.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_probabilistic_modeling_stochastic_processes_plot_create_stationary_covmodel.py>`     to download the full example code
    .. rst-class:: sphx-glr-example-title

    .. _sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_create_stationary_covmodel.py:


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

This use case illustrates how the User can create a covariance function from parametric models. The library implements the *multivariate Exponential model* as a parametric model for the covariance function where the spatial covariance function :math:`\rho` writes:

.. math::
   \rho(\underline{s}, \underline{t} ) = e^{-\left\| \underline{s}- \underline{t} \right\|_2} \quad \forall (\underline{s}, \underline{t}) \in \mathcal{D}

It is possible to define the exponential model from the spatial covariance matrix :math:`\underline{\underline{C}}^{spat}` rather than the correlation matrix :math:`\underline{\underline{R}}`:

.. math::
    \forall \underline{t} \in \mathcal{D},\quad \underline{\underline{C}}^{spat} = \mathbb{E} \left[ X_{\underline{t}} X^t_{\underline{t}} \right] = \underline{\underline{A}}\,\underline{\underline{R}}\, \underline{\underline{A}}

with:

.. math::
   \underline{\underline{A}} = \mbox{Diag}(a_1, \dots, a_d)

We call :math:`\underline{a}` the amplitude vector and :math:`\underline{\lambda}` the scale vector.

The library implements the multivariate exponential model thanks to the object *ExponentialModel* which is created from:


- the scale and amplitude vectors :math:`(\underline{\lambda}, \underline{a})`: in that case, by default :math:`\underline{\underline{R}} = \underline{\underline{I}}`;
- the scale and amplitude vectors and the spatial correlation matrix  :math:`(\underline{\lambda}, \underline{a},\underline{\underline{R}})`;
- the scale and amplitude vectors and the spatial covariance matrix  :math:`(\underline{\lambda}, \underline{a},\underline{\underline{C}})`; Then :math:`\underline{\underline{C}}` is mapped into the associated correlation matrix :math:`\underline{\underline{R}}` and the previous constructor is used.



.. code-block:: default

    from __future__ import print_function
    import openturns as ot
    import openturns.viewer as viewer
    from matplotlib import pylab as plt
    import math as m
    ot.Log.Show(ot.Log.NONE)








Create the amplitude vector (output dimension 3)


.. code-block:: default

    amplitude = [1.0, 2.0, 3.0]

    # Scale vector (input dimension 1)
    scale = [4.0]

    # spatialCorrelation
    spatialCorrelation = ot.CorrelationMatrix(3)
    spatialCorrelation[0, 1] = 0.8
    spatialCorrelation[0, 2] = 0.6
    spatialCorrelation[1, 2] = 0.1

    # spatialCovariance
    spatialCovariance = ot.CovarianceMatrix(3)
    spatialCovariance[0, 0] = 4.0
    spatialCovariance[1, 1] = 5.0
    spatialCovariance[2, 2] = 6.0
    spatialCovariance[0, 1] = 1.2
    spatialCovariance[0, 2] = 0.9
    spatialCovariance[1, 2] = -0.2








Create the covariance model from the amplitude and scale, no spatial correlation


.. code-block:: default

    ot.ExponentialModel(scale, amplitude)






.. raw:: html

    <p>ExponentialModel(scale=[4], amplitude=[1,2,3], no spatial correlation)</p>
    <br />
    <br />

or from the amplitude, scale and spatialCovariance


.. code-block:: default

    ot.ExponentialModel(scale, amplitude, spatialCorrelation)






.. raw:: html

    <p>ExponentialModel(scale=[4], amplitude=[1,2,3], spatial correlation=<br>
    [[ 1   0.8 0.6 ]<br>
     [ 0.8 1   0.1 ]<br>
     [ 0.6 0.1 1   ]])</p>
    <br />
    <br />

or from the scale and spatialCovariance


.. code-block:: default

    ot.ExponentialModel(scale, spatialCovariance)





.. raw:: html

    <p>ExponentialModel(scale=[4], amplitude=[2,2.23607,2.44949], no spatial correlation)</p>
    <br />
    <br />


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

   **Total running time of the script:** ( 0 minutes  0.002 seconds)


.. _sphx_glr_download_auto_probabilistic_modeling_stochastic_processes_plot_create_stationary_covmodel.py:


.. only :: html

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



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

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



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

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


.. only:: html

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

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