"""Gaussian processes regression."""

# Authors: Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>
# Modified by: Pete Green <p.l.green@liverpool.ac.uk>
# 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 ..preprocessing._data import _handle_zeros_in_scale
from ..utils import check_random_state
from ..utils.optimize import _check_optimize_result

GPR_CHOLESKY_LOWER = True


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

    The implementation is based on Algorithm 2.1 of [1]_.

    In addition to standard scikit-learn estimator API,
    :class:`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 ``ConstantKernel(1.0, constant_value_bounds="fixed"
        * RBF(1.0, length_scale_bounds="fixed")`` is used as default. Note that
        the kernel hyperparameters are optimized during fitting unless the
        bounds are marked as "fixed".

    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 :class:`~sklearn.linear_model.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': the objective function to be minimized, which
                #   takes the hyperparameters theta as a 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-BFGS-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,
        the remaining ones (if any) from thetas sampled log-uniform randomly
        from the space of allowed theta-values. If greater than 0, all bounds
        must be finite. Note that `n_restarts_optimizer == 0` implies that one
        run is performed.

    normalize_y : bool, default=False
        Whether or not to normalized the target values `y` by removing the mean
        and scaling to unit-variance. This is recommended for cases where
        zero-mean, unit-variance priors are used. Note that, in this
        implementation, the normalisation is reversed before the GP predictions
        are reported.

        .. versionchanged:: 0.23

    copy_X_train : bool, default=True
        If True, a persistent copy of the training data is stored in the
        object. Otherwise, just a reference to the training data is stored,
        which might cause predictions to change if the data is modified
        externally.

    random_state : int, RandomState instance or None, default=None
        Determines random number generation used to initialize the centers.
        Pass an int for reproducible results across multiple function calls.
        See :term:`Glossary <random_state>`.

    Attributes
    ----------
    X_train_ : array-like of shape (n_samples, n_features) or list of object
        Feature vectors or other representations of training data (also
        required for prediction).

    y_train_ : array-like of shape (n_samples,) or (n_samples, n_targets)
        Target values in training data (also required for prediction).

    kernel_ : kernel instance
        The kernel used for prediction. The structure of the kernel is the
        same as the one passed as parameter but with optimized hyperparameters.

    L_ : array-like of shape (n_samples, n_samples)
        Lower-triangular Cholesky decomposition of the kernel in ``X_train_``.

    alpha_ : array-like of shape (n_samples,)
        Dual coefficients of training data points in kernel space.

    log_marginal_likelihood_value_ : float
        The log-marginal-likelihood of ``self.kernel_.theta``.

    n_features_in_ : int
        Number of features seen during :term:`fit`.

        .. versionadded:: 0.24

    References
    ----------
    .. [1] `Rasmussen, Carl Edward.
       "Gaussian processes in machine learning."
       Summer school on machine learning. Springer, Berlin, Heidelberg, 2003
       <http://www.gaussianprocess.org/gpml/chapters/RW.pdf>`_.

    Examples
    --------
    >>> from sklearn.datasets import make_friedman2
    >>> from sklearn.gaussian_process import GaussianProcessRegressor
    >>> from sklearn.gaussian_process.kernels import DotProduct, WhiteKernel
    >>> X, y = make_friedman2(n_samples=500, noise=0, random_state=0)
    >>> kernel = DotProduct() + WhiteKernel()
    >>> gpr = GaussianProcessRegressor(kernel=kernel,
    ...         random_state=0).fit(X, y)
    >>> gpr.score(X, y)
    0.3680...
    >>> gpr.predict(X[:2,:], return_std=True)
    (array([653.0..., 592.1...]), array([316.6..., 316.6...]))
    """

    def __init__(
        self,
        kernel=None,
        *,
        alpha=1e-10,
        optimizer="fmin_l_bfgs_b",
        n_restarts_optimizer=0,
        normalize_y=False,
        copy_X_train=True,
        random_state=None,
    ):
        self.kernel = kernel
        self.alpha = alpha
        self.optimizer = optimizer
        self.n_restarts_optimizer = n_restarts_optimizer
        self.normalize_y = normalize_y
        self.copy_X_train = copy_X_train
        self.random_state = random_state

    def fit(self, X, y):
        """Fit Gaussian process regression model.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features) or list of object
            Feature vectors or other representations of training data.

        y : array-like of shape (n_samples,) or (n_samples, n_targets)
            Target values.

        Returns
        -------
        self : returns an instance of self.
        """
        if self.kernel is None:  # Use an RBF kernel as default
            self.kernel_ = C(1.0, constant_value_bounds="fixed") * RBF(
                1.0, length_scale_bounds="fixed"
            )
        else:
            self.kernel_ = clone(self.kernel)

        self._rng = check_random_state(self.random_state)

        if self.kernel_.requires_vector_input:
            dtype, ensure_2d = "numeric", True
        else:
            dtype, ensure_2d = None, False
        X, y = self._validate_data(
            X,
            y,
            multi_output=True,
            y_numeric=True,
            ensure_2d=ensure_2d,
            dtype=dtype,
        )

        # Normalize target value
        if self.normalize_y:
            self._y_train_mean = np.mean(y, axis=0)
            self._y_train_std = _handle_zeros_in_scale(np.std(y, axis=0), copy=False)

            # Remove mean and make unit variance
            y = (y - self._y_train_mean) / self._y_train_std

        else:
            self._y_train_mean = np.zeros(1)
            self._y_train_std = 1

        if np.iterable(self.alpha) and self.alpha.shape[0] != y.shape[0]:
            if self.alpha.shape[0] == 1:
                self.alpha = self.alpha[0]
            else:
                raise ValueError(
                    "alpha must be a scalar or an array with same number of "
                    f"entries as y. ({self.alpha.shape[0]} != {y.shape[0]})"
                )

        self.X_train_ = np.copy(X) if self.copy_X_train else X
        self.y_train_ = np.copy(y) if self.copy_X_train else y

        if self.optimizer is not None and self.kernel_.n_dims > 0:
            # Choose hyperparameters based on maximizing the log-marginal
            # likelihood (potentially starting from several initial values)
            def obj_func(theta, eval_gradient=True):
                if eval_gradient:
                    lml, grad = self.log_marginal_likelihood(
                        theta, eval_gradient=True, clone_kernel=False
                    )
                    return -lml, -grad
                else:
                    return -self.log_marginal_likelihood(theta, clone_kernel=False)

            # First optimize starting from theta specified in kernel
            optima = [
                (
                    self._constrained_optimization(
                        obj_func, self.kernel_.theta, self.kernel_.bounds
                    )
                )
            ]

            # Additional runs are performed from log-uniform chosen initial
            # theta
            if self.n_restarts_optimizer > 0:
                if not np.isfinite(self.kernel_.bounds).all():
                    raise ValueError(
                        "Multiple optimizer restarts (n_restarts_optimizer>0) "
                        "requires that all bounds are finite."
                    )
                bounds = self.kernel_.bounds
                for iteration in range(self.n_restarts_optimizer):
                    theta_initial = self._rng.uniform(bounds[:, 0], bounds[:, 1])
                    optima.append(
                        self._constrained_optimization(obj_func, theta_initial, bounds)
                    )
            # Select result from run with minimal (negative) log-marginal
            # likelihood
            lml_values = list(map(itemgetter(1), optima))
            self.kernel_.theta = optima[np.argmin(lml_values)][0]
            self.kernel_._check_bounds_params()

            self.log_marginal_likelihood_value_ = -np.min(lml_values)
        else:
            self.log_marginal_likelihood_value_ = self.log_marginal_likelihood(
                self.kernel_.theta, clone_kernel=False
            )

        # Precompute quantities required for predictions which are independent
        # of actual query points
        # Alg. 2.1, page 19, line 2 -> L = cholesky(K + sigma^2 I)
        K = self.kernel_(self.X_train_)
        K[np.diag_indices_from(K)] += self.alpha
        try:
            self.L_ = cholesky(K, lower=GPR_CHOLESKY_LOWER, check_finite=False)
        except np.linalg.LinAlgError as exc:
            exc.args = (
                f"The kernel, {self.kernel_}, is not returning a positive "
                "definite matrix. Try gradually increasing the 'alpha' "
                "parameter of your GaussianProcessRegressor estimator.",
            ) + exc.args
            raise
        # Alg 2.1, page 19, line 3 -> alpha = L^T \ (L \ y)
        self.alpha_ = cho_solve(
            (self.L_, GPR_CHOLESKY_LOWER),
            self.y_train_,
            check_finite=False,
        )
        return self

    def predict(self, X, return_std=False, return_cov=False):
        """Predict using the Gaussian process regression model

        We can also predict based on an unfitted model by using the GP prior.
        In addition to the mean of the predictive distribution, optionally also
        returns its standard deviation (`return_std=True`) or covariance
        (`return_cov=True`). Note that at most one of the two can be requested.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features) or list of object
            Query points where the GP is evaluated.

        return_std : bool, default=False
            If True, the standard-deviation of the predictive distribution at
            the query points is returned along with the mean.

        return_cov : bool, default=False
            If True, the covariance of the joint predictive distribution at
            the query points is returned along with the mean.

        Returns
        -------
        y_mean : ndarray of shape (n_samples,) or (n_samples, n_targets)
            Mean of predictive distribution a query points.

        y_std : ndarray of shape (n_samples,), optional
            Standard deviation of predictive distribution at query points.
            Only returned when `return_std` is True.

        y_cov : ndarray of shape (n_samples, n_samples), optional
            Covariance of joint predictive distribution a query points.
            Only returned when `return_cov` is True.
        """
        if return_std and return_cov:
            raise RuntimeError(
                "At most one of return_std or return_cov can be requested."
            )

        if self.kernel is None or self.kernel.requires_vector_input:
            dtype, ensure_2d = "numeric", True
        else:
            dtype, ensure_2d = None, False

        X = self._validate_data(X, ensure_2d=ensure_2d, dtype=dtype, reset=False)

        if not hasattr(self, "X_train_"):  # Unfitted;predict based on GP prior
            if self.kernel is None:
                kernel = C(1.0, constant_value_bounds="fixed") * RBF(
                    1.0, length_scale_bounds="fixed"
                )
            else:
                kernel = self.kernel
            y_mean = np.zeros(X.shape[0])
            if return_cov:
                y_cov = kernel(X)
                return y_mean, y_cov
            elif return_std:
                y_var = kernel.diag(X)
                return y_mean, np.sqrt(y_var)
            else:
                return y_mean
        else:  # Predict based on GP posterior
            # Alg 2.1, page 19, line 4 -> f*_bar = K(X_test, X_train) . alpha
            K_trans = self.kernel_(X, self.X_train_)
            y_mean = K_trans @ self.alpha_

            # undo normalisation
            y_mean = self._y_train_std * y_mean + self._y_train_mean

            # Alg 2.1, page 19, line 5 -> v = L \ K(X_test, X_train)^T
            V = solve_triangular(
                self.L_, K_trans.T, lower=GPR_CHOLESKY_LOWER, check_finite=False
            )

            if return_cov:
                # Alg 2.1, page 19, line 6 -> K(X_test, X_test) - v^T. v
                y_cov = self.kernel_(X) - V.T @ V

                # undo normalisation
                y_cov = y_cov * self._y_train_std ** 2

                return y_mean, y_cov
            elif return_std:
                # Compute variance of predictive distribution
                # Use einsum to avoid explicitly forming the large matrix
                # V^T @ V just to extract its diagonal afterward.
                y_var = self.kernel_.diag(X)
                y_var -= np.einsum("ij,ji->i", V.T, V)

                # Check if any of the variances is negative because of
                # numerical issues. If yes: set the variance to 0.
                y_var_negative = y_var < 0
                if np.any(y_var_negative):
                    warnings.warn(
                        "Predicted variances smaller than 0. "
                        "Setting those variances to 0."
                    )
                    y_var[y_var_negative] = 0.0

                # undo normalisation
                y_var = y_var * self._y_train_std ** 2

                return y_mean, np.sqrt(y_var)
            else:
                return y_mean

    def sample_y(self, X, n_samples=1, random_state=0):
        """Draw samples from Gaussian process and evaluate at X.

        Parameters
        ----------
        X : array-like of shape (n_samples_X, n_features) or list of object
            Query points where the GP is evaluated.

        n_samples : int, default=1
            Number of samples drawn from the Gaussian process per query point

        random_state : int, RandomState instance or None, default=0
            Determines random number generation to randomly draw samples.
            Pass an int for reproducible results across multiple function
            calls.
            See :term:`Glossary <random_state>`.

        Returns
        -------
        y_samples : ndarray of shape (n_samples_X, n_samples), or \
            (n_samples_X, n_targets, n_samples)
            Values of n_samples samples drawn from Gaussian process and
            evaluated at query points.
        """
        rng = check_random_state(random_state)

        y_mean, y_cov = self.predict(X, return_cov=True)
        if y_mean.ndim == 1:
            y_samples = rng.multivariate_normal(y_mean, y_cov, n_samples).T
        else:
            y_samples = [
                rng.multivariate_normal(y_mean[:, i], y_cov, n_samples).T[:, np.newaxis]
                for i in range(y_mean.shape[1])
            ]
            y_samples = np.hstack(y_samples)
        return y_samples

    def log_marginal_likelihood(
        self, theta=None, eval_gradient=False, clone_kernel=True
    ):
        """Returns log-marginal likelihood of theta for training data.

        Parameters
        ----------
        theta : array-like of shape (n_kernel_params,) default=None
            Kernel hyperparameters for which the log-marginal likelihood is
            evaluated. If None, the precomputed log_marginal_likelihood
            of ``self.kernel_.theta`` is returned.

        eval_gradient : bool, default=False
            If True, the gradient of the log-marginal likelihood with respect
            to the kernel hyperparameters at position theta is returned
            additionally. If True, theta must not be None.

        clone_kernel : bool, default=True
            If True, the kernel attribute is copied. If False, the kernel
            attribute is modified, but may result in a performance improvement.

        Returns
        -------
        log_likelihood : float
            Log-marginal likelihood of theta for training data.

        log_likelihood_gradient : ndarray of shape (n_kernel_params,), optional
            Gradient of the log-marginal likelihood with respect to the kernel
            hyperparameters at position theta.
            Only returned when eval_gradient is True.
        """
        if theta is None:
            if eval_gradient:
                raise ValueError("Gradient can only be evaluated for theta!=None")
            return self.log_marginal_likelihood_value_

        if clone_kernel:
            kernel = self.kernel_.clone_with_theta(theta)
        else:
            kernel = self.kernel_
            kernel.theta = theta

        if eval_gradient:
            K, K_gradient = kernel(self.X_train_, eval_gradient=True)
        else:
            K = kernel(self.X_train_)

        # Alg. 2.1, page 19, line 2 -> L = cholesky(K + sigma^2 I)
        K[np.diag_indices_from(K)] += self.alpha
        try:
            L = cholesky(K, lower=GPR_CHOLESKY_LOWER, check_finite=False)
        except np.linalg.LinAlgError:
            return (-np.inf, np.zeros_like(theta)) if eval_gradient else -np.inf

        # Support multi-dimensional output of self.y_train_
        y_train = self.y_train_
        if y_train.ndim == 1:
            y_train = y_train[:, np.newaxis]

        # Alg 2.1, page 19, line 3 -> alpha = L^T \ (L \ y)
        alpha = cho_solve((L, GPR_CHOLESKY_LOWER), y_train, check_finite=False)

        # Alg 2.1, page 19, line 7
        # -0.5 . y^T . alpha - sum(log(diag(L))) - n_samples / 2 log(2*pi)
        # y is originally thought to be a (1, n_samples) row vector. However,
        # in multioutputs, y is of shape (n_samples, 2) and we need to compute
        # y^T . alpha for each output, independently using einsum. Thus, it
        # is equivalent to:
        # for output_idx in range(n_outputs):
        #     log_likelihood_dims[output_idx] = (
        #         y_train[:, [output_idx]] @ alpha[:, [output_idx]]
        #     )
        log_likelihood_dims = -0.5 * np.einsum("ik,ik->k", y_train, alpha)
        log_likelihood_dims -= np.log(np.diag(L)).sum()
        log_likelihood_dims -= K.shape[0] / 2 * np.log(2 * np.pi)
        # the log likehood is sum-up across the outputs
        log_likelihood = log_likelihood_dims.sum(axis=-1)

        if eval_gradient:
            # Eq. 5.9, p. 114, and footnote 5 in p. 114
            # 0.5 * trace((alpha . alpha^T - K^-1) . K_gradient)
            # alpha is supposed to be a vector of (n_samples,) elements. With
            # multioutputs, alpha is a matrix of size (n_samples, n_outputs).
            # Therefore, we want to construct a matrix of
            # (n_samples, n_samples, n_outputs) equivalent to
            # for output_idx in range(n_outputs):
            #     output_alpha = alpha[:, [output_idx]]
            #     inner_term[..., output_idx] = output_alpha @ output_alpha.T
            inner_term = np.einsum("ik,jk->ijk", alpha, alpha)
            # compute K^-1 of shape (n_samples, n_samples)
            K_inv = cho_solve(
                (L, GPR_CHOLESKY_LOWER), np.eye(K.shape[0]), check_finite=False
            )
            # create a new axis to use broadcasting between inner_term and
            # K_inv
            inner_term -= K_inv[..., np.newaxis]
            # Since we are interested about the trace of
            # inner_term @ K_gradient, we don't explicitly compute the
            # matrix-by-matrix operation and instead use an einsum. Therefore
            # it is equivalent to:
            # for param_idx in range(n_kernel_params):
            #     for output_idx in range(n_output):
            #         log_likehood_gradient_dims[param_idx, output_idx] = (
            #             inner_term[..., output_idx] @
            #             K_gradient[..., param_idx]
            #         )
            log_likelihood_gradient_dims = 0.5 * np.einsum(
                "ijl,jik->kl", inner_term, K_gradient
            )
            # the log likehood gradient is the sum-up across the outputs
            log_likelihood_gradient = log_likelihood_gradient_dims.sum(axis=-1)

        if eval_gradient:
            return log_likelihood, log_likelihood_gradient
        else:
            return log_likelihood

    def _constrained_optimization(self, obj_func, initial_theta, bounds):
        if self.optimizer == "fmin_l_bfgs_b":
            opt_res = scipy.optimize.minimize(
                obj_func,
                initial_theta,
                method="L-BFGS-B",
                jac=True,
                bounds=bounds,
            )
            _check_optimize_result("lbfgs", opt_res)
            theta_opt, func_min = opt_res.x, opt_res.fun
        elif callable(self.optimizer):
            theta_opt, func_min = self.optimizer(obj_func, initial_theta, bounds=bounds)
        else:
            raise ValueError(f"Unknown optimizer {self.optimizer}.")

        return theta_opt, func_min

    def _more_tags(self):
        return {"requires_fit": False}
