.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_stochastic_processes/plot_covariance_model_nugget.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

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

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

.. _sphx_glr_auto_stochastic_processes_plot_covariance_model_nugget.py:


Create a covariance model with and without a nugget effect
==========================================================

.. GENERATED FROM PYTHON SOURCE LINES 7-19

This example shows the impact of a nugget effect on the covariance function of a process:

- we create two covariance functions: the second one adds a nugget effect to the first one,
- we draw the covariance functions.

The nugget effect allows us to model a noise observed in the output values of a process. This noise may be, for
example, a measurement noise from a sensor with finite precision.

The noise is characterized by the scalar
:math:`\varepsilon_{nugget}` which is added to the correlation function evaluated on each :math:`(\vect{s}, \vect{s})` where
:math:`\vect{s} \in \cD`. See :ref:`Covariance models <covariance_model>` to get more details on covariance models and
the introduction of a nugget factor, and in particular see equation :eq:`Css`.

.. GENERATED FROM PYTHON SOURCE LINES 21-26

.. code-block:: Python

    import openturns as ot
    import openturns.viewer as otv

    # sphinx_gallery_thumbnail_number = 1








.. GENERATED FROM PYTHON SOURCE LINES 27-32

First we create the covariance function of a process defined on :math:`\cD \in \Rset` and with scalar
outputs. We use the :class:`~openturns.SquaredExponential` with unit scale, unit amplitude. By default, the
nugget factor is equal to :math:`10^{-12}` (defined through the `CovarianceModel-DefaultNuggetFactor`
key in the class :class:`~openturns.ResourceMap`) which allows the regularization of the covariance
matrix. We set it to zero.

.. GENERATED FROM PYTHON SOURCE LINES 32-35

.. code-block:: Python

    cov_model = ot.SquaredExponential()
    cov_model.setNuggetFactor(0.0)








.. GENERATED FROM PYTHON SOURCE LINES 36-38

Then, we create a second covariance function which adds a nugget factor to the first one with.
We take the nugget factor :math:`\varepsilon_{nugget} = 1`.

.. GENERATED FROM PYTHON SOURCE LINES 38-42

.. code-block:: Python

    cov_model_with_nugget = ot.SquaredExponential()
    epsilon_nugget = 1
    cov_model_with_nugget.setNuggetFactor(epsilon_nugget)








.. GENERATED FROM PYTHON SOURCE LINES 43-44

We draw the covariance function with and without the nugget effect.

.. GENERATED FROM PYTHON SOURCE LINES 44-58

.. code-block:: Python

    g = cov_model.draw()
    g.setBoundingBox(ot.Interval([-5, 0.0], [5.0, 2.0]))
    g.setTitle("Without nugget effect")
    g_nugget = cov_model_with_nugget.draw()
    g_nugget.setBoundingBox(ot.Interval([-5, 0.0], [5.0, 2.0]))
    g_nugget.setTitle("With nugget effect")

    grid = ot.GridLayout(1, 2)
    grid.setGraph(0, 0, g)
    grid.setGraph(0, 1, g_nugget)
    view = otv.View(grid)
    otv.View.ShowAll()





.. image-sg:: /auto_stochastic_processes/images/sphx_glr_plot_covariance_model_nugget_001.svg
   :alt: , Without nugget effect, With nugget effect
   :srcset: /auto_stochastic_processes/images/sphx_glr_plot_covariance_model_nugget_001.svg
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 59-60

Display all figures.

.. GENERATED FROM PYTHON SOURCE LINES 60-61

.. code-block:: Python

    otv.View.ShowAll()








.. _sphx_glr_download_auto_stochastic_processes_plot_covariance_model_nugget.py:

.. only:: html

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

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

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

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

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

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

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