.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/numerical_analysis/regression/plot_01_polynomial_regression.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_numerical_analysis_regression_plot_01_polynomial_regression.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_numerical_analysis_regression_plot_01_polynomial_regression.py:


Least-squares polynomial regression and its conditioning
=============================================================

:class:`~mathematicskit.numerical_analysis.systems.regression.PolynomialRegression`
fits a degree-``d`` polynomial by solving the normal equations
:math:`V^T V c = V^T y`. This script fits noisy cubic data, checks the
:math:`R^2` and residuals, and then shows the fit's condition number
(estimated via power iteration on the normal-equations matrix) growing
sharply with degree -- squaring the design matrix's own condition number
is exactly why the normal-equations approach is eventually replaced by a
QR-based least-squares solve in :mod:`mathematicskit.linalg`.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.numerical_analysis import PolynomialRegression








.. GENERATED FROM PYTHON SOURCE LINES 22-24

Fit noisy cubic data
-----------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-44

.. code-block:: Python


    rng = np.random.default_rng(0)
    x = np.linspace(-1.0, 1.0, 60)
    y_true = 2.0 * x**3 - x + 0.5
    y = y_true + rng.normal(0.0, 0.05, x.shape)

    model = PolynomialRegression(x, y, degree=3)
    result = model.fit()
    print(f"coefficients (highest power first): {result.coefficients}")
    print(f"R^2 = {result.r_squared:.5f}, adjusted R^2 = {result.adjusted_r_squared:.5f}")

    x_fine = np.linspace(-1.0, 1.0, 300)
    fig1, ax1 = plt.subplots(figsize=(6, 4))
    ax1.scatter(x, y, s=12, color="gray", label="noisy data")
    ax1.plot(x_fine, model.predict(x_fine), color="firebrick", label="degree-3 fit")
    ax1.plot(x_fine, 2.0 * x_fine**3 - x_fine + 0.5, "--", color="black", label="true cubic")
    ax1.legend()
    ax1.set_title("Least-squares polynomial fit")
    fig1.tight_layout()




.. image-sg:: /api/gallery/numerical_analysis/regression/images/sphx_glr_plot_01_polynomial_regression_001.png
   :alt: Least-squares polynomial fit
   :srcset: /api/gallery/numerical_analysis/regression/images/sphx_glr_plot_01_polynomial_regression_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    coefficients (highest power first): [ 1.90017422 -0.00562603 -0.92442784  0.50580371]
    R^2 = 0.98524, adjusted R^2 = 0.98445




.. GENERATED FROM PYTHON SOURCE LINES 45-49

Conditioning worsens with degree
------------------------------------
Fitting increasingly high-degree polynomials to the same data makes the
Vandermonde normal-equations matrix more ill-conditioned.

.. GENERATED FROM PYTHON SOURCE LINES 49-63

.. code-block:: Python


    degrees = [1, 2, 4, 6, 8, 10]
    conditions = [PolynomialRegression(x, y, degree=d).fit().condition_number for d in degrees]
    for d, c in zip(degrees, conditions):
        print(f"degree={d:2d}  condition number~{c:.3e}")

    fig2, ax2 = plt.subplots(figsize=(6, 4))
    ax2.semilogy(degrees, conditions, "o-", color="steelblue")
    ax2.set_xlabel("polynomial degree")
    ax2.set_ylabel("condition number of V^T V")
    ax2.set_title("Normal-equations conditioning worsens with degree")
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/numerical_analysis/regression/images/sphx_glr_plot_01_polynomial_regression_002.png
   :alt: Normal-equations conditioning worsens with degree
   :srcset: /api/gallery/numerical_analysis/regression/images/sphx_glr_plot_01_polynomial_regression_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    degree= 1  condition number~1.703e+00
    degree= 2  condition number~3.666e+00
    degree= 4  condition number~1.817e+01
    degree= 6  condition number~9.583e+01
    degree= 8  condition number~5.204e+02
    degree=10  condition number~2.884e+03





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

   **Total running time of the script:** (0 minutes 0.087 seconds)


.. _sphx_glr_download_api_gallery_numerical_analysis_regression_plot_01_polynomial_regression.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_polynomial_regression.ipynb <plot_01_polynomial_regression.ipynb>`

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

      :download:`Download Python source code: plot_01_polynomial_regression.py <plot_01_polynomial_regression.py>`

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

      :download:`Download zipped: plot_01_polynomial_regression.zip <plot_01_polynomial_regression.zip>`


.. only:: html

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

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