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

.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_meta_modeling_general_purpose_metamodels_plot_taylor_approximation.py>`
        to download the full example code

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

.. _sphx_glr_auto_meta_modeling_general_purpose_metamodels_plot_taylor_approximation.py:


Taylor approximations
=====================

.. GENERATED FROM PYTHON SOURCE LINES 6-18

In this example we are going to build a local approximation of a model using the Taylor decomposition:

Here is the decomposition at the first order:

.. math::
   \vect{y} \, \approx \, \widehat{h}(\vect{x}) \,
         = \, h(\vect{x}_0) \, +
        \, \sum_{i=1}^{n_{X}} \; \frac{\partial h}{\partial x_i}(\vect{x}_0).\left(x_i - x_{0,i} \right)


Here :math:`h(x) = [cos(x_1 + x_2), (x_2 + 1) e^{x_1 - 2 x_2}]`.


.. GENERATED FROM PYTHON SOURCE LINES 20-36

.. code-block:: default

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

    # Prepare some data.
    formulas = ['cos(x1 + x2)', '(x2 + 1) * exp(x1 - 2 * x2)']
    model = ot.SymbolicFunction(['x1', 'x2'], formulas)

    # Center of the approximation.
    x0 = [-0.4, -0.4]

    # Drawing bounds.
    a = -0.4
    b = 0.0








.. GENERATED FROM PYTHON SOURCE LINES 37-38

Create a linear (first-order) Taylor approximation.

.. GENERATED FROM PYTHON SOURCE LINES 38-42

.. code-block:: default

    algo = ot.LinearTaylor(x0, model)
    algo.run()
    responseSurface = algo.getMetaModel()








.. GENERATED FROM PYTHON SOURCE LINES 43-44

Plot the second output of our model with :math:`x_1=x_{0,1}`.

.. GENERATED FROM PYTHON SOURCE LINES 44-55

.. code-block:: default

    graph = ot.ParametricFunction(
        responseSurface, [0], [x0[1]]).getMarginal(1).draw(a, b)
    graph.setLegends(['taylor'])
    curve = ot.ParametricFunction(model, [0], [x0[1]]).getMarginal(
        1).draw(a, b).getDrawable(0)
    curve.setColor('red')
    curve.setLegend('model')
    graph.add(curve)
    graph.setLegendPosition('topright')
    view = viewer.View(graph)




.. image-sg:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_taylor_approximation_001.png
   :alt: y1 as a function of x2
   :srcset: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_taylor_approximation_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 56-63

Here is the decomposition at the second order:

.. math::\vect{y} \, \approx \, \widehat{h}(\vect{x}) \, = \,
        h(\vect{x}_0) \, + \, \sum_{i=1}^{n_{X}} \;
      \frac{\partial h}{\partial x_i}(\vect{x}_0).\left(x_i - x_{0,i} \right) \, +
     \, \frac{1}{2} \; \sum_{i,j=1}^{n_X} \;
      \frac{\partial^2 h}{\partial x_i \partial x_j}(\vect{x}_0).\left(x_i - x_{0,i} \right).\left(x_j - x_{0,j} \right):math:``

.. GENERATED FROM PYTHON SOURCE LINES 65-66

Create a quadratic (second-order) Taylor approximation.

.. GENERATED FROM PYTHON SOURCE LINES 66-70

.. code-block:: default

    algo = ot.QuadraticTaylor(x0, model)
    algo.run()
    responseSurface = algo.getMetaModel()








.. GENERATED FROM PYTHON SOURCE LINES 71-72

Plot second output of our model with :math:`x_1=x_{0,1}`.

.. GENERATED FROM PYTHON SOURCE LINES 72-83

.. code-block:: default

    graph = ot.ParametricFunction(
        responseSurface, [0], [x0[1]]).getMarginal(1).draw(a, b)
    graph.setLegends(['taylor'])
    curve = ot.ParametricFunction(model, [0], [x0[1]]).getMarginal(
        1).draw(a, b).getDrawable(0)
    curve.setColor('red')
    curve.setLegend('model')
    graph.add(curve)
    graph.setLegendPosition('topright')
    view = viewer.View(graph)
    plt.show()



.. image-sg:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_taylor_approximation_002.png
   :alt: y1 as a function of x2
   :srcset: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_taylor_approximation_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** ( 0 minutes  0.143 seconds)


.. _sphx_glr_download_auto_meta_modeling_general_purpose_metamodels_plot_taylor_approximation.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_taylor_approximation.py <plot_taylor_approximation.py>`



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


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