasattr(scores, 'item'):
              with suppress(ValueError):
                  # e.g. unwrap memmapped scalars
                  scores = scores.item()
          if not isinstance(scores, numbers.Number):
              raise ValueError(error_msg % (scores, type(scores), scorer))
      return scores
  
  
  @_deprecate_positional_args
  def cross_val_predict(estimator, X, y=None, *, groups=None, cv=None,
                        n_jobs=None, verbose=0, fit_params=None,
                        pre_dispatch='2*n_jobs', method='predict'):
      """Generate cross-validated estimates for each input data point
  
      The data is split according to the cv parameter. Each sample belongs
      to exactly one test set, and its prediction is computed with an
      estimator fitted on the corresponding training set.
  
      Passing these predictions into an evaluation metric may not be a valid
      way to measure generalization performance. Results can differ from
      :func:`cross_validate` and :func:`cross_val_score` unless all tests sets
      have equal size and the metric decomposes over samples.
  
      Read more in the :ref:`User Guide <cross_validation>`.
  
      Parameters
      ----------
      estimator : estimator object implementing 'fit' and 'predict'
          The object to use to fit the data.
  
      X : array-like of shape (n_samples, n_features)
          The data to fit. Can be, for example a list, or an array at least 2d.
  
      y : array-like of shape (n_samples,) or (n_samples, n_outputs), \
              default=None
          The target variable to try to predict in the case of
          supervised learning.
  
      groups : array-like of shape (n_samples,), default=None
          Group labels for the samples used while splitting the dataset into
          train/test set. Only used in conjunction with a "Group" :term:`cv`
          instance (e.g., :class:`GroupKFold`).
  
      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,
          - int, to specify the number of folds in a `(Stratified)KFold`,
          - :term:`CV splitter`,
          - An iterable yielding (train, test) splits as arrays of indices.
  
          For int/None inputs, if the estimator is a classifier and ``y`` is
          either binary or multiclass, :class:`StratifiedKFold` is used. In all
          other cases, :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.
  
      n_jobs : int, default=None
          Number of jobs to run in parallel. Training the estimator and
          predicting are parallelized over the cross-validation splits.
          ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
          ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
          for more details.
  
      verbose : int, default=0
          The verbosity level.
  
      fit_params : dict, defualt=None
          Parameters to pass to the fit method of the estimator.
  
      pre_dispatch : int or str, default='2*n_jobs'
          Controls the number of jobs that get dispatched during parallel
          execution. Reducing this number can be useful to avoid an
          explosion of memory consumption when more jobs get dispatched
          than CPUs can process. This parameter can be:
  
              - None, in which case all the jobs are immediately
                created and spawned. Use this for lightweight and
                fast-running jobs, to avoid delays due to on-demand
                spawning of the jobs
  
              - An int, giving the exact number of total jobs that are
                spawned
  
              - A str, giving an expression as a function of n_jobs,
                as in '2*n_jobs'
  
      method : {'predict', 'predict_proba', 'predict_log_proba', \
                'decision_function'}, default='predict'
          The method to be invoked by `estimator`.
  
      Returns
      -------
      predictions : ndarray
          This is the result of calling `method`. Shape:
  
              - When `method` is 'predict' and in special case where `method` is
                'decision_function' and the target is binary: (n_samples,)
              - When `method` is one of {'predict_proba', 'predict_log_proba',
                'decision_function'} (unless special case above):
                (n_samples, n_classes)
              - If `estimator` is :term:`multioutput`, an extra dimension
                'n_outputs' is added to the end of each shape above.
  
      See Also
      --------
      cross_val_score : Calculate score for each CV split.
      cross_validate : Calculate one or more scores and timings for each CV
          split.
  
      Notes
      -----
      In the case that one or more classes are absent in a training portion, a
      default score needs to be assigned to all instances for that class if
      ``method`` produces columns per class, as in {'decision_function',
      'predict_proba', 'predict_log_proba'}.  For ``predict_proba`` this value is
      0.  In order to ensure finite output, we approximate negative infinity by
      the minimum finite float value for the dtype in other cases.
  
      Examples
      --------
      >>> from sklearn import datasets, linear_model
      >>> from sklearn.model_selection import cross_val_predict
      >>> diabetes = datasets.load_diabetes()
      >>> X = diabetes.data[:150]
      >>> y = diabetes.target[:150]
      >>> lasso = linear_model.Lasso()
      >>> y_pred = cross_val_predict(lasso, X, y, cv=3)
      """
      X, y, groups = indexable(X, y, groups)
  
      cv = check_cv(cv, y, classifier=is_classifier(estimator))
  
      # If classification methods produce multiple columns of output,
      # we need to manually encode classes to ensure consistent column ordering.
      encode = method in ['decision_function', 'predict_proba',
                          'predict_log_proba'] and y is not None
      if encode:
          y = np.asarray(y)
          if y.ndim == 1:
              le = LabelEncoder()
              y = le.fit_transform(y)
          elif y.ndim == 2:
              y_enc = np.zeros_like(y, dtype=int)
              for i_label in range(y.shape[1]):
                  y_enc[:, i_label] = LabelEncoder().fit_transform(y[:, i_label])
              y = y_enc
  
      # We clone the estimator to make sure that all the folds are
      # independent, and that it is pickle-able.
      parallel = Parallel(n_jobs=n_jobs, verbose=verbose,
                          pre_dispatch=pre_dispatch)
      prediction_blocks = parallel(delayed(_fit_and_predict)(
          clone(estimator), X, y, train, test, verbose, fit_params, method)
          for train, test in cv.split(X, y, groups))
  
      # Concatenate the predictions
      predictions = [pred_block_i for pred_block_i, _ in prediction_blocks]
      test_indices = np.concatenate([indices_i
                                     for _, indices_i in prediction_blocks])
  
      if not _check_is_permutation(test_indices, _num_samples(X)):
          raise ValueError('cross_val_predict only works for partitions')
  
      inv_test_indices = np.empty(len(test_indices), dtype=int)
      inv_test_indices[test_indices] = np.arange(len(test_indices))
  
      if sp.issparse(predictions[0]):
          predictions = sp.vstack(predictions, format=predictions[0].format)
      elif encode and isinstance(predictions[0], list):
          # `predictions` is a list of method outputs from each fold.
          # If each of those is also a list, then treat this as a
          # multioutput-multiclass task. We need to separately concatenate
          # the method outputs for each label into an `n_labels` long list.
          n_labels = y.shape[1]
          concat_pred = []
          for i_label in range(n_labels):
              label_preds = np.concatenate([p[i_label] for p in predictions])
              concat_pred.append(label_preds)
          predictions = concat_pred
      else:
          predictions = np.concatenate(predictions)
  
      if isinstance(predictions, list):
          return [p[inv_test_indices] for p in predictions]
      else:
          return predictions[inv_test_indices]
  
  
  def _fit_and_predict(estimator, X, y, train, test, verbose, fit_params,
                       method):
      """Fit estimator and predict values for a given dataset split.
  
      Read more in the :ref:`User Guide <cross_validation>`.
  
      Parameters
      ----------
      estimator : estimator object implementing 'fit' and 'predict'
          The object to use to fit the data.
  
      X : array-like of shape (n_samples, n_features)
          The data to fit.
  
          .. versionchanged:: 0.20
              X is only required to be an object with finite length or shape now
  
      y : array-like of shape (n_samples,) or (n_samples, n_outputs) or None
          The target variable to try to predict in the case of
          supervised learning.
  
      train : array-like of shape (n_train_samples,)
          Indices of training samples.
  
      test : array-like of shape (n_test_samples,)
          Indices of test samples.
  
      verbose : int
          The verbosity level.
  
      fit_params : dict or None
          Parameters that will be passed to ``estimator.fit``.
  
      method : str
          Invokes the passed method name of the passed estimator.
  
      Returns
      -------
      predictions : sequence
          Result of calling 'estimator.method'
  
      test : array-like
          This is the value of the test parameter
      """
      # Adjust length of sample weights
      fit_params = fit_params if fit_params is not None else {}
      fit_params = _check_fit_params(X, fit_params, train)
  
      X_train, y_train = _safe_split(estimator, X, y, train)
      X_test, _ = _safe_split(estimator, X, y, test, train)
  
      if y_train is None:
          estimator.fit(X_train, **fit_params)
      else:
          estimator.fit(X_train, y_train, **fit_params)
      func = getattr(estimator, method)
      predictions = func(X_test)
  
      encode = method in ['decision_function', 'predict_proba',
                          'predict_log_proba'] and y is not None
  
      if encode:
          if isinstance(predictions, list):
              predictions = [_enforce_prediction_order(
                  estimator.classes_[i_label], predictions[i_label],
                  n_classes=len(set(y[:, i_label])), method=method)
                  for i_label in range(len(predictions))]
          else:
              # A 2D y array should be a binary label indicator matrix
              n_classes = len(set(y)) if y.ndim == 1 else y.shape[1]
              predictions = _enforce_prediction_order(
                  estimator.classes_, predictions, n_classes, method)
      return predictions, test
  
  
  def _enforce_prediction_order(classes, predictions, n_classes, method):
      """Ensure that prediction arrays have correct column order
  
      When doing cross-validation, if one or more classes are
      not present in the subset of data used for training,
      then the output prediction array might not have the same
      columns as other folds. Use the list of class names
      (assumed to be ints) to enforce the correct column order.
  
      Note that `classes` is the list of classes in this fold
      (a subset of the classes in the full training set)
      and `n_classes` is the number of classes in the full training set.
      """
      if n_classes != len(classes):
          recommendation = (
              'To fix this, use a cross-validation '
              'technique resulting in properly '
              'stratified folds')
          warnings.warn('Number of classes in training fold ({}) does '
                        'not match total number of classes ({}). '
                        'Results may not be appropriate for your use case. '
                        '{}'.format(len(classes), n_classes, recommendation),
                        RuntimeWarning)
          if method == 'decision_function':
              if (predictions.ndim == 2 and
                      predictions.shape[1] != len(classes)):
                  # This handles the case when the shape of predictions
                  # does not match the number of classes used to train
                  # it with. This case is found when sklearn.svm.SVC is
                  # set to `decision_function_shape='ovo'`.
                  raise ValueError('Output shape {} of {} does not match '
                                   'number of classes ({}) in fold. '
                                   'Irregular decision_function outputs '
                                   'are not currently supported by '
                                   'cross_val_predict'.format(
                                      predictions.shape, method, len(classes)))
              if len(classes) <= 2:
                  # In this special case, `predictions` contains a 1D array.
                  raise ValueError('Only {} class/es in training fold, but {} '
                                   'in overall dataset. This '
                                   'is not supported for decision_function '
                                   'with imbalanced folds. {}'.format(
                                      len(classes), n_classes, recommendation))
  
          float_min = np.finfo(predictions.dtype).min
          default_values = {'decision_function': float_min,
                            'predict_log_proba': float_min,
                            'predict_proba': 0}
          predictions_for_all_classes = np.full((_num_samples(predictions),
                                                 n_classes),
                                                default_values[method],
                                                dtype=predictions.dtype)
          predictions_for_all_classes[:, classes] = predictions
          predictions = predictions_for_all_classes
      return predictions
  
  
  def _check_is_permutation(indices, n_samples):
      """Check whether indices is a reordering of the array np.arange(n_samples)
  
      Parameters
      ----------
      indices : ndarray
          int array to test
      n_samples : int
          number of expected elements
  
      Returns
      -------
      is_partition : bool
          True iff sorted(indices) is np.arange(n)
      """
      if len(indices) != n_samples:
          return False
      hit = np.zeros(n_samples, dtype=bool)
      hit[indices] = True
      if not np.all(hit):
          return False
      return True
  
  
  @_deprecate_positional_args
  def permutation_test_score(estimator, X, y, *, groups=None, cv=None,
                             n_permutations=100, n_jobs=None, random_state=0,
                             verbose=0, scoring=None):
      """Evaluates the significance of a cross-validated score using permutations
  
      Permutes targets to generate 'randomized data' and compute the empirical
      p-value against the null hypothesis that features and targets are
      independent.
  
      The p-value represents the fraction of randomized data sets where the
      estimator performed as well or better than in the original data. A small
      p-value suggests that there is a real dependency between features and
      targets which has been used by the estimator to give good predictions.
      A large p-value may be due to lack of real dependency between features
      and targets or the estimator was not able to use the dependency to
      give good predictions.
  
      Read more in the :ref:`User Guide <permutation_test_score>`.
  
      Parameters
      ----------
      estimator : estimator object implementing 'fit'
          The object to use to fit the data.
  
      X : array-like of shape at least 2D
          The data to fit.
  
      y : array-like of shape (n_samples,) or (n_samples, n_outputs) or None
          The target variable to try to predict in the case of
          supervised learning.
  
      groups : array-like of shape (n_samples,), default=None
          Labels to constrain permutation within groups, i.e. ``y`` values
          are permuted among samples with the same group identifier.
          When not specified, ``y`` values are permuted among all samples.
  
          When a grouped cross-validator is used, the group labels are
          also passed on to the ``split`` method of the cross-validator. The
          cross-validator uses them for grouping the samples  while splitting
          the dataset into train/test set.
  
      scoring : str or callable, default=None
          A single str (see :ref:`scoring_parameter`) or a callable
          (see :ref:`scoring`) to evaluate the predictions on the test set.
  
          If None the estimator's score method is used.
  
      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,
          - int, to specify the number of folds in a `(Stratified)KFold`,
          - :term:`CV splitter`,
          - An iterable yielding (train, test) splits as arrays of indices.
  
          For int/None inputs, if the estimator is a classifier and ``y`` is
          either binary or multiclass, :class:`StratifiedKFold` is used. In all
          other cases, :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.
  
      n_permutations : int, default=100
          Number of times to permute ``y``.
  
      n_jobs : int, default=None
          Number of jobs to run in parallel. Training the estimator and computing
          the cross-validated score are parallelized over the permutations.
          ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
          ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
          for more details.
  
      random_state : int, RandomState instance or None, default=0
          Pass an int for reproducible output for permutation of
          ``y`` values among samples. See :term:`Glossary <random_state>`.
  
      verbose : int, default=0
          The verbosity level.
  
      Returns
      -------
      score : float
          The true score without permuting targets.
  
      permutation_scores : array of shape (n_permutations,)
          The scores obtained for each permutations.
  
      pvalue : float
          The p-value, which approximates the probability that the score would
          be obtained by chance. This is calculated as:
  
          `(C + 1) / (n_permutations + 1)`
  
          Where C is the number of permutations whose score >= the true score.
  
          The best possible p-value is 1/(n_permutations + 1), the worst is 1.0.
  
      Notes
      -----
      This function implements Test 1 in:
  
          Ojala and Garriga. `Permutation Tests for Studying Classifier
          Performance
          <http://www.jmlr.org/papers/volume11/ojala10a/ojala10a.pdf>`_. The
          Journal of Machine Learning Research (2010) vol. 11
  
      """
      X, y, groups = indexable(X, y, groups)
  
      cv = check_cv(cv, y, classifier=is_classifier(estimator))
      scorer = check_scoring(estimator, scoring=scoring)
      random_state = check_random_state(random_state)
  
      # We clone the estimator to make sure that all the folds are
      # independent, and that it is pickle-able.
      score = _permutation_test_score(clone(estimator), X, y, groups, cv, scorer)
      permutation_scores = Parallel(n_jobs=n_jobs, verbose=verbose)(
          delayed(_permutation_test_score)(
              clone(estimator), X, _shuffle(y, groups, random_state),
              groups, cv, scorer)
          for _ in range(n_permutations))
      permutation_scores = np.array(permutation_scores)
      pvalue = (np.sum(permutation_scores >= score) + 1.0) / (n_permutations + 1)
      return score, permutation_scores, pvalue
  
  
  def _permutation_test_score(estimator, X, y, groups, cv, scorer):
      """Auxiliary function for permutation_test_score"""
      avg_score = []
      for train, test in cv.split(X, y, groups):
          X_train, y_train = _safe_split(estimator, X, y, train)
          X_test, y_test = _safe_split(estimator, X, y, test, train)
          estimator.fit(X_train, y_train)
          avg_score.append(scorer(estimator, X_test, y_test))
      return np.mean(avg_score)
  
  
  def _shuffle(y, groups, random_state):
      """Return a shuffled copy of y eventually shuffle among same groups."""
      if groups is None:
          indices = random_state.permutation(len(y))
      else:
          indices = np.arange(len(groups))
          for group in np.unique(groups):
              this_mask = (groups == group)
              indices[this_mask] = random_state.permutation(indices[this_mask])
      return _safe_indexing(y, indices)
  
  
  @_deprecate_positional_args
  def learning_curve(estimator, X, y, *, groups=None,
                     train_sizes=np.linspace(0.1, 1.0, 5), cv=None,
                     scoring=None, exploit_incremental_learning=False,
                     n_jobs=None, pre_dispatch="all", verbose=0, shuffle=False,
                     random_state=None, error_score=np.nan, return_times=False):
      """Learning curve.
  
      Determines cross-validated training and test scores for different training
      set sizes.
  
      A cross-validation generator splits the whole dataset k times in training
      and test data. Subsets of the training set with varying sizes will be used
      to train the estimator and a score for each training subset size and the
      test set will be computed. Afterwards, the scores will be averaged over
      all k runs for each training subset size.
  
      Read more in the :ref:`User Guide <learning_curve>`.
  
      Parameters
      ----------
      estimator : object type that implements the "fit" and "predict" methods
          An object of that type which is cloned for each validation.
  
      X : array-like of shape (n_samples, n_features)
          Training vector, where n_samples is the number of samples and
          n_features is the number of features.
  
      y : array-like of shape (n_samples,) or (n_samples, n_outputs)
          Target relative to X for classification or regression;
          None for unsupervised learning.
  
      groups : array-like of  shape (n_samples,), default=None
          Group labels for the samples used while splitting the dataset into
          train/test set. Only used in conjunction with a "Group" :term:`cv`
          instance (e.g., :class:`GroupKFold`).
  
      train_sizes : array-like of shape (n_ticks,), \
              default=np.linspace(0.1, 1.0, 5)
          Relative or absolute numbers of training examples that will be used to
          generate the learning curve. If the dtype is float, it is regarded as a
          fraction of the maximum size of the training set (that is determined
          by the selected validation method), i.e. it has to be within (0, 1].
          Otherwise it is interpreted as absolute sizes of the training sets.
          Note that for classification the number of samples usually have to
          be big enough to contain at least one sample from each class.
  
      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,
          - int, to specify the number of folds in a `(Stratified)KFold`,
          - :term:`CV splitter`,
          - An iterable yielding (train, test) splits as arrays of indices.
  
          For int/None inputs, if the estimator is a classifier and ``y`` is
          either binary or multiclass, :class:`StratifiedKFold` is used. In all
          other cases, :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.
  
      scoring : str or callable, default=None
          A str (see model evaluation documentation) or
          a scorer callable object / function with signature
          ``scorer(estimator, X, y)``.
  
      exploit_incremental_learning : bool, default=False
          If the estimator supports incremental learning, this will be
          used to speed up fitting for different training set sizes.
  
      n_jobs : int, default=None
          Number of jobs to run in parallel. Training the estimator and computing
          the score are parallelized over the different training and test sets.
          ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
          ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
          for more details.
  
      pre_dispatch : int or str, default='all'
          Number of predispatched jobs for parallel execution (default is
          all). The option can reduce the allocated memory. The str can
          be an expression like '2*n_jobs'.
  
      verbose : int, default=0
          Controls the verbosity: the higher, the more messages.
  
      shuffle : bool, default=False
          Whether to shuffle training data before taking prefixes of it
          based on``train_sizes``.
  
      random_state : int, RandomState instance or None, default=None
          Used when ``shuffle`` is True. Pass an int for reproducible
          output across multiple function calls.
          See :term:`Glossary <random_state>`.
  
      error_score : 'raise' or numeric, default=np.nan
          Value to assign to the score if an error occurs in estimator fitting.
          If set to 'raise', the error is raised.
          If a numeric value is given, FitFailedWarning is raised. This parameter
          does not affect the refit step, which will always raise the error.
  
          .. versionadded:: 0.20
  
      return_times : bool, default=False
          Whether to return the fit and score times.
  
      Returns
      -------
      train_sizes_abs : array of shape (n_unique_ticks,)
          Numbers of training examples that has been used to generate the
          learning curve. Note that the number of ticks might be less
          than n_ticks because duplicate entries will be removed.
  
      train_scores : array of shape (n_ticks, n_cv_folds)
          Scores on training sets.
  
      test_scores : array of shape (n_ticks, n_cv_folds)
          Scores on test set.
  
      fit_times : array of shape (n_ticks, n_cv_folds)
          Times spent for fitting in seconds. Only present if ``return_times``
          is True.
  
      score_times : array of shape (n_ticks, n_cv_folds)
          Times spent for scoring in seconds. Only present if ``return_times``
          is True.
  
      Notes
      -----
      See :ref:`examples/model_selection/plot_learning_curve.py
      <sphx_glr_auto_examples_model_selection_plot_learning_curve.py>`
      """
      if exploit_incremental_learning and not hasattr(estimator, "partial_fit"):
          raise ValueError("An estimator must support the partial_fit interface "
                           "to exploit incremental learning")
      X, y, groups = indexable(X, y, groups)
  
      cv = check_cv(cv, y, classifier=is_classifier(estimator))
      # Store it as list as we will be iterating over the list multiple times
      cv_iter = list(cv.split(X, y, groups))
  
      scorer = check_scoring(estimator, scoring=scoring)
  
      n_max_training_samples = len(cv_iter[0][0])
      # Because the lengths of folds can be significantly different, it is
      # not guaranteed that we use all of the available training data when we
      # use the first 'n_max_training_samples' samples.
      train_sizes_abs = _translate_train_sizes(train_sizes,
                                               n_max_training_samples)
      n_unique_ticks = train_sizes_abs.shape[0]
      if verbose > 0:
          print("[learning_curve] Training set sizes: " + str(train_sizes_abs))
  
      parallel = Parallel(n_jobs=n_jobs, pre_dispatch=pre_dispatch,
                          verbose=verbose)
  
      if shuffle:
          rng = check_random_state(random_state)
          cv_iter = ((rng.permutation(train), test) for train, test in cv_iter)
  
      if exploit_incremental_learning:
          classes = np.unique(y) if is_classifier(estimator) else None
          out = parallel(delayed(_incremental_fit_estimator)(
              clone(estimator), X, y, classes, train, test, train_sizes_abs,
              scorer, verbose, return_times) for train, test in cv_iter)
          out = np.asarray(out).transpose((2, 1, 0))
      else:
          train_test_proportions = []
          for train, test in cv_iter:
              for n_train_samples in train_sizes_abs:
                  train_test_proportions.append((train[:n_train_samples], test))
  
          results = parallel(delayed(_fit_and_score)(
              clone(estimator), X, y, scorer, train, test, verbose,
              parameters=None, fit_params=None, return_train_score=True,
              error_score=error_score, return_times=return_times)
              for train, test in train_test_proportions)
          results = _aggregate_score_dicts(results)
          train_scores = results["train_scores"].reshape(-1, n_unique_ticks).T
          test_scores = results["test_scores"].reshape(-1, n_unique_ticks).T
          out = [train_scores, test_scores]
  
          if return_times:
              fit_times = results["fit_time"].reshape(-1, n_unique_ticks).T
              score_times = results["score_time"].reshape(-1, n_unique_ticks).T
              out.extend([fit_times, score_times])
  
      ret = train_sizes_abs, out[0], out[1]
  
      if return_times:
          ret = ret + (out[2], out[3])
  
      return ret
  
  
  def _translate_train_sizes(train_sizes, n_max_training_samples):
      """Determine absolute sizes of training subsets and validate 'train_sizes'.
  
      Examples:
          _translate_train_sizes([0.5, 1.0], 10) -> [5, 10]
          _translate_train_sizes([5, 10], 10) -> [5, 10]
  
      Parameters
      ----------
      train_sizes : array-like of shape (n_ticks,)
          Numbers of training examples that will be used to generate the
          learning curve. If the dtype is float, it is regarded as a
          fraction of 'n_max_training_samples', i.e. it has to be within (0, 1].
  
      n_max_training_samples : int
          Maximum number of training samples (upper bound of 'train_sizes').
  
      Returns
      -------
      train_sizes_abs : array of shape (n_unique_ticks,)
          Numbers of training examples that will be used to generate the
          learning curve. Note that the number of ticks might be less
          than n_ticks because duplicate entries will be removed.
      """
      train_sizes_abs = np.asarray(train_sizes)
      n_ticks = train_sizes_abs.shape[0]
      n_min_required_samples = np.min(train_sizes_abs)
      n_max_required_samples = np.max(train_sizes_abs)
      if np.issubdtype(train_sizes_abs.dtype, np.floating):
          if n_min_required_samples <= 0.0 or n_max_required_samples > 1.0:
              raise ValueError("train_sizes has been interpreted as fractions "
                               "of the maximum number of training samples and "
                               "must be within (0, 1], but is within [%f, %f]."
                               % (n_min_required_samples,
                                  n_max_required_samples))
          train_sizes_abs = (train_sizes_abs * n_max_training_samples).astype(
                               dtype=int, copy=False)
          train_sizes_abs = np.clip(train_sizes_abs, 1,
                                    n_max_training_samples)
      else:
          if (n_min_required_samples <= 0 or
                  n_max_required_samples > n_max_training_samples):
              raise ValueError("train_sizes has been interpreted as absolute "
                               "numbers of training samples and must be within "
                               "(0, %d], but is within [%d, %d]."
                               % (n_max_training_samples,
                                  n_min_required_samples,
                                  n_max_required_samples))
  
      train_sizes_abs = np.unique(train_sizes_abs)
      if n_ticks > train_sizes_abs.shape[0]:
          warnings.warn("Removed duplicate entries from 'train_sizes'. Number "
                        "of ticks will be less than the size of "
                        "'train_sizes' %d instead of %d)."
                        % (train_sizes_abs.shape[0], n_ticks), RuntimeWarning)
  
      return train_sizes_abs
  
  
  def _incremental_fit_estimator(estimator, X, y, classes, train, test,
                                 train_sizes, scorer, verbose, return_times):
      """Train estimator on training subsets incrementally and compute scores."""
      train_scores, test_scores, fit_times, score_times = [], [], [], []
      partitions = zip(train_sizes, np.split(train, train_sizes)[:-1])
      for n_train_samples, partial_train in partitions:
          train_subset = train[:n_train_samples]
          X_train, y_train = _safe_split(estimator, X, y, train_subset)
          X_partial_train, y_partial_train = _safe_split(estimator, X, y,
                                                         partial_train)
          X_test, y_test = _safe_split(estimator, X, y, test, train_subset)
          start_fit = time.time()
          if y_partial_train is None:
              estimator.partial_fit(X_partial_train, classes=classes)
          else:
              estimator.partial_fit(X_partial_train, y_partial_train,
                                    classes=classes)
          fit_time = time.time() - start_fit
          fit_times.append(fit_time)
  
          start_score = time.time()
  
          test_scores.append(_score(estimator, X_test, y_test, scorer))
          train_scores.append(_score(estimator, X_train, y_train, scorer))
  
          score_time = time.time() - start_score
          score_times.append(score_time)
  
      ret = ((train_scores, test_scores, fit_times, score_times)
             if return_times else (train_scores, test_scores))
  
      return np.array(ret).T
  
  
  @_deprecate_positional_args
  def validation_curve(estimator, X, y, *, param_name, param_range, groups=None,
                       cv=None, scoring=None, n_jobs=None, pre_dispatch="all",
                       verbose=0, error_score=np.nan):
      """Validation curve.
  
      Determine training and test scores for varying parameter values.
  
      Compute scores for an estimator with different values of a specified
      parameter. This is similar to grid search with one parameter. However, this
      will also compute training scores and is merely a utility for plotting the
      results.
  
      Read more in the :ref:`User Guide <validation_curve>`.
  
      Parameters
      ----------
      estimator : object type that implements the "fit" and "predict" methods
          An object of that type which is cloned for each validation.
  
      X : array-like of shape (n_samples, n_features)
          Training vector, where n_samples is the number of samples and
          n_features is the number of features.
  
      y : array-like of shape (n_samples,) or (n_samples, n_outputs) or None
          Target relative to X for classification or regression;
          None for unsupervised learning.
  
      param_name : str
          Name of the parameter that will be varied.
  
      param_range : array-like of shape (n_values,)
          The values of the parameter that will be evaluated.
  
      groups : array-like of shape (n_samples,), default=None
          Group labels for the samples used while splitting the dataset into
          train/test set. Only used in conjunction with a "Group" :term:`cv`
          instance (e.g., :class:`GroupKFold`).
  
      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,
          - int, to specify the number of folds in a `(Stratified)KFold`,
          - :term:`CV splitter`,
          - An iterable yielding (train, test) splits as arrays of indices.
  
          For int/None inputs, if the estimator is a classifier and ``y`` is
          either binary or multiclass, :class:`StratifiedKFold` is used. In all
          other cases, :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.
  
      scoring : str or callable, default=None
          A str (see model evaluation documentation) or
          a scorer callable object / function with signature
          ``scorer(estimator, X, y)``.
  
      n_jobs : int, default=None
          Number of jobs to run in parallel. Training the estimator and computing
          the score are parallelized over the combinations of each parameter
          value and each cross-validation split.
          ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
          ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
          for more details.
  
      pre_dispatch : int or str, default='all'
          Number of predispatched jobs for parallel execution (default is
          all). The option can reduce the allocated memory. The str can
          be an expression like '2*n_jobs'.
  
      verbose : int, default=0
          Controls the verbosity: the higher, the more messages.
  
      error_score : 'raise' or numeric, default=np.nan
          Value to assign to the score if an error occurs in estimator fitting.
          If set to 'raise', the error is raised.
          If a numeric value is given, FitFailedWarning is raised. This parameter
          does not affect the refit step, which will always raise the error.
  
          .. versionadded:: 0.20
  
      Returns
      -------
      train_scores : array of shape (n_ticks, n_cv_folds)
          Scores on training sets.
  
      test_scores : array of shape (n_ticks, n_cv_folds)
          Scores on test set.
  
      Notes
      -----
      See :ref:`sphx_glr_auto_examples_model_selection_plot_validation_curve.py`
  
      """
      X, y, groups = indexable(X, y, groups)
  
      cv = check_cv(cv, y, classifier=is_classifier(estimator))
      scorer = check_scoring(estimator, scoring=scoring)
  
      parallel = Parallel(n_jobs=n_jobs, pre_dispatch=pre_dispatch,
                          verbose=verbose)
      results = parallel(delayed(_fit_and_score)(
          clone(estimator), X, y, scorer, train, test, verbose,
          parameters={param_name: v}, fit_params=None, return_train_score=True,
          error_score=error_score)
          # NOTE do not change order of iteration to allow one time cv splitters
          for train, test in cv.split(X, y, groups) for v in param_range)
      n_params = len(param_range)
  
      results = _aggregate_score_dicts(results)
      train_scores = results["train_scores"].reshape(-1, n_params).T
      test_scores = results["test_scores"].reshape(-1, n_params).T
  
      return train_scores, test_scores
  
  
  def _aggregate_score_dicts(scores):
      """Aggregate the list of dict to dict of np ndarray
  
      The aggregated output of _aggregate_score_dicts will be a list of dict
      of form [{'prec': 0.1, 'acc':1.0}, {'prec': 0.1, 'acc':1.0}, ...]
      Convert it to a dict of array {'prec': np.array([0.1 ...]), ...}
  
      Parameters
      ----------
  
      scores : list of dict
          List of dicts of the scores for all scorers. This is a flat list,
          assumed originally to be of row major order.
  
      Example
      -------
  
      >>> scores = [{'a': 1, 'b':10}, {'a': 2, 'b':2}, {'a': 3, 'b':3},
      ...           {'a': 10, 'b': 10}]                         # doctest: +SKIP
      >>> _aggregate_score_dicts(scores)                        # doctest: +SKIP
      {'a': array([1, 2, 3, 10]),
       'b': array([10, 2, 3, 10])}
      """
      return {
          key: np.asarray([score[key] for score in scores])
          if isinstance(scores[0][key], numbers.Number)
          else [score[key] for score in scores]
          for key in scores[0]
      }
  