.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/reliability/lifetimes/plot_03_weibull_distribution.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_reliability_lifetimes_plot_03_weibull_distribution.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_reliability_lifetimes_plot_03_weibull_distribution.py:


Weibull's distribution and the Weibull plot (1951)
==================================================

Waloddi Weibull (1951) proposed :math:`R(t) = e^{-(t/\eta)^\beta}`
(:func:`~engineeringkit.reliability.weibull_reliability`) for anything from
the strength of steel to the life of bearings. The shape :math:`\beta` sets
the hazard (:func:`~engineeringkit.reliability.weibull_hazard`): falling for
:math:`\beta < 1`, constant at 1, rising above. Because
:math:`\ln(-\ln R) = \beta \ln t - \beta \ln \eta`, failure data plotted on
those axes fall on a line whose slope is :math:`\beta`. The example fits a
small life test that way and compares the mean life with
:func:`~engineeringkit.reliability.weibull_mttf`.

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.reliability import weibull_cdf, weibull_hazard, weibull_mttf, weibull_reliability








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Three shapes, three hazards
---------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-37

.. code-block:: Python


    t = np.linspace(0.01, 3000, 400)
    eta = 1000.0
    fig, axes = plt.subplots(1, 2, figsize=(9, 3.5))
    for beta in (0.5, 1.0, 3.0):
        axes[0].plot(t, weibull_reliability(t, eta, beta), label=rf"$\beta$ = {beta}")
        axes[1].plot(t, weibull_hazard(t, eta, beta) * 1e3, label=rf"$\beta$ = {beta}")
    axes[0].set(xlabel="time [h]", ylabel="reliability")
    axes[1].set(xlabel="time [h]", ylabel="hazard [1/1000 h]", ylim=(0, 6))
    axes[0].legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/reliability/lifetimes/images/sphx_glr_plot_03_weibull_distribution_001.png
   :alt: plot 03 weibull distribution
   :srcset: /api/gallery/reliability/lifetimes/images/sphx_glr_plot_03_weibull_distribution_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 38-44

Fit a life test on Weibull paper
--------------------------------

Thirty bearings with :math:`\eta` = 1000 h and :math:`\beta` = 2.5 run to
failure. Bernard's median ranks :math:`(i - 0.3)/(n + 0.4)` estimate
:math:`F` at the ordered failure times.

.. GENERATED FROM PYTHON SOURCE LINES 44-65

.. code-block:: Python


    rng = np.random.default_rng(1951)
    true_eta, true_beta, n = 1000.0, 2.5, 30
    life = np.sort(true_eta * rng.weibull(true_beta, n))
    F = (np.arange(1, n + 1) - 0.3) / (n + 0.4)
    slope, intercept = np.polyfit(np.log(life), np.log(-np.log(1 - F)), 1)
    beta_hat, eta_hat = slope, np.exp(-intercept / slope)
    print(f"fitted beta = {beta_hat:.2f}, eta = {eta_hat:.0f} h; MTTF = {weibull_mttf(eta_hat, beta_hat):.0f} h")

    fig, ax = plt.subplots(figsize=(5.5, 4))
    ax.plot(np.log(life), np.log(-np.log(1 - F)), "o", label="median ranks")
    x = np.linspace(np.log(life[0]), np.log(life[-1]))
    ax.plot(x, np.log(-np.log(1 - weibull_cdf(np.exp(x), eta_hat, beta_hat))), label=rf"fit: $\beta$ = {beta_hat:.2f}")
    ax.set(xlabel="ln(time)", ylabel="ln(-ln(1 - F))", title="Weibull plot")
    ax.legend()
    fig.tight_layout()

    assert abs(beta_hat - true_beta) < 0.6 and abs(eta_hat / true_eta - 1) < 0.15
    assert np.isclose(weibull_mttf(eta, 1.0), eta)

    plt.show()



.. image-sg:: /api/gallery/reliability/lifetimes/images/sphx_glr_plot_03_weibull_distribution_002.png
   :alt: Weibull plot
   :srcset: /api/gallery/reliability/lifetimes/images/sphx_glr_plot_03_weibull_distribution_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    fitted beta = 2.47, eta = 886 h; MTTF = 786 h





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

   **Total running time of the script:** (0 minutes 0.067 seconds)


.. _sphx_glr_download_api_gallery_reliability_lifetimes_plot_03_weibull_distribution.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/reliability/lifetimes/plot_03_weibull_distribution.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_weibull_distribution.ipynb <plot_03_weibull_distribution.ipynb>`

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

      :download:`Download Python source code: plot_03_weibull_distribution.py <plot_03_weibull_distribution.py>`

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

      :download:`Download zipped: plot_03_weibull_distribution.zip <plot_03_weibull_distribution.zip>`


.. only:: html

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

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