.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        Click :ref:`here <sphx_glr_download_auto_examples_gaussian_process_plot_gpr_noisy_targets.py>`     to download the full example code or to run this example in your browser via Binder
    .. rst-class:: sphx-glr-example-title

    .. _sphx_glr_auto_examples_gaussian_process_plot_gpr_noisy_targets.py:


=========================================================
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.



.. rst-class:: sphx-glr-horizontal


    *

      .. image:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_noisy_targets_001.png
          :alt: plot gpr noisy targets
          :class: sphx-glr-multi-img

    *

      .. image:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_noisy_targets_002.png
          :alt: plot gpr noisy targets
          :class: sphx-glr-multi-img






.. code-block:: default

    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

    # Instantiate 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
    plt.figure()
    plt.plot(x, f(x), 'r:', label=r'$f(x) = x\,\sin(x)$')
    plt.plot(X, y, 'r.', markersize=10, label='Observations')
    plt.plot(x, y_pred, 'b-', label='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

    # Instantiate a Gaussian Process model
    gp = GaussianProcessRegressor(kernel=kernel, alpha=dy ** 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
    plt.figure()
    plt.plot(x, f(x), 'r:', label=r'$f(x) = x\,\sin(x)$')
    plt.errorbar(X.ravel(), y, dy, fmt='r.', markersize=10, label='Observations')
    plt.plot(x, y_pred, 'b-', label='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()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 0 minutes  0.535 seconds)


.. _sphx_glr_download_auto_examples_gaussian_process_plot_gpr_noisy_targets.py:


.. only :: html

 .. container:: sphx-glr-footer
    :class: sphx-glr-footer-example


  .. container:: binder-badge

    .. image:: https://mybinder.org/badge_logo.svg
      :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/0.23.X?urlpath=lab/tree/notebooks/auto_examples/gaussian_process/plot_gpr_noisy_targets.ipynb
      :width: 150 px


  .. container:: sphx-glr-download sphx-glr-download-python

     :download:`Download Python source code: plot_gpr_noisy_targets.py <plot_gpr_noisy_targets.py>`



  .. container:: sphx-glr-download sphx-glr-download-jupyter

     :download:`Download Jupyter notebook: plot_gpr_noisy_targets.ipynb <plot_gpr_noisy_targets.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_