.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/linear_model/plot_polynomial_interpolation.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_examples_linear_model_plot_polynomial_interpolation.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_linear_model_plot_polynomial_interpolation.py:


========================
Polynomial interpolation
========================

This example demonstrates how to approximate a function with a polynomial of
degree n_degree by using ridge regression. Concretely, from n_samples 1d
points, it suffices to build the Vandermonde matrix, which is n_samples x
n_degree+1 and has the following form:

[[1, x_1, x_1 ** 2, x_1 ** 3, ...],
 [1, x_2, x_2 ** 2, x_2 ** 3, ...],
 ...]

Intuitively, this matrix can be interpreted as a matrix of pseudo features (the
points raised to some power). The matrix is akin to (but different from) the
matrix induced by a polynomial kernel.

This example shows that you can do non-linear regression with a linear model,
using a pipeline to add non-linear features. Kernel methods extend this idea
and can induce very high (even infinite) dimensional feature spaces.

.. GENERATED FROM PYTHON SOURCE LINES 24-73



.. image:: /auto_examples/linear_model/images/sphx_glr_plot_polynomial_interpolation_001.png
    :alt: plot polynomial interpolation
    :class: sphx-glr-single-img





.. code-block:: default

    print(__doc__)

    # Author: Mathieu Blondel
    #         Jake Vanderplas
    # License: BSD 3 clause

    import numpy as np
    import matplotlib.pyplot as plt

    from sklearn.linear_model import Ridge
    from sklearn.preprocessing import PolynomialFeatures
    from sklearn.pipeline import make_pipeline


    def f(x):
        """ function to approximate by polynomial interpolation"""
        return x * np.sin(x)


    # generate points used to plot
    x_plot = np.linspace(0, 10, 100)

    # generate points and keep a subset of them
    x = np.linspace(0, 10, 100)
    rng = np.random.RandomState(0)
    rng.shuffle(x)
    x = np.sort(x[:20])
    y = f(x)

    # create matrix versions of these arrays
    X = x[:, np.newaxis]
    X_plot = x_plot[:, np.newaxis]

    colors = ['teal', 'yellowgreen', 'gold']
    lw = 2
    plt.plot(x_plot, f(x_plot), color='cornflowerblue', linewidth=lw,
             label="ground truth")
    plt.scatter(x, y, color='navy', s=30, marker='o', label="training points")

    for count, degree in enumerate([3, 4, 5]):
        model = make_pipeline(PolynomialFeatures(degree), Ridge())
        model.fit(X, y)
        y_plot = model.predict(X_plot)
        plt.plot(x_plot, y_plot, color=colors[count], linewidth=lw,
                 label="degree %d" % degree)

    plt.legend(loc='lower left')

    plt.show()


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

   **Total running time of the script:** ( 0 minutes  0.127 seconds)


.. _sphx_glr_download_auto_examples_linear_model_plot_polynomial_interpolation.py:


.. only :: html

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


  .. container:: binder-badge

    .. image:: images/binder_badge_logo.svg
      :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/0.24.X?urlpath=lab/tree/notebooks/auto_examples/linear_model/plot_polynomial_interpolation.ipynb
      :alt: Launch binder
      :width: 150 px


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

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



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

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


.. only:: html

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

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