.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/model_selection/plot_underfitting_overfitting.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_model_selection_plot_underfitting_overfitting.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_model_selection_plot_underfitting_overfitting.py:


============================
Underfitting vs. Overfitting
============================

This example demonstrates the problems of underfitting and overfitting and
how we can use linear regression with polynomial features to approximate
nonlinear functions. The plot shows the function that we want to approximate,
which is a part of the cosine function. In addition, the samples from the
real function and the approximations of different models are displayed. The
models have polynomial features of different degrees. We can see that a
linear function (polynomial with degree 1) is not sufficient to fit the
training samples. This is called **underfitting**. A polynomial of degree 4
approximates the true function almost perfectly. However, for higher degrees
the model will **overfit** the training data, i.e. it learns the noise of the
training data.
We evaluate quantitatively **overfitting** / **underfitting** by using
cross-validation. We calculate the mean squared error (MSE) on the validation
set, the higher, the less likely the model generalizes correctly from the
training data.

.. GENERATED FROM PYTHON SOURCE LINES 23-83



.. image-sg:: /auto_examples/model_selection/images/sphx_glr_plot_underfitting_overfitting_001.png
   :alt: Degree 1 MSE = 4.08e-01(+/- 4.25e-01), Degree 4 MSE = 4.32e-02(+/- 7.08e-02), Degree 15 MSE = 1.82e+08(+/- 5.46e+08)
   :srcset: /auto_examples/model_selection/images/sphx_glr_plot_underfitting_overfitting_001.png
   :class: sphx-glr-single-img





.. 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.linear_model import LinearRegression
    from sklearn.model_selection import cross_val_score
    from sklearn.pipeline import Pipeline
    from sklearn.preprocessing import PolynomialFeatures


    def true_fun(X):
        return np.cos(1.5 * np.pi * X)


    np.random.seed(0)

    n_samples = 30
    degrees = [1, 4, 15]

    X = np.sort(np.random.rand(n_samples))
    y = true_fun(X) + np.random.randn(n_samples) * 0.1

    plt.figure(figsize=(14, 5))
    for i in range(len(degrees)):
        ax = plt.subplot(1, len(degrees), i + 1)
        plt.setp(ax, xticks=(), yticks=())

        polynomial_features = PolynomialFeatures(degree=degrees[i], include_bias=False)
        linear_regression = LinearRegression()
        pipeline = Pipeline(
            [
                ("polynomial_features", polynomial_features),
                ("linear_regression", linear_regression),
            ]
        )
        pipeline.fit(X[:, np.newaxis], y)

        # Evaluate the models using crossvalidation
        scores = cross_val_score(
            pipeline, X[:, np.newaxis], y, scoring="neg_mean_squared_error", cv=10
        )

        X_test = np.linspace(0, 1, 100)
        plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label="Model")
        plt.plot(X_test, true_fun(X_test), label="True function")
        plt.scatter(X, y, edgecolor="b", s=20, label="Samples")
        plt.xlabel("x")
        plt.ylabel("y")
        plt.xlim((0, 1))
        plt.ylim((-2, 2))
        plt.legend(loc="best")
        plt.title(
            "Degree {}\nMSE = {:.2e}(+/- {:.2e})".format(
                degrees[i], -scores.mean(), scores.std()
            )
        )
    plt.show()


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

   **Total running time of the script:** (0 minutes 0.216 seconds)


.. _sphx_glr_download_auto_examples_model_selection_plot_underfitting_overfitting.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/model_selection/plot_underfitting_overfitting.ipynb
        :alt: Launch binder
        :width: 150 px

    .. container:: lite-badge

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

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

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

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

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

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

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


.. include:: plot_underfitting_overfitting.recommendations


.. only:: html

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

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