.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/reliability/lifetimes/plot_04_epstein_sobel_life_testing.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_04_epstein_sobel_life_testing.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_04_epstein_sobel_life_testing.py:


Epstein and Sobel's exponential life testing (1953)
===================================================

How long must a test run, and with how many units, to demonstrate an MTBF?
Epstein and Sobel (1953) showed that for exponential lives the *total time
on test* :math:`T`, summed over failed and surviving units, carries all the
information: :math:`\hat\theta = T/r` after :math:`r` failures, and
:math:`2T/\theta` is chi-square with :math:`2r` degrees of freedom. That
gives exact confidence bounds
(:func:`~engineeringkit.reliability.mtbf_confidence_bounds`). The example
stops a test of 12 units at the fourth failure, many times over, and checks
that the 90% bounds catch the true MTBF 90% of the time.

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.reliability import exponential_mttf, exponential_reliability, failure_rate_from_mtbf, mtbf_confidence_bounds








.. GENERATED FROM PYTHON SOURCE LINES 23-29

One test
--------

Components with a failure rate of :math:`2\times10^{-4}` per hour (time
unit: hours). Twelve go on test; the test stops at the fourth failure, so
eight units are still running.

.. GENERATED FROM PYTHON SOURCE LINES 29-41

.. code-block:: Python


    lam = 2e-4
    theta = exponential_mttf(lam)
    n, r = 12, 4
    rng = np.random.default_rng(1953)
    life = np.sort(rng.exponential(theta, n))
    total = life[:r].sum() + (n - r) * life[r - 1]
    low, high = mtbf_confidence_bounds(total, r, confidence=0.9)
    print(f"test stopped at {life[r - 1]:.0f} h, total time on test {total:.0f} h")
    print(f"MTBF estimate {total / r:.0f} h, 90% bounds [{low:.0f}, {high:.0f}] h (true {theta:.0f} h)")
    print(f"estimated failure rate {failure_rate_from_mtbf(total / r):.2e}/h")





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

 .. code-block:: none

    test stopped at 2502 h, total time on test 23477 h
    MTBF estimate 5869 h, 90% bounds [3028, 17183] h (true 5000 h)
    estimated failure rate 1.70e-04/h




.. GENERATED FROM PYTHON SOURCE LINES 42-44

Repeat the test 300 times
-------------------------

.. GENERATED FROM PYTHON SOURCE LINES 44-69

.. code-block:: Python


    tests = 300
    bounds = []
    for _ in range(tests):
        life = np.sort(rng.exponential(theta, n))
        bounds.append(mtbf_confidence_bounds(life[:r].sum() + (n - r) * life[r - 1], r, 0.9))
    bounds = np.array(bounds)
    covered = (bounds[:, 0] <= theta) & (theta <= bounds[:, 1])
    print(f"intervals containing the true MTBF: {covered.mean():.1%}")

    fig, axes = plt.subplots(1, 2, figsize=(9.5, 3.8))
    show = np.arange(60)
    axes[0].vlines(show, bounds[show, 0], bounds[show, 1], colors=np.where(covered[show], "C0", "C3"))
    axes[0].axhline(theta, color="k", lw=1)
    axes[0].set(yscale="log", xlabel="test", ylabel="MTBF [h]", title="90% bounds, 60 of the tests")
    t = np.linspace(0, 3 * theta, 200)
    axes[1].plot(t, exponential_reliability(t, lam), label="true")
    axes[1].fill_between(t, exponential_reliability(t, 1 / low), exponential_reliability(t, 1 / high), alpha=0.3, label="90% band, first test")
    axes[1].set(xlabel="time [h]", ylabel="reliability")
    axes[1].legend()
    fig.tight_layout()

    assert abs(covered.mean() - 0.9) < 0.05

    plt.show()



.. image-sg:: /api/gallery/reliability/lifetimes/images/sphx_glr_plot_04_epstein_sobel_life_testing_001.png
   :alt: 90% bounds, 60 of the tests
   :srcset: /api/gallery/reliability/lifetimes/images/sphx_glr_plot_04_epstein_sobel_life_testing_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    intervals containing the true MTBF: 90.0%





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

   **Total running time of the script:** (0 minutes 0.071 seconds)


.. _sphx_glr_download_api_gallery_reliability_lifetimes_plot_04_epstein_sobel_life_testing.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_04_epstein_sobel_life_testing.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_epstein_sobel_life_testing.ipynb <plot_04_epstein_sobel_life_testing.ipynb>`

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

      :download:`Download Python source code: plot_04_epstein_sobel_life_testing.py <plot_04_epstein_sobel_life_testing.py>`

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

      :download:`Download zipped: plot_04_epstein_sobel_life_testing.zip <plot_04_epstein_sobel_life_testing.zip>`


.. only:: html

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

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