.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/sensitivity/sobol/sobol_additive.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_sobol_sobol_additive.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_sobol_sobol_additive.py:


Additive function
==============================================

We introduce the variance-based Sobol indices using an elementary example. 
For more details, refer [1]_.

.. math::
    f(x) = a \cdot X_1 + b \cdot X_2, \quad X_1, X_2 \sim \mathcal{N}(0, 1), \quad a,b \in \mathbb{R}

.. [1] Saltelli A, T. (2008). Global sensitivity analysis: The primer. John Wiley.

.. GENERATED FROM PYTHON SOURCE LINES 17-28

.. code-block:: default

    import numpy as np

    from UQpy.run_model.RunModel import RunModel
    from UQpy.run_model.model_execution.PythonModel import PythonModel
    from UQpy.distributions import Normal
    from UQpy.distributions.collection.JointIndependent import JointIndependent
    from UQpy.sensitivity.SobolSensitivity import SobolSensitivity
    from UQpy.sensitivity.PostProcess import *

    np.random.seed(123)


.. GENERATED FROM PYTHON SOURCE LINES 29-30

**Define the model and input distributions**

.. GENERATED FROM PYTHON SOURCE LINES 30-50

.. code-block:: default


    # Create Model object
    a, b = 1, 2

    model = PythonModel(
        model_script="local_additive.py",
        model_object_name="evaluate",
        var_names=[
            "X_1",
            "X_2",
        ],
        delete_files=True,
        params=[a, b],
    )

    runmodel_obj = RunModel(model=model)

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


.. GENERATED FROM PYTHON SOURCE LINES 51-52

**Compute Sobol indices**

.. GENERATED FROM PYTHON SOURCE LINES 54-58

.. code-block:: default

    SA = SobolSensitivity(runmodel_obj, dist_object)

    SA.run(n_samples=50_000)


.. GENERATED FROM PYTHON SOURCE LINES 59-66

**First order Sobol indices**

Expected first order Sobol indices:

:math:`\mathrm{S}_1 = \frac{a^2 \cdot \mathbb{V}[X_1]}{a^2 \cdot \mathbb{V}[X_1] + b^2 \cdot \mathbb{V}[X_2]} = \frac{1^2 \cdot 1}{1^2 \cdot 1 + 2^2 \cdot 1} = 0.2`

:math:`\mathrm{S}_2 = \frac{b^2 \cdot \mathbb{V}[X_2]}{a^2 \cdot \mathbb{V}[X_1] + b^2 \cdot \mathbb{V}[X_2]} = \frac{2^2 \cdot 1}{1^2 \cdot 1 + 2^2 \cdot 1} = 0.8`

.. GENERATED FROM PYTHON SOURCE LINES 68-70

.. code-block:: default

    SA.first_order_indices


.. GENERATED FROM PYTHON SOURCE LINES 71-72

**Plot the first and total order sensitivity indices**

.. GENERATED FROM PYTHON SOURCE LINES 72-79

.. code-block:: default

    fig1, ax1 = plot_index_comparison(
        SA.first_order_indices[:, 0],
        SA.total_order_indices[:, 0],
        label_1="First order Sobol indices",
        label_2="Total order Sobol indices",
        plot_title="First and Total order Sobol indices",
    )


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

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


.. _sphx_glr_download_auto_examples_sensitivity_sobol_sobol_additive.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/sobol/sobol_additive.ipynb
        :alt: Launch binder
        :width: 150 px



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

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

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

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


.. only:: html

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

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