.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_probabilistic_modeling/stochastic_processes/plot_discrete_markov_chain_process.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_discrete_markov_chain_process.py: Create a discrete Markov chain process ====================================== .. GENERATED FROM PYTHON SOURCE LINES 6-27 This example details first how to create and manipulate a discrete Markov chain. A discrete Markov chain :math:`X: \Omega \times \mathcal{D} \rightarrow E`, where :math:`E = [\![ 0,...,p-1]\!]` is a process where :math:`\mathcal{D}=\mathbb{R}` discretized on the time grid :math:`(t_k)_{k \geq 0}` such that: .. math:: \begin{aligned} \forall k > 0,\: \mathbb{P} ( X_{t_k} \> | \> X_{t_0},...X_{t_{k-1}} ) = \mathbb{P} ( X_{t_k} \> | \> X_{t_{k-1}} ) \end{aligned} The transition matrix of the process :math:`\mathcal{M} = (m_{i,j})` can be defined such that: .. math:: \begin{aligned} \forall t_k \in \mathcal{D}, \forall i,j < p , \> m_{i+1,j+1} = \mathbb{P} (X_{t_{k+1}} = j \> | \> X_{t_{k}} = i) \end{aligned} The library proposes to model it through the object *DiscreteMarkovChain* defined thanks to the origin :math:`X_{t_0}` (which can be either deterministic or uncertain), the transition matrix :math:`\mathcal{M}` and the time grid. .. GENERATED FROM PYTHON SOURCE LINES 29-35 .. code-block:: Python import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) .. GENERATED FROM PYTHON SOURCE LINES 36-37 Define the origin .. GENERATED FROM PYTHON SOURCE LINES 37-39 .. code-block:: Python origin = ot.Dirac(0.0) .. GENERATED FROM PYTHON SOURCE LINES 40-41 Define the transition matrix .. GENERATED FROM PYTHON SOURCE LINES 41-43 .. code-block:: Python transition = ot.SquareMatrix([[0.1, 0.3, 0.6], [0.7, 0.1, 0.2], [0.5, 0.3, 0.2]]) .. GENERATED FROM PYTHON SOURCE LINES 44-45 Define an 1-d mesh .. GENERATED FROM PYTHON SOURCE LINES 45-47 .. code-block:: Python tgrid = ot.RegularGrid(0.0, 1.0, 50) .. GENERATED FROM PYTHON SOURCE LINES 48-49 Markov chain definition and realization .. GENERATED FROM PYTHON SOURCE LINES 49-55 .. code-block:: Python process = ot.DiscreteMarkovChain(origin, transition, tgrid) real = process.getRealization() graph = real.drawMarginal(0) graph.setTitle("Discrete Markov chain") view = viewer.View(graph) .. image-sg:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_discrete_markov_chain_process_001.png :alt: Discrete Markov chain :srcset: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_discrete_markov_chain_process_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 56-57 Get several realizations .. GENERATED FROM PYTHON SOURCE LINES 57-63 .. code-block:: Python process.setTimeGrid(ot.RegularGrid(0.0, 1.0, 20)) reals = process.getSample(3) graph = reals.drawMarginal(0) graph.setTitle("Discrete Markov chain, 3 realizations") view = viewer.View(graph) .. image-sg:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_discrete_markov_chain_process_002.png :alt: Discrete Markov chain, 3 realizations :srcset: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_discrete_markov_chain_process_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 64-65 Markov chain future 10 steps .. GENERATED FROM PYTHON SOURCE LINES 65-70 .. code-block:: Python future = process.getFuture(10) graph = future.drawMarginal(0) graph.setTitle("Markov chain future 10 steps") view = viewer.View(graph) .. image-sg:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_discrete_markov_chain_process_003.png :alt: Markov chain future 10 steps :srcset: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_discrete_markov_chain_process_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 71-72 Markov chain 3 different futures .. GENERATED FROM PYTHON SOURCE LINES 72-77 .. code-block:: Python futures = process.getFuture(10, 3) graph = futures.drawMarginal(0) graph.setTitle("Three Markov chain futures, 10 steps") view = viewer.View(graph) plt.show() .. image-sg:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_discrete_markov_chain_process_004.png :alt: Three Markov chain futures, 10 steps :srcset: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_discrete_markov_chain_process_004.png :class: sphx-glr-single-img .. _sphx_glr_download_auto_probabilistic_modeling_stochastic_processes_plot_discrete_markov_chain_process.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_discrete_markov_chain_process.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_discrete_markov_chain_process.py `