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

.. only:: html

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

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

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

.. _sphx_glr_auto_meta_modeling_kriging_metamodel_plot_gpr_likelihood.py:


Gaussian process regression: draw the likelihood
================================================

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Abstract
--------

In this short example we draw the log-likelihood as a function of the scale
parameter of a covariance model.

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.. code-block:: Python

    import openturns as ot
    import openturns.experimental as otexp
    import openturns.viewer as otv









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We define the exact model with a :class:`~openturns.SymbolicFunction` :

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.. code-block:: Python

    f = ot.SymbolicFunction(["x"], ["x*sin(x)"])








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We use the following input and output training samples :

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.. code-block:: Python

    inputSample = ot.Sample([[1.0], [3.0], [5.0], [6.0], [7.0], [8.0]])
    outputSample = f(inputSample)








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We choose a constant basis for the trend of the metamodel :

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.. code-block:: Python

    basis = ot.ConstantBasisFactory().build()
    covarianceModel = ot.SquaredExponential(1)








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For the covariance model, we use a Matern model with :math:`\nu = 1.5` :

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.. code-block:: Python

    covarianceModel = ot.MaternModel([1.0], 1.5)








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We are now ready to fit the Gaussian Process parameters and store the result :

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.. code-block:: Python

    fitter = otexp.GaussianProcessFitter(inputSample, outputSample, covarianceModel, basis)
    fitter.run()
    fitter_result = fitter.getResult()









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We can retrieve the covariance model from the result object and then access
the scale of the model :

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.. code-block:: Python

    theta = fitter_result.getCovarianceModel().getScale()
    print("Scale of the covariance model : %.3e" % theta[0])





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

 .. code-block:: none

    Scale of the covariance model : 1.978e+00




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This hyperparameter is calibrated thanks to a maximization of the log-likelihood. We get this log-likehood as a function of :math:`\theta` :

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.. code-block:: Python

    ot.ResourceMap.SetAsBool("GaussianProcessFitter-UseAnalyticalAmplitudeEstimate", True)
    reducedLogLikelihoodFunction = fitter.getReducedLogLikelihoodFunction()








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We draw the reduced log-likelihood :math:`\mathcal{L}(\theta)` as a function
of the parameter :math:`\theta`.

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.. code-block:: Python

    graph = reducedLogLikelihoodFunction.draw(0.1, 10.0, 100)
    graph.setXTitle(r"$\theta$")
    graph.setYTitle(r"$\mathcal{L}(\theta)$")
    graph.setTitle(r"Log-likelihood as a function of $\theta$")








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We represent the estimated parameter as a point on the log-likelihood curve :

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.. code-block:: Python

    L_theta = reducedLogLikelihoodFunction(theta)
    cloud = ot.Cloud(theta, L_theta)
    cloud.setColor("red")
    cloud.setPointStyle("fsquare")
    graph.add(cloud)
    graph.setLegends([r"Matern $\nu = 1.5$", r"$\theta$ estimate"])








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We verify on the previous graph that the estimate of :math:`\theta` maximizes
the log-likelihood.

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Display figures

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.. code-block:: Python

    view = otv.View(graph)
    otv.View.ShowAll()



.. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_gpr_likelihood_001.svg
   :alt: Log-likelihood as a function of $\theta$
   :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_gpr_likelihood_001.svg
   :class: sphx-glr-single-img






.. _sphx_glr_download_auto_meta_modeling_kriging_metamodel_plot_gpr_likelihood.py:

.. only:: html

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

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

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

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

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

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

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