class_docstr: Implementation of Gaussian process model. The Gaussian process hyperparameters are obtained by optimizing
  the marginal log likelihood.
  
  This code is based on the implementation of RoBO:
  Klein, A. and Falkner, S. and Mansur, N. and Hutter, F.
  RoBO: A Flexible and Robust Bayesian Optimization Framework in Python
  In: NIPS 2017 Bayesian Optimization Workshop
  
  Parameters
  ----------
  configspace : ConfigurationSpace
  kernel : Kernel
      Kernel which is used for the Gaussian process.
  n_restarts : int, defaults to 10
      Number of restarts for the Gaussian process hyperparameter optimization.
  normalize_y : bool, defaults to True
      Zero mean unit variance normalization of the output values.
  instance_features : dict[str, list[int | float]] | None, defaults to None
      Features (list of int or floats) of the instances (str). The features are incorporated into the X data,
      on which the model is trained on.
  pca_components : float, defaults to 7
      Number of components to keep when using PCA to reduce dimensionality of instance features.
  seed : int
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docstr: Computes the Cholesky decomposition of the covariance of X and estimates the GP
  hyperparameters by optimizing the marginal log likelihood. The prior mean of the GP is set to
  the empirical mean of X.
  
  Parameters
  ----------
  X : np.ndarray [#samples, #hyperparameters + #features]
      Input data points.
  Y : np.ndarray [#samples, #objectives]
      The corresponding target values.
  optimize_hyperparameters: boolean
      If set to true, the hyperparameters are optimized, otherwise the default hyperparameters of the kernel are
      used.
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code: def _train(
      self: Self,
      X: np.ndarray,
      y: np.ndarray,
      optimize_hyperparameters: bool = True,
  ) -> Self:
      """Computes the Cholesky decomposition of the covariance of X and estimates the GP
      hyperparameters by optimizing the marginal log likelihood. The prior mean of the GP is set to
      the empirical mean of X.
  
      Parameters
      ----------
      X : np.ndarray [#samples, #hyperparameters + #features]
          Input data points.
      Y : np.ndarray [#samples, #objectives]
          The corresponding target values.
      optimize_hyperparameters: boolean
          If set to true, the hyperparameters are optimized, otherwise the default hyperparameters of the kernel are
          used.
      """
      if self._normalize_y:
          y = self._normalize(y)
  
      X = self._impute_inactive(X)
      y = y.flatten()
  
      n_tries = 10
      for i in range(n_tries):
          try:
              self._gp = self._get_gaussian_process()
              self._gp.fit(X, y)
              break
          except np.linalg.LinAlgError as e:
              if i == n_tries:
                  raise e
  
              # Assume that the last entry of theta is the noise
              theta = np.exp(self._kernel.theta)
              theta[-1] += 1
              self._kernel.theta = np.log(theta)
  
      if optimize_hyperparameters:
          self._all_priors = self._get_all_priors(
              add_bound_priors=False
          )
          self._hypers = self._optimize()
          self._gp.kernel.theta = self._hypers
          self._gp.fit(X, y)
      else:
          self._hypers = self._gp.kernel.theta
  
      # Set the flag
      self._is_trained = True
  
      return self
  
imports: from __future__ import annotations
  from typing import Any, Optional, TypeVar, cast
  import logging
  import numpy
  from ConfigSpace import ConfigurationSpace
  from scipy import optimize
  from sklearn.gaussian_process import GaussianProcessRegressor
  from sklearn.gaussian_process.kernels import Kernel
  from smac.constants import VERY_SMALL_NUMBER
  from smac.model.gaussian_process.abstract_gaussian_process import AbstractGaussianProcess
  from smac.model.gaussian_process.priors.abstract_prior import AbstractPrior
label_desc: Use image node_id 3 for calling the GaussianProcess obj's underlying member method code with example usage: obj._train(X, y, optimize_hyperparameters) and returns: self