.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_functional_modeling/functional_basis/plot_multidimensional_basis.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_functional_basis_plot_multidimensional_basis.py>`
        to download the full example code

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

.. _sphx_glr_auto_functional_modeling_functional_basis_plot_multidimensional_basis.py:


============================================================================
Create a multivariate basis of functions from scalar multivariable functions
============================================================================

.. GENERATED FROM PYTHON SOURCE LINES 7-53

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

We want to build a finite basis of functions :math:`\mathcal{B}=\{f_1,...,f_n\}` where each
:math:`f_i: \mathbb{R}^d \mapsto \mathbb{R}^q` with :math:`d>1, q>1`, from given scalar multivariable
functions :math:`g_i: \mathbb{R}^d \mapsto \mathbb{R}`.

In other words, we want to create a basis of functions having both a multivariate input *and* output from
functions having a multivariate input and a scalar output.

In this example, we do that by the aggregation of the given functions :math:`g_i`,
using the :class:`~openturns.AggregatedFunction` class. This class allows one to create a multivariate output
function from a list of functions having a univariate output dimension. Note that we can only aggregate functions
that share the same input space.

We illustrate two cases that differ from the way the functions :math:`g_i` have been created:

- Case 1: we suppose that the functions :math:`g_i` have no particular origin,
- Case 2: we suppose that the functions :math:`g_i` have been built from particular bases.

In both cases, we use the :class:`~openturns.Basis` class to create the basis and the
:class:`~openturns.AggregatedFunction` class to create a multivariate output function from a
list of scalar multivariable functions.

Case 1: Aggregation of given functions
======================================

In that case, we have some functions :math:`g_i: \mathbb{R}^d \mapsto \mathbb{R}` and we aggregate them
in order to build the basis :math:`\mathcal{B}`.

For example, we set :math:`d=2`, :math:`q=3` and we use the functions :math:`(g_1, g_2, g_3)` defined by:

.. math::
    \begin{array}{lcl}
      g_1(x_1, x_2) & = & 1\\
      g_2(x_1, x_2) & = & x_1 + x_2\\
      g_3(x_1, x_2) & = & x_1^2 + x_2^2
    \end{array}

and we want to build the finite basis :math:`\mathcal{B}=\{f_1,f_2\}` defined by:

.. math::
    \begin{array}{lcl}
      f_1(x_1, x_2) & = & (1, 1, 1)\\
      f_2(x_1, x_2) & = & (1, x_1 + x_2, x_1^2 + x_2^2)
    \end{array}

.. GENERATED FROM PYTHON SOURCE LINES 53-69

.. code-block:: Python

    import openturns as ot

    d = 2
    q = 3

    # Create the functions g_i
    g_1 = ot.SymbolicFunction(["x1", "x2"], ["1"])
    g_2 = ot.SymbolicFunction(["x1", "x2"], ["x1+x2"])
    g_3 = ot.SymbolicFunction(["x1", "x2"], ["x1^2+x2^2"])

    # Create the multivariate basis
    f_1 = ot.AggregatedFunction([g_1] * q)
    f_2 = ot.AggregatedFunction([g_1, g_2, g_3])
    basis = ot.Basis([f_1, f_2])
    print("Fonction f_1 : ", basis.build(0))





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

 .. code-block:: none

    Fonction f_1 :  [[x1,x2]->[1],[x1,x2]->[1],[x1,x2]->[1]]




.. GENERATED FROM PYTHON SOURCE LINES 70-93

Case 2: Aggregation of functions built from given bases
=======================================================

In that case, we use some functions :math:`g_i: \mathbb{R}^d \mapsto \mathbb{R}` that have been
built from particular bases. For example, they are members of polynomial families which are orthonormal
to given distributions.

In this example, we use both polynomial families:

- :math:`(g_1^i)_{i\geq 0}` which is a orthonormal multivariate polynomial basis with respect to the uniform
  distributipn on :math:`[-1,1]^d` (it is obtained through a tensor product of univariate Legendre polynomials
  orthonormal with respect to the uniform distribution on :math:`[-1,1]`),
- :math:`(g_2^i)_{i\geq 0}` which is a orthonormal multivariate polynomial basis with respect to the multivariate
  standard normal distribution (it is obtained through a tensor product of univariate Hermite polynomials
  orthonormal with respect to the standard normal univariate distribution).

We consider the linear
enumeration of the polynomial family. The functions :math:`f_i` are built as the aggregation of
the :math:`i`-th polynomials of the families :math:`(g_1^k)_k` and :math:`(g_2^k)_k`.
We only consider :math:`g_1^i` and :math:`g_2^i` of total degree less than 2:
then we have 6 polynomials :math:`f_i`.

We still set :math:`d=2`, :math:`q=3`.

.. GENERATED FROM PYTHON SOURCE LINES 93-107

.. code-block:: Python

    d = 2

    legendre_builder = ot.OrthogonalProductPolynomialFactory([ot.LegendreFactory()] * d)
    hermite_builder = ot.OrthogonalProductPolynomialFactory([ot.HermiteFactory()] * d)

    # Generate the multivariate functions f_i (quadratic)
    total_deg = 2
    basis_size = ot.LinearEnumerateFunction(d).getStrataCumulatedCardinal(total_deg)
    basis = []
    for i in range(basis_size):
        f_i = ot.AggregatedFunction([legendre_builder.build(i), hermite_builder.build(i)])
        basis.append(f_i)

    basis = ot.Basis(basis)








.. _sphx_glr_download_auto_functional_modeling_functional_basis_plot_multidimensional_basis.py:

.. only:: html

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

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

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

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

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