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

    Click :ref:`here <sphx_glr_download_auto_examples_linear_model_plot_ridge_path.py>` to download the full example code
.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_linear_model_plot_ridge_path.py:


===========================================================
Plot Ridge coefficients as a function of the regularization
===========================================================

Shows the effect of collinearity in the coefficients of an estimator.

.. currentmodule:: sklearn.linear_model

:class:`Ridge` Regression is the estimator used in this example.
Each color represents a different feature of the
coefficient vector, and this is displayed as a function of the
regularization parameter.

This example also shows the usefulness of applying Ridge regression
to highly ill-conditioned matrices. For such matrices, a slight
change in the target variable can cause huge variances in the
calculated weights. In such cases, it is useful to set a certain
regularization (alpha) to reduce this variation (noise).

When alpha is very large, the regularization effect dominates the
squared loss function and the coefficients tend to zero.
At the end of the path, as alpha tends toward zero
and the solution tends towards the ordinary least squares, coefficients
exhibit big oscillations. In practise it is necessary to tune alpha
in such a way that a balance is maintained between both.




.. image:: /auto_examples/linear_model/images/sphx_glr_plot_ridge_path_001.png
    :class: sphx-glr-single-img





.. code-block:: python


    # Author: Fabian Pedregosa -- <fabian.pedregosa@inria.fr>
    # License: BSD 3 clause

    print(__doc__)

    import numpy as np
    import matplotlib.pyplot as plt
    from sklearn import linear_model

    # X is the 10x10 Hilbert matrix
    X = 1. / (np.arange(1, 11) + np.arange(0, 10)[:, np.newaxis])
    y = np.ones(10)

    # #############################################################################
    # Compute paths

    n_alphas = 200
    alphas = np.logspace(-10, -2, n_alphas)

    coefs = []
    for a in alphas:
        ridge = linear_model.Ridge(alpha=a, fit_intercept=False)
        ridge.fit(X, y)
        coefs.append(ridge.coef_)

    # #############################################################################
    # Display results

    ax = plt.gca()

    ax.plot(alphas, coefs)
    ax.set_xscale('log')
    ax.set_xlim(ax.get_xlim()[::-1])  # reverse axis
    plt.xlabel('alpha')
    plt.ylabel('weights')
    plt.title('Ridge coefficients as a function of the regularization')
    plt.axis('tight')
    plt.show()

**Total running time of the script:** ( 0 minutes  0.161 seconds)


.. _sphx_glr_download_auto_examples_linear_model_plot_ridge_path.py:


.. only :: html

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



  .. container:: sphx-glr-download

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



  .. container:: sphx-glr-download

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


.. only:: html

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

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