.. 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_propagate_gpr_ishigami.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_propagate_gpr_ishigami.py>`
        to download the full example code.

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

.. _sphx_glr_auto_meta_modeling_kriging_metamodel_plot_propagate_gpr_ishigami.py:


Gaussian Process Regression: propagate uncertainties
====================================================

In this example we propagate uncertainties through a GP metamodel of the :ref:`Ishigami model<use-case-ishigami>`.

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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 first build the metamodel and then compute its mean with a Monte-Carlo
computation.

We load the Ishigami model from the usecases module:

.. GENERATED FROM PYTHON SOURCE LINES 18-23

.. code-block:: Python


    from openturns.usecases import ishigami_function

    im = ishigami_function.IshigamiModel()








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We build a design of experiments with a Latin Hypercube Sampling (LHS) for the three input variables
supposed independent.

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.. code-block:: Python

    experiment = ot.LHSExperiment(im.inputDistribution, 30, False, True)
    xdata = experiment.generate()








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We get the exact model and evaluate it at the input training data `xdata` to
build the output data `ydata`.

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.. code-block:: Python

    model = im.model
    ydata = model(xdata)








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We define our GP process:

 - a constant basis in :math:`\mathbb{R}^3` ;
 - a squared exponential covariance function.


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.. code-block:: Python

    dimension = 3
    basis = ot.ConstantBasisFactory(dimension).build()
    covarianceModel = ot.SquaredExponential([0.1] * dimension, [1.0])
    fitter = otexp.GaussianProcessFitter(xdata, ydata, covarianceModel, basis)
    fitter.run()
    fitter_result = fitter.getResult()
    algo = otexp.GaussianProcessRegression(fitter_result)
    algo.run()
    result = algo.getResult()








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We finally get the metamodel to use with Monte-Carlo.

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.. code-block:: Python

    metamodel = result.getMetaModel()








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We want to estmate the mean of the Ishigami model with Monte-Carlo using the
metamodel instead of the exact model.

We first create a random vector following the input distribution :

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.. code-block:: Python

    X = ot.RandomVector(im.inputDistribution)








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And then we create a random vector from the image of the input random vector
by the metamodel :

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.. code-block:: Python

    Y = ot.CompositeRandomVector(metamodel, X)








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We now set our :class:`~openturns.ExpectationSimulationAlgorithm` object :

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.. code-block:: Python

    algo = ot.ExpectationSimulationAlgorithm(Y)
    algo.setMaximumOuterSampling(50000)
    algo.setBlockSize(1)
    algo.setCoefficientOfVariationCriterionType("NONE")








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We run it and store the results :

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.. code-block:: Python

    algo.run()
    result = algo.getResult()








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The expectation ( :math:`\mathbb{E}(Y)` mean ) is obtained with :

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.. code-block:: Python

    expectation = result.getExpectationEstimate()








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The mean estimate of the metamodel is

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.. code-block:: Python

    print("Mean of the Ishigami metamodel : %.3e" % expectation[0])





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

 .. code-block:: none

    Mean of the Ishigami metamodel : 3.532e+00




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We draw the convergence history.

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.. code-block:: Python

    graph = algo.drawExpectationConvergence()
    view = otv.View(graph)




.. image-sg:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_propagate_gpr_ishigami_001.svg
   :alt: Expectation convergence graph at level 0.95
   :srcset: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_propagate_gpr_ishigami_001.svg
   :class: sphx-glr-single-img





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For reference, the exact mean of the Ishigami model is :

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.. code-block:: Python

    print("Mean of the Ishigami model : %.3e" % im.expectation)





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

 .. code-block:: none

    Mean of the Ishigami model : 3.500e+00




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.. code-block:: Python

    otv.View.ShowAll()








.. _sphx_glr_download_auto_meta_modeling_kriging_metamodel_plot_propagate_gpr_ishigami.py:

.. only:: html

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

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

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

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

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

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

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