.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/reliability/lifetimes/plot_05_kaplan_meier.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_05_kaplan_meier.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_05_kaplan_meier.py:


Kaplan and Meier's product-limit estimator (1958)
=================================================

Field data are incomplete: some units are still working when the data are
collected, others were removed for reasons other than failure. Dropping
them biases the life estimate low; counting them as failures biases it too.
Kaplan and Meier (1958) multiplied the conditional survival
:math:`1 - d_j/n_j` at each failure, where :math:`n_j` counts only units
still under observation
(:func:`~engineeringkit.reliability.kaplan_meier`). The example censors a
Weibull sample at random and recovers the true survival curve.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.reliability import kaplan_meier, weibull_reliability








.. GENERATED FROM PYTHON SOURCE LINES 22-28

Censored field data
-------------------

200 pumps with Weibull lives (:math:`\eta` = 5000 h, :math:`\beta` =
1.8). Each enters service at a random date, so its observation ends at a
uniformly distributed time up to 8000 h (time unit: hours).

.. GENERATED FROM PYTHON SOURCE LINES 28-38

.. code-block:: Python


    rng = np.random.default_rng(1958)
    eta, beta, n = 5000.0, 1.8, 200
    life = eta * rng.weibull(beta, n)
    window = rng.uniform(0, 8000, n)
    times = np.minimum(life, window)
    failed = (life <= window).astype(int)
    km = kaplan_meier(times, failed)
    print(km.summary())





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

 .. code-block:: none

    {'n': 200, 'failures': 96, 'censored': 104, 'final_survival': 0.21319158745210554}




.. GENERATED FROM PYTHON SOURCE LINES 39-43

Compare with the truth and with naive estimates
-----------------------------------------------

"Ignore the censored units" treats only the failures as the sample.

.. GENERATED FROM PYTHON SOURCE LINES 43-62

.. code-block:: Python


    t = np.linspace(0, 6000, 300)
    naive = np.array([np.mean(times[failed == 1] > x) for x in t])
    error = np.abs(km.survival_at(t) - weibull_reliability(t, eta, beta))
    band = 2 * km.std_error
    fig, ax = km.plot(label="Kaplan-Meier")
    ax.fill_between(km.times, km.survival - band, km.survival + band, step="post", alpha=0.2, label=r"$\pm 2$ s.e. (Greenwood)")
    ax.plot(t, weibull_reliability(t, eta, beta), "k--", label="true Weibull")
    ax.plot(t, naive, color="C3", label="failures only (biased)")
    ax.set(xlabel="time [h]", xlim=(0, 6000))
    ax.legend()
    fig.tight_layout()
    naive_error = np.abs(naive - weibull_reliability(t, eta, beta))
    print(f"largest error of Kaplan-Meier: {error.max():.3f}; of the naive estimate: {naive_error.max():.3f}")

    assert error.max() < 0.08
    assert naive_error.max() > 2 * error.max()

    plt.show()



.. image-sg:: /api/gallery/reliability/lifetimes/images/sphx_glr_plot_05_kaplan_meier_001.png
   :alt: plot 05 kaplan meier
   :srcset: /api/gallery/reliability/lifetimes/images/sphx_glr_plot_05_kaplan_meier_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    largest error of Kaplan-Meier: 0.057; of the naive estimate: 0.334





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

   **Total running time of the script:** (0 minutes 0.032 seconds)


.. _sphx_glr_download_api_gallery_reliability_lifetimes_plot_05_kaplan_meier.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_05_kaplan_meier.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_kaplan_meier.ipynb <plot_05_kaplan_meier.ipynb>`

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

      :download:`Download Python source code: plot_05_kaplan_meier.py <plot_05_kaplan_meier.py>`

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

      :download:`Download zipped: plot_05_kaplan_meier.zip <plot_05_kaplan_meier.zip>`


.. only:: html

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

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