.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_examples_ensemble_plot_gradient_boosting_quantile.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_ensemble_plot_gradient_boosting_quantile.py:


=====================================================
Prediction Intervals for Gradient Boosting Regression
=====================================================

This example shows how quantile regression can be used
to create prediction intervals.



.. image:: /auto_examples/ensemble/images/sphx_glr_plot_gradient_boosting_quantile_001.png
    :alt: plot gradient boosting quantile
    :class: sphx-glr-single-img






.. code-block:: default


    import numpy as np
    import matplotlib.pyplot as plt

    from sklearn.ensemble import GradientBoostingRegressor

    np.random.seed(1)


    def f(x):
        """The function to predict."""
        return x * np.sin(x)

    #----------------------------------------------------------------------
    #  First the noiseless case
    X = np.atleast_2d(np.random.uniform(0, 10.0, size=100)).T
    X = X.astype(np.float32)

    # Observations
    y = f(X).ravel()

    dy = 1.5 + 1.0 * np.random.random(y.shape)
    noise = np.random.normal(0, dy)
    y += noise
    y = y.astype(np.float32)

    # Mesh the input space for evaluations of the real function, the prediction and
    # its MSE
    xx = np.atleast_2d(np.linspace(0, 10, 1000)).T
    xx = xx.astype(np.float32)

    alpha = 0.95

    clf = GradientBoostingRegressor(loss='quantile', alpha=alpha,
                                    n_estimators=250, max_depth=3,
                                    learning_rate=.1, min_samples_leaf=9,
                                    min_samples_split=9)

    clf.fit(X, y)

    # Make the prediction on the meshed x-axis
    y_upper = clf.predict(xx)

    clf.set_params(alpha=1.0 - alpha)
    clf.fit(X, y)

    # Make the prediction on the meshed x-axis
    y_lower = clf.predict(xx)

    clf.set_params(loss='ls')
    clf.fit(X, y)

    # Make the prediction on the meshed x-axis
    y_pred = clf.predict(xx)

    # Plot the function, the prediction and the 95% confidence interval based on
    # the MSE
    fig = plt.figure()
    plt.plot(xx, f(xx), 'g:', label=r'$f(x) = x\,\sin(x)$')
    plt.plot(X, y, 'b.', markersize=10, label=u'Observations')
    plt.plot(xx, y_pred, 'r-', label=u'Prediction')
    plt.plot(xx, y_upper, 'k-')
    plt.plot(xx, y_lower, 'k-')
    plt.fill(np.concatenate([xx, xx[::-1]]),
             np.concatenate([y_upper, y_lower[::-1]]),
             alpha=.5, fc='b', ec='None', label='95% prediction 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  0.452 seconds)


.. _sphx_glr_download_auto_examples_ensemble_plot_gradient_boosting_quantile.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/ensemble/plot_gradient_boosting_quantile.ipynb
      :width: 150 px


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

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



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

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


.. only:: html

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

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