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

.. only:: html

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

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

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

.. _sphx_glr_auto_functional_modeling_vectorial_functions_plot_functions_inputDim.py:


==========================================
Increase the input dimension of a function
==========================================

.. GENERATED FROM PYTHON SOURCE LINES 7-22

Description
===========

We want to build a function :math:`f : \mathbb{R}^d \mapsto \mathbb{R}` from
*p* functions :math:`f_i: \mathbb{R} \mapsto \mathbb{R}`.

We can do that:

- Case 1: using the tensor product of the functions :math:`f_i`,
- Case 2: by addition of the functions :math:`f_i`.

We need to implement both basic steps:

- Step 1: creation of the projection function: :math:`\pi_i : (x_1, \dots, x_d) \mapsto x_i`,
- Step 2: creation of the composed function: :math:`g_i = f_i \circ \pi_i : (x_1, \dots, x_d) \mapsto f_i(x_i)`.

.. GENERATED FROM PYTHON SOURCE LINES 25-38

Step 1: Creation of the  projection function
============================================

The projection function is defined by:

.. math::

     \pi_i : (x_1, \dots, x_d) \mapsto x_i

We can do that using:

- a :class:`~openturns.SymbolicFunction` class,
- a :class:`~openturns.LinearFunction` class.

.. GENERATED FROM PYTHON SOURCE LINES 40-41

**Method 1**: We use the :class:`~openturns.SymbolicFunction` class.

.. GENERATED FROM PYTHON SOURCE LINES 41-60

.. code-block:: Python

    import openturns as ot


    def buidProjSymbolic(p, i):
        # R^p --> R
        # (x1, ..., xp) --> xi
        inputVar = ot.Description.BuildDefault(p, "x")
        return ot.SymbolicFunction(inputVar, [inputVar[i]])


    d = 2
    all_projections = [buidProjSymbolic(d, i) for i in range(d)]
    print(
        "Input dimension = ",
        all_projections[0].getInputDimension(),
        "Output dimension = ",
        all_projections[0].getOutputDimension(),
    )





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

 .. code-block:: none

    Input dimension =  2 Output dimension =  1




.. GENERATED FROM PYTHON SOURCE LINES 61-64

**Method 2**: We use the :class:`~openturns.LinearFunction` class.

The function :math:`\pi_i(\vect{x}) = \mat{A}\left(\vect{x}-\vect{c}\right) + \vect{b}`.

.. GENERATED FROM PYTHON SOURCE LINES 64-78

.. code-block:: Python



    def buildProjLinear(d, i):
        # R^d --> R
        # (x1, ..., xd) --> xi
        matA = ot.Matrix(1, d)
        matA[0, i] = 1.0
        cVect = [0.0] * d
        bVect = [0.0]
        return ot.LinearFunction(cVect, bVect, matA)


    all_projections = [buildProjLinear(d, i) for i in range(d)]








.. GENERATED FROM PYTHON SOURCE LINES 79-89

Step 2: Creation of the  composed function
==========================================

The composed function is defined by: :math:`g_i = f_i \circ \pi_i` defined by:

.. math::

     g_i: (x_1, \dots, x_d) \mapsto f_i(x_i)

We use the :class:`~openturns.ComposedFunction` class.

.. GENERATED FROM PYTHON SOURCE LINES 89-96

.. code-block:: Python


    f1 = ot.SymbolicFunction(["x1"], ["x1^2"])
    f2 = ot.SymbolicFunction(["x2"], ["3*x2"])
    fi_list = [f1, f2]
    all_g = [ot.ComposedFunction(f, proj) for (f, proj) in zip(fi_list, all_projections)]
    print(all_g[0].getInputDimension(), all_g[0].getOutputDimension())





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

 .. code-block:: none

    2 1




.. GENERATED FROM PYTHON SOURCE LINES 97-108

Case 1: Tensor product
======================

We want to build the function :math:`f : \mathbb{R}^d \mapsto \mathbb{R}` defined by:

.. math::

    f: (x_1, \dots, x_d) \mapsto \prod_{i=1}^d f_i(x_i)

As the operator :math:`*` can only be applied to functions sharing the same input space, we need to
use the projection function :math:`\pi_i` and the functions :math:`g_i` all defined on :math:`\mathbb{R}^d`.

.. GENERATED FROM PYTHON SOURCE LINES 108-122

.. code-block:: Python



    def tensorProduct(factors):
        prod = factors[0]
        for i in range(1, len(factors)):
            prod = prod * factors[i]
        return prod


    f = tensorProduct(all_g)
    print("input dimension =", f.getInputDimension())
    print("output dimension =", f.getOutputDimension())
    print("f([1.0, 2.0]) = ", f([1.0, 2.0]))





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

 .. code-block:: none

    input dimension = 2
    output dimension = 1
    f([1.0, 2.0]) =  [6]




.. GENERATED FROM PYTHON SOURCE LINES 123-133

Case 2: Sum
===========

We want to build the function :math:`f : \mathbb{R}^d \mapsto \mathbb{R}` defined by:

.. math::

    f: (x_1, \dots, x_d) \mapsto \sum_{i=1}^d f_i(x_i)

We use the :class:`~openturns.LinearCombinationFunction` class.

.. GENERATED FROM PYTHON SOURCE LINES 133-138

.. code-block:: Python


    coef = [1.0, 1.0]
    f = ot.LinearCombinationFunction(all_g, [1.0] * len(all_g))
    print("input dimension =", f.getInputDimension())
    print("output dimension =", f.getOutputDimension())




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

 .. code-block:: none

    input dimension = 2
    output dimension = 1





.. _sphx_glr_download_auto_functional_modeling_vectorial_functions_plot_functions_inputDim.py:

.. only:: html

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

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

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

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

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