<commit_before>"""Gaussian processes regression."""

# Authors: <NAME> <<EMAIL>>
# Modified by: <NAME> <<EMAIL>>
# License: BSD 3 clause

import warnings
from operator import itemgetter

import numpy as np
from scipy.linalg import cholesky, cho_solve, solve_triangular
import scipy.optimize

from ..base import BaseEstimator, RegressorMixin, clone
from ..base import MultiOutputMixin
from .kernels import RBF, ConstantKernel as C
from ..utils import check_random_state
from ..utils.validation import check_array
from ..utils.optimize import _check_optimize_result
from ..utils.validation import _deprecate_positional_args


class GaussianProcessRegressor(MultiOutputMixin,
                               RegressorMixin, BaseEstimator):
    """Gaussian process regression (GPR).

    The implementation is based on Algorithm 2.1 of Gaussian Processes
    for Machine Learning (GPML) by Rasmussen and Williams.

    In addition to standard scikit-learn estimator API,
    GaussianProcessRegressor:

       * allows prediction without prior fitting (based on the GP prior)
       * provides an additional method sample_y(X), which evaluates samples
         drawn from the GPR (prior or posterior) at given inputs
       * exposes a method log_marginal_likelihood(theta), which can be used
         externally for other ways of selecting hyperparameters, e.g., via
         Markov chain Monte Carlo.

    Read more in the :ref:`User Guide <gaussian_process>`.

    .. versionadded:: 0.18

    Parameters
    ----------
    kernel : kernel instance, default=None
        The kernel specifying the covariance function of the GP. If None is
        passed, the kernel "1.0 * RBF(1.0)" is used as default. Note that
        the kernel's hyperparameters are optimized during fitting.

    alpha : float or array-like of shape (n_samples), default=1e-10
        Value added to the diagonal of the kernel matrix during fitting.
        Larger values correspond to increased noise level in the observations.
        This can also prevent a potential numerical issue during fitting, by
        ensuring that the calculated values form a positive definite matrix.
        If an array is passed, it must have the same number of entries as the
        data used for fitting and is used as datapoint-dependent noise level.
        Note that this is equivalent to adding a WhiteKernel with c=alpha.
        Allowing to specify the noise level directly as a parameter is mainly
        for convenience and for consistency with Ridge.

    optimizer : "fmin_l_bfgs_b" or callable, default="fmin_l_bfgs_b"
        Can either be one of the internally supported optimizers for optimizing
        the kernel's parameters, specified by a string, or an externally
        defined optimizer passed as a callable. If a callable is passed, it
        must have the signature::

            def optimizer(obj_func, initial_theta, bounds):
                # * 'obj_func' is the objective function to be minimized, which
                #   takes the hyperparameters theta as parameter and an
                #   optional flag eval_gradient, which determines if the
                #   gradient is returned additionally to the function value
                # * 'initial_theta': the initial value for theta, which can be
                #   used by local optimizers
                # * 'bounds': the bounds on the values of theta
                ....
                # Returned are the best found hyperparameters theta and
                # the corresponding value of the target function.
                return theta_opt, func_min

        Per default, the 'L-BGFS-B' algorithm from scipy.optimize.minimize
        is used. If None is passed, the kernel's parameters are kept fixed.
        Available internal optimizers are::

            'fmin_l_bfgs_b'

    n_restarts_optimizer : int, default=0
        The number of restarts of the optimizer for finding the kernel's
        parameters which maximize the log-marginal likelihood. The first run
        of the optimizer is performed from the kernel's initial parameters,<commit_msg>DOC Resolve confusion about the role of alpha in Gaussian Process Regression (#15990)<commit_after>"""Gaussian processes regression."""

# Authors: <NAME> <<EMAIL>>
# Modified by: <NAME> <<EMAIL>>
# License: BSD 3 clause

import warnings
from operator import itemgetter

import numpy as np
from scipy.linalg import cholesky, cho_solve, solve_triangular
import scipy.optimize

from ..base import BaseEstimator, RegressorMixin, clone
from ..base import MultiOutputMixin
from .kernels import RBF, ConstantKernel as C
from ..utils import check_random_state
from ..utils.validation import check_array
from ..utils.optimize import _check_optimize_result
from ..utils.validation import _deprecate_positional_args


class GaussianProcessRegressor(MultiOutputMixin,
                               RegressorMixin, BaseEstimator):
    """Gaussian process regression (GPR).

    The implementation is based on Algorithm 2.1 of Gaussian Processes
    for Machine Learning (GPML) by <NAME> Williams.

    In addition to standard scikit-learn estimator API,
    GaussianProcessRegressor:

       * allows prediction without prior fitting (based on the GP prior)
       * provides an additional method sample_y(X), which evaluates samples
         drawn from the GPR (prior or posterior) at given inputs
       * exposes a method log_marginal_likelihood(theta), which can be used
         externally for other ways of selecting hyperparameters, e.g., via
         Markov chain Monte Carlo.

    Read more in the :ref:`User Guide <gaussian_process>`.

    .. versionadded:: 0.18

    Parameters
    ----------
    kernel : kernel instance, default=None
        The kernel specifying the covariance function of the GP. If None is
        passed, the kernel "1.0 * RBF(1.0)" is used as default. Note that
        the kernel's hyperparameters are optimized during fitting.

    alpha : float or ndarray of shape (n_samples,), default=1e-10
        Value added to the diagonal of the kernel matrix during fitting.
        This can prevent a potential numerical issue during fitting, by
        ensuring that the calculated values form a positive definite matrix.
        It can also be interpreted as the variance of additional Gaussian
        measurement noise on the training observations. Note that this is
        different from using a `WhiteKernel`. If an array is passed, it must
        have the same number of entries as the data used for fitting and is
        used as datapoint-dependent noise level. Allowing to specify the
        noise level directly as a parameter is mainly for convenience and
        for consistency with Ridge.

    optimizer : "fmin_l_bfgs_b" or callable, default="fmin_l_bfgs_b"
        Can either be one of the internally supported optimizers for optimizing
        the kernel's parameters, specified by a string, or an externally
        defined optimizer passed as a callable. If a callable is passed, it
        must have the signature::

            def optimizer(obj_func, initial_theta, bounds):
                # * 'obj_func' is the objective function to be minimized, which
                #   takes the hyperparameters theta as parameter and an
                #   optional flag eval_gradient, which determines if the
                #   gradient is returned additionally to the function value
                # * 'initial_theta': the initial value for theta, which can be
                #   used by local optimizers
                # * 'bounds': the bounds on the values of theta
                ....
                # Returned are the best found hyperparameters theta and
                # the corresponding value of the target function.
                return theta_opt, func_min

        Per default, the 'L-BGFS-B' algorithm from scipy.optimize.minimize
        is used. If None is passed, the kernel's parameters are kept fixed.
        Available internal optimizers are::

            'fmin_l_bfgs_b'

    n_restarts_optimizer : int, default=0
        The number of restarts of the optimizer for finding the kernel's
        parameters which maximize the log-marginal likelihood. The first run
        of the optimizer is performed from the kernel's initial parameters,
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