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

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_linear_model_plot_ridge_path.py>`
        to download the full example code. or to run this example in your browser via JupyterLite or Binder

.. 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.

.. GENERATED FROM PYTHON SOURCE LINES 29-42

.. code-block:: Python


    # Authors: The scikit-learn developers
    # SPDX-License-Identifier: BSD-3-Clause

    import matplotlib.pyplot as plt
    import numpy as np

    from sklearn import linear_model

    # X is the 10x10 Hilbert matrix
    X = 1.0 / (np.arange(1, 11) + np.arange(0, 10)[:, np.newaxis])
    y = np.ones(10)








.. GENERATED FROM PYTHON SOURCE LINES 43-45

Compute paths
-------------

.. GENERATED FROM PYTHON SOURCE LINES 45-55

.. code-block:: Python


    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_)








.. GENERATED FROM PYTHON SOURCE LINES 56-58

Display results
---------------

.. GENERATED FROM PYTHON SOURCE LINES 58-69

.. code-block:: Python


    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()



.. image-sg:: /auto_examples/linear_model/images/sphx_glr_plot_ridge_path_001.png
   :alt: Ridge coefficients as a function of the regularization
   :srcset: /auto_examples/linear_model/images/sphx_glr_plot_ridge_path_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.338 seconds)


.. _sphx_glr_download_auto_examples_linear_model_plot_ridge_path.py:

.. only:: html

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

    .. container:: binder-badge

      .. image:: images/binder_badge_logo.svg
        :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/1.6.X?urlpath=lab/tree/notebooks/auto_examples/linear_model/plot_ridge_path.ipynb
        :alt: Launch binder
        :width: 150 px

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../lite/lab/index.html?path=auto_examples/linear_model/plot_ridge_path.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

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

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

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

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_ridge_path.zip <plot_ridge_path.zip>`


.. include:: plot_ridge_path.recommendations


.. only:: html

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

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