content: """
  =========================================================
  Gaussian Processes regression: basic introductory example
  =========================================================
  
  A simple one-dimensional regression example computed in two different ways:
  
  1. A noise-free case
  2. A noisy case with known noise-level per datapoint
  
  In both cases, the kernel's parameters are estimated using the maximum
  likelihood principle.
  
  The figures illustrate the interpolating property of the Gaussian Process
  model as well as its probabilistic nature in the form of a pointwise 95%
  confidence interval.
  
  Note that the parameter ``alpha`` is applied as a Tikhonov
  regularization of the assumed covariance between the training points.
  """
  print(__doc__)
  
  # Author: Vincent Dubourg <vincent.dubourg@gmail.com>
  #         Jake Vanderplas <vanderplas@astro.washington.edu>
  #         Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>s
  # License: BSD 3 clause
  
  import numpy as np
  from matplotlib import pyplot as plt
  
  from sklearn.gaussian_process import GaussianProcessRegressor
  from sklearn.gaussian_process.kernels import RBF, ConstantKernel as C
  
  np.random.seed(1)
  
  
  def f(x):
      """The function to predict."""
      return x * np.sin(x)
  
  # ----------------------------------------------------------------------
  #  First the noiseless case
  X = np.atleast_2d([1., 3., 5., 6., 7., 8.]).T
  
  # Observations
  y = f(X).ravel()
  
  # Mesh the input space for evaluations of the real function, the prediction and
  # its MSE
  x = np.atleast_2d(np.linspace(0, 10, 1000)).T
  
  # Instanciate a Gaussian Process model
  kernel = C(1.0, (1e-3, 1e3)) * RBF(10, (1e-2, 1e2))
  gp = GaussianProcessRegressor(kernel=kernel, n_restarts_optimizer=9)
  
  # Fit to data using Maximum Likelihood Estimation of the parameters
  gp.fit(X, y)
  
  # Make the prediction on the meshed x-axis (ask for MSE as well)
  y_pred, sigma = gp.predict(x, return_std=True)
  
  # Plot the function, the prediction and the 95% confidence interval based on
  # the MSE
  fig = plt.figure()
  plt.plot(x, f(x), 'r:', label=u'$f(x) = x\,\sin(x)$')
  plt.plot(X, y, 'r.', markersize=10, label=u'Observations')
  plt.plot(x, y_pred, 'b-', label=u'Prediction')
  plt.fill(np.concatenate([x, x[::-1]]),
           np.concatenate([y_pred - 1.9600 * sigma,
                          (y_pred + 1.9600 * sigma)[::-1]]),
           alpha=.5, fc='b', ec='None', label='95% confidence interval')
  plt.xlabel('$x$')
  plt.ylabel('$f(x)$')
  plt.ylim(-10, 20)
  plt.legend(loc='upper left')
  
  # ----------------------------------------------------------------------
  # now the noisy case
  X = np.linspace(0.1, 9.9, 20)
  X = np.atleast_2d(X).T
  
  # Observations and noise
  y = f(X).ravel()
  dy = 0.5 + 1.0 * np.random.random(y.shape)
  noise = np.random.normal(0, dy)
  y += noise
  
  # Instanciate a Gaussian Process model
  gp = GaussianProcessRegressor(kernel=kernel, alpha=(dy / y) ** 2,
                                n_restarts_optimizer=10)
  
  # Fit to data using Maximum Likelihood Estimation of the parameters
  gp.fit(X, y)
  
  # Make the prediction on the meshed x-axis (ask for MSE as well)
  y_pred, sigma = gp.predict(x, return_std=True)
  
  # Plot the function, the prediction and the 95% confidence interval based on
  # the MSE
  fig = plt.figure()
  plt.plot(x, f(x), 'r:', label=u'$f(x) = x\,\sin(x)$')
  plt.errorbar(X.ravel(), y, dy, fmt='r.', markersize=10, label=u'Observations')
  plt.plot(x, y_pred, 'b-', label=u'Prediction')
  plt.fill(np.concatenate([x, x[::-1]]),
           np.concatenate([y_pred - 1.9600 * sigma,
                          (y_pred + 1.9600 * sigma)[::-1]]),
           alpha=.5, fc='b', ec='None', label='95% confidence interval')
  plt.xlabel('$x$')
  plt.ylabel('$f(x)$')
  plt.ylim(-10, 20)
  plt.legend(loc='upper left')
  
  plt.show()
  