.. 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_create_linear_least_squares_model.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_general_purpose_metamodels_plot_create_linear_least_squares_model.py>`
        to download the full example code

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

.. _sphx_glr_auto_meta_modeling_general_purpose_metamodels_plot_create_linear_least_squares_model.py:


Create a linear least squares model
===================================

.. GENERATED FROM PYTHON SOURCE LINES 6-51

In this example we are going to create a global approximation of a model
response based on a linear function using the :class:`~openturns.LinearLeastSquares` class.

We consider the function :math:`h : \Rset^2 \rightarrow \Rset^2` is defined by:

.. math::
   h(x_1, x_2) = \left(\cos(x_1 + x_2), (x_2 + 1) e^{x_1 - 2 x_2} \right)

for any :math:`\vect{x} \in \Rset^2`.
Since the output is a dimension 2 vector, the model has vector
coefficients.
We use the linear model:

.. math::
    \vect{y} \, \approx \, \widehat{\vect{h}}(\vect{x}) \,
    = \, \sum_{k=0}^2 \; \vect{a}_k \; \psi_k(\vect{x})


for any :math:`\vect{x} \in \Rset^2` where :math:`\left\{\vect{a}_k \in \Rset^2\right\}_{k = 0,..., 2}`
are the vector coefficients and :math:`\left\{\psi_k : \Rset^2 \rightarrow \Rset\right\}_{k = 0, ..., 2}`
are the basis functions.
This implies that each marginal output :math:`\widehat{h}_i(\vect{x})`
is approximated by the linear model:

.. math::
    \widehat{h}_i(\vect{x}) \,
    = \, \sum_{k=0}^2 \; a_{ki} \; \psi_k(\vect{x})

for any :math:`\vect{x} \in \Rset^2` and any :math:`i = 1, 2`
where :math:`a_{ki}` is the :math:`k`-th coefficient of the :math:`i`-th output
marginal:

.. math::
     \vect{a}_k = \begin{pmatrix}a_{k1} \\ a_{k2} \end{pmatrix}

for :math:`k = 0, 1, 2`.
We consider the basis functions:

.. math::

     \psi_0(\vect{x}) & = 1 \\
     \psi_1(\vect{x}) & = x_1 \\
     \psi_2(\vect{x}) & = x_2 \\

for any :math:`\vect{x} \in \Rset^2`.

.. GENERATED FROM PYTHON SOURCE LINES 54-60

.. code-block:: Python

    import openturns as ot
    import openturns.viewer as viewer
    from matplotlib import pylab as plt

    ot.Log.Show(ot.Log.NONE)








.. GENERATED FROM PYTHON SOURCE LINES 61-63

Define the model
~~~~~~~~~~~~~~~~

.. GENERATED FROM PYTHON SOURCE LINES 65-67

Prepare an input sample.
Each point is a pair of coordinates :math:`(x_1, x_2)`.

.. GENERATED FROM PYTHON SOURCE LINES 69-75

.. code-block:: Python

    inputTrain = [[0.5, 0.5], [-0.5, -0.5], [-0.5, 0.5], [0.5, -0.5]]
    inputTrain += [[0.25, 0.25], [-0.25, -0.25], [-0.25, 0.25], [0.25, -0.25]]
    inputTrain = ot.Sample(inputTrain)
    inputTrain.setDescription(["x1", "x2"])
    inputTrain






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <table>
      <tr><td></td><th>x1</th><th>x2</th></tr>
      <tr><th>0</th><td>0.5</td><td>0.5</td></tr>
      <tr><th>1</th><td>-0.5</td><td>-0.5</td></tr>
      <tr><th>2</th><td>-0.5</td><td>0.5</td></tr>
      <tr><th>3</th><td>0.5</td><td>-0.5</td></tr>
      <tr><th>4</th><td>0.25</td><td>0.25</td></tr>
      <tr><th>5</th><td>-0.25</td><td>-0.25</td></tr>
      <tr><th>6</th><td>-0.25</td><td>0.25</td></tr>
      <tr><th>7</th><td>0.25</td><td>-0.25</td></tr>
    </table>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 76-77

Compute the output sample from the input sample and a function.

.. GENERATED FROM PYTHON SOURCE LINES 79-85

.. code-block:: Python

    formulas = ["cos(x1 + x2)", "(x2 + 1) * exp(x1 - 2 * x2)"]
    model = ot.SymbolicFunction(["x1", "x2"], formulas)
    model.setOutputDescription(["y1", "y2"])
    outputTrain = model(inputTrain)
    outputTrain






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <table>
      <tr><td></td><th>y1</th><th>y2</th></tr>
      <tr><th>0</th><td>0.5403023</td><td>0.909796</td></tr>
      <tr><th>1</th><td>0.5403023</td><td>0.8243606</td></tr>
      <tr><th>2</th><td>1</td><td>0.3346952</td></tr>
      <tr><th>3</th><td>1</td><td>2.240845</td></tr>
      <tr><th>4</th><td>0.8775826</td><td>0.973501</td></tr>
      <tr><th>5</th><td>0.8775826</td><td>0.9630191</td></tr>
      <tr><th>6</th><td>1</td><td>0.5904582</td></tr>
      <tr><th>7</th><td>1</td><td>1.58775</td></tr>
    </table>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 86-88

Linear least squares
~~~~~~~~~~~~~~~~~~~~

.. GENERATED FROM PYTHON SOURCE LINES 90-91

Create a linear least squares model.

.. GENERATED FROM PYTHON SOURCE LINES 93-96

.. code-block:: Python

    algo = ot.LinearLeastSquares(inputTrain, outputTrain)
    algo.run()








.. GENERATED FROM PYTHON SOURCE LINES 97-98

Get the linear term.

.. GENERATED FROM PYTHON SOURCE LINES 100-102

.. code-block:: Python

    algo.getLinear()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <p>[[  9.93014e-17  0.998189    ]<br>
     [  4.96507e-17 -0.925648    ]]</p>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 103-104

Get the constant term.

.. GENERATED FROM PYTHON SOURCE LINES 106-108

.. code-block:: Python

    algo.getConstant()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=2 values=[0.854471,1.05305]
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 109-110

Get the metamodel.

.. GENERATED FROM PYTHON SOURCE LINES 110-112

.. code-block:: Python

    responseSurface = algo.getMetaModel()








.. GENERATED FROM PYTHON SOURCE LINES 113-114

Plot the second output of our model with :math:`x_1=0.5`.

.. GENERATED FROM PYTHON SOURCE LINES 116-131

.. code-block:: Python

    graph = ot.ParametricFunction(model, [0], [0.5]).getMarginal(1).draw(-0.5, 0.5)
    graph.setLegends(["Model"])
    curve = (
        ot.ParametricFunction(responseSurface, [0], [0.5])
        .getMarginal(1)
        .draw(-0.5, 0.5)
        .getDrawable(0)
    )
    curve.setLineStyle("dashed")
    curve.setLegend("Linear L.S.")
    graph.add(curve)
    graph.setLegendPosition("upper right")
    graph.setColors(ot.Drawable.BuildDefaultPalette(2))
    view = viewer.View(graph)
    plt.show()



.. image-sg:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_create_linear_least_squares_model_001.png
   :alt: y2 as a function of x2
   :srcset: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_create_linear_least_squares_model_001.png
   :class: sphx-glr-single-img






.. _sphx_glr_download_auto_meta_modeling_general_purpose_metamodels_plot_create_linear_least_squares_model.py:

.. only:: html

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

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

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

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

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