.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/probability/markov_chain/plot_02_continuous_time_chain.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_probability_markov_chain_plot_02_continuous_time_chain.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_probability_markov_chain_plot_02_continuous_time_chain.py:


Kolmogorov's equations: a continuous-time Markov chain
========================================================

A machine is working (0), degraded (1), or broken (2), jumping between
states at the rates in the generator :math:`Q`. The transition matrix
:math:`P(t) = e^{tQ}` solves Kolmogorov's forward and backward equations
and relaxes to the stationary distribution :math:`\pi`, the solution of
:math:`\pi Q = 0`.

.. GENERATED FROM PYTHON SOURCE LINES 13-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.probability import ctmc_stationary_distribution, ctmc_transition_matrix

    Q = np.array(
        [
            [-0.5, 0.4, 0.1],
            [0.0, -1.0, 1.0],
            [2.0, 0.0, -2.0],
        ]
    )








.. GENERATED FROM PYTHON SOURCE LINES 27-29

State probabilities starting from "working"
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-43

.. code-block:: Python


    times = np.linspace(0.0, 8.0, 200)
    probs = np.array([ctmc_transition_matrix(Q, t)[0] for t in times])
    pi = ctmc_stationary_distribution(Q)
    print(f"stationary distribution: {pi.round(4)}")

    fig, ax = plt.subplots()
    for j, name in enumerate(["working", "degraded", "broken"]):
        (line,) = ax.plot(times, probs[:, j], label=name)
        ax.axhline(pi[j], color=line.get_color(), ls=":")
    ax.set_xlabel("t")
    ax.set_ylabel(r"$P_{0j}(t)$")
    ax.legend()
    ax.set_title(r"$P(t) = e^{tQ}$ converging to $\pi$")



.. image-sg:: /api/gallery/probability/markov_chain/images/sphx_glr_plot_02_continuous_time_chain_001.png
   :alt: $P(t) = e^{tQ}$ converging to $\pi$
   :srcset: /api/gallery/probability/markov_chain/images/sphx_glr_plot_02_continuous_time_chain_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    stationary distribution: [0.6061 0.2424 0.1515]

    Text(0.5, 1.0, '$P(t) = e^{tQ}$ converging to $\\pi$')




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

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


.. _sphx_glr_download_api_gallery_probability_markov_chain_plot_02_continuous_time_chain.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_continuous_time_chain.ipynb <plot_02_continuous_time_chain.ipynb>`

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

      :download:`Download Python source code: plot_02_continuous_time_chain.py <plot_02_continuous_time_chain.py>`

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

      :download:`Download zipped: plot_02_continuous_time_chain.zip <plot_02_continuous_time_chain.zip>`


.. only:: html

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

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