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

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_sensitivity_generalised_sobol_generalised_sobol_multioutput.py>`
        to download the full example code or to run this example in your browser via Binder

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

.. _sphx_glr_auto_examples_sensitivity_generalised_sobol_generalised_sobol_multioutput.py:


Toy multioutput function
==============================================

In this example, we demonstrate the computation of the Generalised Sobol indices using 
the toy example in [1]_.

.. math::
    Y = f (X_{1}, X_{2}) := \left(\begin{array}{c}
                                X_{1}+X_{2}+X_{1} X_{2} \\
                                2 X_{1}+3 X_{1} X_{2}+X_{2}
                                \end{array}\right)

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

.. math::
    \text{case 2: } X_1, X_2 \sim \mathcal{U}(0, 1)

.. [1] Gamboa, F., Janon, A., Klein, T., & Lagnoux, A. (2014). Sensitivity analysis for multidimensional and functional outputs. Electronic Journal of Statistics, 8(1), 575-603.

.. GENERATED FROM PYTHON SOURCE LINES 26-35

.. code-block:: default

    from UQpy.run_model.RunModel import RunModel
    from UQpy.run_model.model_execution.PythonModel import PythonModel
    from UQpy.distributions import Uniform, Normal
    from UQpy.distributions.collection.JointIndependent import JointIndependent
    from UQpy.sensitivity.GeneralisedSobolSensitivity import GeneralisedSobolSensitivity
    from UQpy.sensitivity.PostProcess import *

    np.random.seed(123)


.. GENERATED FROM PYTHON SOURCE LINES 36-37

**Define the model and input distributions**

.. GENERATED FROM PYTHON SOURCE LINES 37-51

.. code-block:: default


    # Create Model object
    model = PythonModel(
        model_script="local_multioutput.py",
        model_object_name="evaluate",
        var_names=[r"X_1$", r"X_2"],
        delete_files=True,
    )

    runmodel_obj = RunModel(model=model)

    # Define distribution object
    dist_object_1 = JointIndependent([Normal(0, 1)] * 2)


.. GENERATED FROM PYTHON SOURCE LINES 52-53

**Compute generalised Sobol indices**

.. GENERATED FROM PYTHON SOURCE LINES 55-61

.. code-block:: default

    SA = GeneralisedSobolSensitivity(runmodel_obj, dist_object_1)

    SA.run(
        n_samples=20_000, confidence_level=0.95, n_bootstrap_samples=5_00
    )


.. GENERATED FROM PYTHON SOURCE LINES 62-71

**First order Generalised Sobol indices**

Expected generalised Sobol indices:

Gaussian case

:math:`GS_1` = 0.2941

:math:`GS_2` = 0.1179

.. GENERATED FROM PYTHON SOURCE LINES 73-83

.. code-block:: default

    SA.generalized_first_order_indices

    # **Plot the first order sensitivity indices**
    fig1, ax1 = plot_sensitivity_index(
        SA.generalized_first_order_indices[:, 0],
        confidence_interval=SA.first_order_confidence_interval,
        plot_title="First order Generalised Sobol indices",
        color="C0",
    )


.. GENERATED FROM PYTHON SOURCE LINES 84-97

.. code-block:: default

    SA.generalized_total_order_indices

    # **Plot the first and total order sensitivity indices**
    fig2, ax2 = plot_index_comparison(
        SA.generalized_first_order_indices[:, 0],
        SA.generalized_total_order_indices[:, 0],
        confidence_interval_1=SA.first_order_confidence_interval,
        confidence_interval_2=SA.total_order_confidence_interval,
        label_1="First order",
        label_2="Total order",
        plot_title="First and Total order Generalised Sobol indices",
    )


.. GENERATED FROM PYTHON SOURCE LINES 98-99

**Compute generalised Sobol indices**

.. GENERATED FROM PYTHON SOURCE LINES 101-109

.. code-block:: default

    dist_object_2 = JointIndependent([Uniform(0, 1)] * 2)

    SA = GeneralisedSobolSensitivity(runmodel_obj, dist_object_2)

    SA.run(
        n_samples=20_000, confidence_level=0.95, n_bootstrap_samples=5_00
    )


.. GENERATED FROM PYTHON SOURCE LINES 110-119

**First order Generalised Sobol indices**

Expected generalised Sobol indices:

Uniform case

:math:`GS_1` = 0.6084

:math:`GS_2` = 0.3566

.. GENERATED FROM PYTHON SOURCE LINES 121-131

.. code-block:: default

    SA.generalized_first_order_indices

    # **Plot the first order sensitivity indices**
    fig3, ax3 = plot_sensitivity_index(
        SA.generalized_first_order_indices[:, 0],
        confidence_interval=SA.first_order_confidence_interval,
        plot_title="First order Generalised Sobol indices",
        color="C0",
    )


.. GENERATED FROM PYTHON SOURCE LINES 132-144

.. code-block:: default

    SA.total_order_confidence_interval

    # **Plot the first and total order sensitivity indices**
    fig4, ax4 = plot_index_comparison(
        SA.generalized_first_order_indices[:, 0],
        SA.total_order_confidence_interval[:, 0],
        confidence_interval_1=SA.first_order_confidence_interval,
        confidence_interval_2=SA.total_order_confidence_interval,
        label_1="First order",
        label_2="Total order",
        plot_title="First and Total order Generalised Sobol indices",
    )


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

   **Total running time of the script:** ( 0 minutes  0.000 seconds)


.. _sphx_glr_download_auto_examples_sensitivity_generalised_sobol_generalised_sobol_multioutput.py:

.. only:: html

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


    .. container:: binder-badge

      .. image:: images/binder_badge_logo.svg
        :target: https://mybinder.org/v2/gh/SURGroup/UQpy/master?urlpath=lab/tree/notebooks/auto_examples/sensitivity/generalised_sobol/generalised_sobol_multioutput.ipynb
        :alt: Launch binder
        :width: 150 px



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

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

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

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


.. only:: html

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

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