_validation.py

"""
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>
#         Michal Karbownik <michakarbownik@gmail.com>
# License: BSD 3 clause


import warnings
import numbers
import time
from traceback import format_exc
from contextlib import suppress
from collections import Counter

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.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",
]


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
        Strategy to evaluate the performance of the cross-validated model on
        the test set.

        If `scoring` represents a single score, one can use:

        - a single string (see :ref:`scoring_parameter`);
        - a callable (see :ref:`scoring`) that returns a single value.

        If `scoring` represents multiple scores, one can use:

        - a list or tuple of unique strings;
        - a callable returning a dictionary where the keys are the metric
          names and the values are the metric scores;
        - a dictionary with metric names as keys and callables a values.

        See :ref:`multimetric_grid_search` for an example.

    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:`.Fold` is used. These splitters are instantiated
        with `shuffle=False` so the splits will be the same across calls.

        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.

        .. 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)
    )

    _warn_about_fit_failures(results, error_score)

    # 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_error"] is not None:
            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}


def _warn_about_fit_failures(results, error_score):
    fit_errors = [
        result["fit_error"] for result in results if result["fit_error"] is not None
    ]
    if fit_errors:
        num_failed_fits = len(fit_errors)
        num_fits = len(results)
        fit_errors_counter = Counter(fit_errors)
        delimiter = "-" * 80 + "\n"
        fit_errors_summary = "\n".join(
            f"{delimiter}{n} fits failed with the following error:\n{error}"
            for error, n in fit_errors_counter.items()
        )

        some_fits_failed_message = (
            f"\n{num_failed_fits} fits failed out of a total of {num_fits}.\n"
            "The score on these train-test partitions for these parameters"
            f" will be set to {error_score}.\n"
            "If these failures are not expected, you can try to debug them "
            "by setting error_score='raise'.\n\n"
            f"Below are more details about the failures:\n{fit_errors_summary}"
        )
        warnings.warn(some_fits_failed_message, FitFailedWarning)


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 that generates (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. These splitters are instantiated
        with `shuffle=False` so the splits will be the same across calls.

        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.

        .. 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.

    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, tuple} of int, default=None
        A list or tuple of format (<current_split_id>, <total_num_of_splits>).

    candidate_progress : {list, tuple} of int, 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_error : str or None
            Traceback str if the fit failed, None if the fit succeeded.
    """
    if not isinstance(error_score, numbers.Number) and error_score != "raise":
        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.)"
        )

    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}/{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:
        # 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
        result["fit_error"] = format_exc()
    else:
        result["fit_error"] = None

        fit_time = time.time() - start_time
        test_scores = _score(estimator, X_test, y_test, scorer, error_score)
        score_time = time.time() - start_time - fit_time
        if return_train_score:
            train_scores = _score(estimator, X_train, y_train, scorer, error_score)

    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:
            if 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})"
            else:
                result_msg += ", score="
                if return_train_score:
                    result_msg += f"(train={train_scores:.3f}, test={test_scores:.3f})"
                else:
                    result_msg += f"{test_scores:.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, error_score="raise"):
    """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)

    try:
        if y_test is None:
            scores = scorer(estimator, X_test)
        else:
            scores = scorer(estimator, X_test, y_test)
    except Exception:
        if error_score == "raise":
            raise
        else:
            if isinstance(scorer, _MultimetricScorer):
                scores = {name: error_score for name in scorer._scorers}
            else:
                scores = error_score
            warnings.warn(
                "Scoring failed. The score on this train-test partition for "
                f"these parameters will be set to {error_score}. Details: \n"
                f"{format_exc()}",
                UserWarning,
            )

    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


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 that generates (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. These splitters are instantiated
        with `shuffle=False` so the splits will be the same across calls.

        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, 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'

    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))
    splits = list(cv.split(X, y, groups))

    test_indices = np.concatenate([test for _, test in splits])
    if not _check_is_permutation(test_indices, _num_samples(X)):
        raise ValueError("cross_val_predict only works for partitions")

    # 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)
    predictions = parallel(
        delayed(_fit_and_predict)(
            clone(estimator), X, y, train, test, verbose, fit_params, method
        )
        for train, test in splits
    )

    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'
    """
    # 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


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


def permutation_test_score(
    estimator,
    X,
    y,
    *,
    groups=None,
    cv=None,
    n_permutations=100,
    n_jobs=None,
    random_state=0,
    verbose=0,
    scoring=None,
    fit_params=None,
):
    """Evaluate the significance of a cross-validated score with 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. These splitters are instantiated
        with `shuffle=False` so the splits will be the same across calls.

        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.

    fit_params : dict, default=None
        Parameters to pass to the fit method of the estimator.

        .. versionadded:: 0.24

    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, fit_params=fit_params
    )
    permutation_scores = Parallel(n_jobs=n_jobs, verbose=verbose)(
        delayed(_permutation_test_score)(
            clone(estimator),
            X,
            _shuffle(y, groups, random_state),
            groups,
            cv,
            scorer,
            fit_params=fit_params,
        )
        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, fit_params):
    """Auxiliary function for permutation_test_score"""
    # Adjust length of sample weights
    fit_params = fit_params if fit_params is not None else {}
    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)
        fit_params = _check_fit_params(X, fit_params, train)
        estimator.fit(X_train, y_train, **fit_params)
        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)


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,
    fit_params=None,
):
    """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. These splitters are instantiated
        with `shuffle=False` so the splits will be the same across calls.

        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.

        .. versionadded:: 0.20

    return_times : bool, default=False
        Whether to return the fit and score times.

    fit_params : dict, default=None
        Parameters to pass to the fit method of the estimator.

        .. versionadded:: 0.24

    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,
                error_score=error_score,
                fit_params=fit_params,
            )
            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=fit_params,
                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,
    error_score,
    fit_params,
):
    """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])
    if fit_params is None:
        fit_params = {}
    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, **fit_params)
        else:
            estimator.partial_fit(
                X_partial_train, y_partial_train, classes=classes, **fit_params
            )
        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, error_score))
        train_scores.append(_score(estimator, X_train, y_train, scorer, error_score))

        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


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,
    fit_params=None,
):
    """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. These splitters are instantiated
        with `shuffle=False` so the splits will be the same across calls.

        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.

    fit_params : dict, default=None
        Parameters to pass to the fit method of the estimator.

        .. versionadded:: 0.24

    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.

        .. 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=fit_params,
            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]
    }