.. _sphx_glr_auto_examples_gaussian_process_plot_gpr_prior_posterior.py:


==========================================================================
Illustration of prior and posterior Gaussian process for different kernels
==========================================================================

This example illustrates the prior and posterior of a GPR with different
kernels. Mean, standard deviation, and 10 samples are shown for both prior
and posterior.



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


    *

      .. image:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_000.png
            :scale: 47

    *

      .. image:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_001.png
            :scale: 47

    *

      .. image:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_002.png
            :scale: 47

    *

      .. image:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_003.png
            :scale: 47

    *

      .. image:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_004.png
            :scale: 47





.. code-block:: python

    print(__doc__)

    # Authors: Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>
    #
    # 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, Matern, RationalQuadratic,
                                                  ExpSineSquared, DotProduct,
                                                  ConstantKernel)


    kernels = [1.0 * RBF(length_scale=1.0, length_scale_bounds=(1e-1, 10.0)),
               1.0 * RationalQuadratic(length_scale=1.0, alpha=0.1),
               1.0 * ExpSineSquared(length_scale=1.0, periodicity=3.0,
                                    length_scale_bounds=(0.1, 10.0),
                                    periodicity_bounds=(1.0, 10.0)),
               ConstantKernel(0.1, (0.01, 10.0))
                   * (DotProduct(sigma_0=1.0, sigma_0_bounds=(0.0, 10.0)) ** 2),
               1.0 * Matern(length_scale=1.0, length_scale_bounds=(1e-1, 10.0),
                            nu=1.5)]

    for fig_index, kernel in enumerate(kernels):
        # Specify Gaussian Process
        gp = GaussianProcessRegressor(kernel=kernel)

        # Plot prior
        plt.figure(fig_index, figsize=(8, 8))
        plt.subplot(2, 1, 1)
        X_ = np.linspace(0, 5, 100)
        y_mean, y_std = gp.predict(X_[:, np.newaxis], return_std=True)
        plt.plot(X_, y_mean, 'k', lw=3, zorder=9)
        plt.fill_between(X_, y_mean - y_std, y_mean + y_std,
                         alpha=0.5, color='k')
        y_samples = gp.sample_y(X_[:, np.newaxis], 10)
        plt.plot(X_, y_samples, lw=1)
        plt.xlim(0, 5)
        plt.ylim(-3, 3)
        plt.title("Prior (kernel:  %s)" % kernel, fontsize=12)

        # Generate data and fit GP
        rng = np.random.RandomState(4)
        X = rng.uniform(0, 5, 10)[:, np.newaxis]
        y = np.sin((X[:, 0] - 2.5) ** 2)
        gp.fit(X, y)

        # Plot posterior
        plt.subplot(2, 1, 2)
        X_ = np.linspace(0, 5, 100)
        y_mean, y_std = gp.predict(X_[:, np.newaxis], return_std=True)
        plt.plot(X_, y_mean, 'k', lw=3, zorder=9)
        plt.fill_between(X_, y_mean - y_std, y_mean + y_std,
                         alpha=0.5, color='k')

        y_samples = gp.sample_y(X_[:, np.newaxis], 10)
        plt.plot(X_, y_samples, lw=1)
        plt.scatter(X[:, 0], y, c='r', s=50, zorder=10)
        plt.xlim(0, 5)
        plt.ylim(-3, 3)
        plt.title("Posterior (kernel: %s)\n Log-Likelihood: %.3f"
                  % (gp.kernel_, gp.log_marginal_likelihood(gp.kernel_.theta)),
                  fontsize=12)
        plt.tight_layout()

    plt.show()

**Total running time of the script:**
(0 minutes 2.178 seconds)



.. container:: sphx-glr-download

    **Download Python source code:** :download:`plot_gpr_prior_posterior.py <plot_gpr_prior_posterior.py>`


.. container:: sphx-glr-download

    **Download IPython notebook:** :download:`plot_gpr_prior_posterior.ipynb <plot_gpr_prior_posterior.ipynb>`