content: """Gaussian processes regression. """
  
  # Authors: Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>
  #
  # License: BSD 3 clause
  
  import warnings
  from operator import itemgetter
  
  import numpy as np
  from scipy.linalg import cholesky, cho_solve, solve_triangular
  from scipy.optimize import fmin_l_bfgs_b
  
  from sklearn.base import BaseEstimator, RegressorMixin, clone
  from sklearn.gaussian_process.kernels import RBF, ConstantKernel as C
  from sklearn.utils import check_random_state
  from sklearn.utils.validation import check_X_y, check_array
  
  
  class GaussianProcessRegressor(BaseEstimator, RegressorMixin):
      """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>`.
  
      Parameters
      ----------
      kernel : kernel object
          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, optional (default: 1e-10)
          Value added to the diagonal of the kernel matrix during fitting.
          Larger values correspond to increased noise level in the observations
          and reduce potential numerical issue during fitting. 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 : string or callable, optional (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 maximized, 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 'fmin_l_bfgs_b' algorithm from scipy.optimize
          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, optional (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: boolean, optional (default: False)
          Whether the target values y are normalized, i.e., the mean of the
          observed target values become zero. This parameter should be set to
          True if the target values' mean is expected to differ considerable from
          zero. When enabled, the normalization effectively modifies the GP's
          prior based on the data, which contradicts the likelihood principle;
          normalization is thus disabled per default.
  
      copy_X_train : bool, optional (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 : integer or numpy.RandomState, optional
          The generator used to initialize the centers. If an integer is
          given, it fixes the seed. Defaults to the global numpy random
          number generator.
  
      Attributes
      ----------
      X_train_ : array-like, shape = (n_samples, n_features)
          Feature values in training data (also required for prediction)
  
      y_train_: array-like, shape = (n_samples, [n_output_dims])
          Target values in training data (also required for prediction)
  
      kernel_: kernel object
          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, shape = (n_samples, n_samples)
          Lower-triangular Cholesky decomposition of the kernel in ``X_train_``
  
      alpha_: array-like, 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``
      """
      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, shape = (n_samples, n_features)
              Training data
  
          y : array-like, shape = (n_samples, [n_output_dims])
              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)
  
          X, y = check_X_y(X, y, multi_output=True, y_numeric=True)
  
          # Normalize target value
          if self.normalize_y:
              self.y_train_mean = np.mean(y, axis=0)
              # demean y
              y = y - self.y_train_mean
          else:
              self.y_train_mean = np.zeros(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 entries as y.(%d != %d)"
                                   % (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)
                      return -lml, -grad
                  else:
                      return -self.log_marginal_likelihood(theta)
  
              # 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.log_marginal_likelihood_value_ = -np.min(lml_values)
          else:
              self.log_marginal_likelihood_value_ = \
                  self.log_marginal_likelihood(self.kernel_.theta)
  
          # Precompute quantities required for predictions which are independent
          # of actual query points
          K = self.kernel_(self.X_train_)
          K[np.diag_indices_from(K)] += self.alpha
          self.L_ = cholesky(K, lower=True)  # Line 2
          self.alpha_ = cho_solve((self.L_, True), self.y_train_)  # Line 3
  
          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, also 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, shape = (n_samples, n_features)
              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 : array, shape = (n_samples, [n_output_dims])
              Mean of predictive distribution a query points
  
          y_std : array, shape = (n_samples,), optional
              Standard deviation of predictive distribution at query points.
              Only returned when return_std is True.
  
          y_cov : array, 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(
                  "Not returning standard deviation of predictions when "
                  "returning full covariance.")
  
          X = check_array(X)
  
          if not hasattr(self, "X_train_"):  # Unfitted;predict based on GP prior
              y_mean = np.zeros(X.shape[0])
              if return_cov:
                  y_cov = self.kernel(X)
                  return y_mean, y_cov
              elif return_std:
                  y_var = self.kernel.diag(X)
                  return y_mean, np.sqrt(y_var)
              else:
                  return y_mean
          else:  # Predict based on GP posterior
              K_trans = self.kernel_(X, self.X_train_)
              y_mean = K_trans.dot(self.alpha_)  # Line 4 (y_mean = f_star)
              y_mean = self.y_train_mean + y_mean  # undo normal.
              if return_cov:
                  v = cho_solve((self.L_, True), K_trans.T)  # Line 5
                  y_cov = self.kernel_(X) - K_trans.dot(v)  # Line 6
                  return y_mean, y_cov
              elif return_std:
                  # compute inverse K_inv of K based on its Cholesky
                  # decomposition L and its inverse L_inv
                  L_inv = solve_triangular(self.L_.T, np.eye(self.L_.shape[0]))
                  K_inv = L_inv.dot(L_inv.T)
                  # Compute variance of predictive distribution
                  y_var = self.kernel_.diag(X)
                  y_var -= np.einsum("ki,kj,ij->k", K_trans, K_trans, K_inv)
  
                  # 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
                  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, shape = (n_samples_X, n_features)
              Query points where the GP samples are evaluated
  
          n_samples : int, default: 1
              The number of samples drawn from the Gaussian process
  
          random_state: RandomState or an int seed (0 by default)
              A random number generator instance
  
          Returns
          -------
          y_samples : array, shape = (n_samples_X, [n_output_dims], 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):
          """Returns log-marginal likelihood of theta for training data.
  
          Parameters
          ----------
          theta : array-like, shape = (n_kernel_params,) or 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.
  
          Returns
          -------
          log_likelihood : float
              Log-marginal likelihood of theta for training data.
  
          log_likelihood_gradient : array, 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_
  
          kernel = self.kernel_.clone_with_theta(theta)
  
          if eval_gradient:
              K, K_gradient = kernel(self.X_train_, eval_gradient=True)
          else:
              K = kernel(self.X_train_)
  
          K[np.diag_indices_from(K)] += self.alpha
          try:
              L = cholesky(K, lower=True)  # Line 2
          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]
  
          alpha = cho_solve((L, True), y_train)  # Line 3
  
          # Compute log-likelihood (compare line 7)
          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)
          log_likelihood = log_likelihood_dims.sum(-1)  # sum over dimensions
  
          if eval_gradient:  # compare Equation 5.9 from GPML
              tmp = np.einsum("ik,jk->ijk", alpha, alpha)  # k: output-dimension
              tmp -= cho_solve((L, True), np.eye(K.shape[0]))[:, :, np.newaxis]
              # Compute "0.5 * trace(tmp.dot(K_gradient))" without
              # constructing the full matrix tmp.dot(K_gradient) since only
              # its diagonal is required
              log_likelihood_gradient_dims = \
                  0.5 * np.einsum("ijl,ijk->kl", tmp, K_gradient)
              log_likelihood_gradient = log_likelihood_gradient_dims.sum(-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":
              theta_opt, func_min, convergence_dict = \
                  fmin_l_bfgs_b(obj_func, initial_theta, bounds=bounds)
              if convergence_dict["warnflag"] != 0:
                  warnings.warn("fmin_l_bfgs_b terminated abnormally with the "
                                " state: %s" % convergence_dict)
          elif callable(self.optimizer):
              theta_opt, func_min = \
                  self.optimizer(obj_func, initial_theta, bounds=bounds)
          else:
              raise ValueError("Unknown optimizer %s." % self.optimizer)
  
          return theta_opt, func_min
  