.. only:: html

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

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

    .. _sphx_glr_auto_meta_modeling_kriging_metamodel_plot_kriging.py:


Create a metamodel using Kriging
================================

In this example we are going to create an approximation of a model response using a kriging model. We consider a bidimensional function with gaussian inputs. Then we create a kriging metamodel with a constant basis and a `SquaredExponential` covariance. 

We consider the function 

.. math::
   g(X) = \cos(X_1 + X_2)


for any :math:`\mathbf{X}\in\mathbb{R}^2`. 
We assume that :math:`X_1` and :math:`X_2` have a gaussian distribution :

.. math::
   X_1 \sim \mathcal{N}(0,1) \textrm{ and } X_2 \sim \mathcal{N}(0,1).



.. code-block:: default

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








We define the model. 


.. code-block:: default

    dimension = 2
    input_names = ['x1', 'x2']
    formulas = ['cos(x1 + x2)']
    model = ot.SymbolicFunction(input_names, formulas)








We generate a simple Monte-Carlo input sample and evaluate the corresponding output sample.


.. code-block:: default

    distribution = ot.Normal(dimension)
    samplesize = 15
    x  = distribution.getSample(samplesize)
    y = model(x)








Then we create a kriging metamodel, using a constant trend and a squared exponential covariance model. 


.. code-block:: default

    basis = ot.ConstantBasisFactory(dimension).build()
    covarianceModel = ot.SquaredExponential([0.1]*dimension, [1.0])
    algo = ot.KrigingAlgorithm(x, y, covarianceModel, basis)
    algo.run()
    result = algo.getResult()
    metamodel = result.getMetaModel()








It is not so easy to visualize a bidimensional function. In order to simplify the graphics, we consider the value of the function at the input :math:`x_{1,ref}=0.5`. This amounts to create a `ParametricFunction` where the first variable :math:`x_1` (at input index 0) is set to :math:`0.5`.


.. code-block:: default

    x1ref = 0.5
    metamodelAtXref = ot.ParametricFunction(metamodel, [0], [x1ref])
    modelAtXref = ot.ParametricFunction(model, [0], [x1ref])








For this given value of :math:`x_1`, we plot the model and the metamodel with :math:`x_2` from its 1% up to its 99% quantile. We configure the X title to "X2" because the default setting would state that this axis is the first value of the parametric function, which default name is "X0".


.. code-block:: default

    x2min = ot.Normal().computeQuantile(0.01)[0]
    x2max = ot.Normal().computeQuantile(0.99)[0]
    graph = metamodelAtXref.draw(x2min, x2max)
    graph.setLegends(["Kriging"])
    curve = modelAtXref.draw(x2min, x2max)
    curve.setLegends(["Model"])
    curve.setColors(['red'])
    graph.add(curve)
    graph.setLegendPosition("topright")
    graph.setTitle("Sample size = %d" % (samplesize))
    graph.setXTitle("X2")
    view = viewer.View(graph)
    plt.show()




.. image:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_001.png
    :alt: Sample size = 15
    :class: sphx-glr-single-img





As we can see, the metamodel is quite accurate in this case. The metamodel is very close to the model in the center of the domain, where the density of the input distribution is highest.


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

   **Total running time of the script:** ( 0 minutes  0.086 seconds)


.. _sphx_glr_download_auto_meta_modeling_kriging_metamodel_plot_kriging.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_kriging.py <plot_kriging.py>`



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

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


.. only:: html

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

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