.. 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_user_stationary_covmodel.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_probabilistic_modeling_stochastic_processes_plot_user_stationary_covmodel.py>`
        to download the full example code

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

.. _sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_user_stationary_covmodel.py:


Create a stationary covariance model
====================================

.. GENERATED FROM PYTHON SOURCE LINES 6-18

This use case illustrates how the user can define his own stationary covariance model thanks to the object *UserDefinedStationaryCovarianceModel* defined from:

- a mesh :math:`\mathcal{M}` of dimension :math:`n` defined by the vertices :math:`(\underline{\tau}_0,\dots, \underline{\tau}_{N-1})` and the associated simplices,
- a collection of covariance matrices stored in the object *CovarianceMatrixCollection* noted :math:`\underline{\underline{C}}_0, \dots, \underline{\underline{C}}_{N-1}`
  where :math:`\underline{\underline{C}}_k` :math:`\in \mathcal{M}_{d \times d}(\mathbb{R})` for :math:`0 \leq k \leq N-1`

Then we build a stationary covariance function which is a piecewise constant function on :math:`\mathcal{D}` defined by:

.. math::
   \forall \underline{\tau} \in \mathcal{D}, \, C^{stat}(\underline{\tau}) =  \underline{\underline{C}}_k

where :math:`k`  is such that :math:`\underline{\tau}_k` is the  vertex of :math:`\mathcal{M}` the nearest to :math:`\underline{t}.`

.. GENERATED FROM PYTHON SOURCE LINES 20-26

.. 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 27-29

We detail the example described in the documentation
Create the time grid

.. GENERATED FROM PYTHON SOURCE LINES 29-48

.. code-block:: Python

    t0 = 0.0
    dt = 0.5
    N = int((20.0 - t0) / dt)
    mesh = ot.RegularGrid(t0, dt, N)

    # Create the covariance function


    def gamma(tau):
        return 1.0 / (1.0 + tau * tau)


    # Create the collection of HermitianMatrix
    coll = ot.SquareMatrixCollection()
    for k in range(N):
        t = mesh.getValue(k)
        matrix = ot.SquareMatrix([[gamma(t)]])
        coll.add(matrix)








.. GENERATED FROM PYTHON SOURCE LINES 49-50

Create the covariance model

.. GENERATED FROM PYTHON SOURCE LINES 50-58

.. code-block:: Python

    covmodel = ot.UserDefinedStationaryCovarianceModel(mesh, coll)

    # One vertex of the mesh
    tau = 1.5

    # Get the covariance function computed at the vertex tau
    covmodel(tau)






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <p>[[ 0.307692 ]]</p>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 59-60

Graph of the spectral function

.. GENERATED FROM PYTHON SOURCE LINES 60-76

.. code-block:: Python

    x = ot.Sample(N, 2)
    for k in range(N):
        t = mesh.getValue(k)
        x[k, 0] = t
        value = covmodel(t)
        x[k, 1] = value[0, 0]

    # Create the curve of the spectral function
    curve = ot.Curve(x, "User Model")

    # Create the graph
    myGraph = ot.Graph("User covariance model", "Time", "Covariance function", True)
    myGraph.add(curve)
    myGraph.setLegendPosition("upper right")
    view = viewer.View(myGraph)
    plt.show()



.. image-sg:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_user_stationary_covmodel_001.png
   :alt: User covariance model
   :srcset: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_user_stationary_covmodel_001.png
   :class: sphx-glr-single-img






.. _sphx_glr_download_auto_probabilistic_modeling_stochastic_processes_plot_user_stationary_covmodel.py:

.. only:: html

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

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

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

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

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