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


Sobol function
==============================================

The Sobol function is non-linear function that is commonly used to benchmark uncertainty 
and senstivity analysis methods. Unlike the Ishigami function which has 3 input 
variables, the Sobol function can have any number of input variables. 

.. math::

    g(x_1, x_2, \ldots, x_D) := \prod_{i=1}^{D} \frac{|4x_i - 2| + a_i}{1 + a_i},

where,

.. math::
    x_i \sim \mathcal{U}(0, 1), \quad a_i \in \mathbb{R}.

This is an example from [1]_, where first order, total order and additionally the second
order indices are computed.

.. [1] Glen, G., & Isaacs, K. (2012). Estimating Sobol sensitivity indices using correlations. Environmental Modelling and Software, 37, 157–166.

.. GENERATED FROM PYTHON SOURCE LINES 27-38

.. 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 Uniform
    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 39-40

**Define the model and input distributions**

.. GENERATED FROM PYTHON SOURCE LINES 40-58

.. code-block:: default


    # Create Model object
    num_vars = 6
    a_vals = np.array([0.0, 0.5, 3.0, 9.0, 99.0, 99.0])

    model = PythonModel(
        model_script="local_sobol_func.py",
        model_object_name="evaluate",
        var_names=["X_" + str(i) for i in range(num_vars)],
        delete_files=True,
        a_values=a_vals,
    )

    runmodel_obj = RunModel(model=model)

    # Define distribution object
    dist_object = JointIndependent([Uniform(0, 1)] * num_vars)


.. GENERATED FROM PYTHON SOURCE LINES 59-60

**Compute Sobol indices**

.. GENERATED FROM PYTHON SOURCE LINES 62-67

.. code-block:: default

    SA = SobolSensitivity(runmodel_obj, dist_object)

    # Compute Sobol indices using the pick and freeze algorithm
    SA.run(n_samples=50_000, estimate_second_order=True)


.. GENERATED FROM PYTHON SOURCE LINES 68-83

**First order Sobol indices**

Expected first order Sobol indices:

:math:`S_1` = 5.86781190e-01

:math:`S_2` = 2.60791640e-01

:math:`S_3` = 3.66738244e-02

:math:`S_4` = 5.86781190e-03

:math:`S_5` = 5.86781190e-05

:math:`S_6` = 5.86781190e-05

.. GENERATED FROM PYTHON SOURCE LINES 85-87

.. code-block:: default

    SA.first_order_indices


.. GENERATED FROM PYTHON SOURCE LINES 88-89

**Plot the first order sensitivity indices**

.. GENERATED FROM PYTHON SOURCE LINES 89-95

.. code-block:: default

    fig1, ax1 = plot_sensitivity_index(
        SA.first_order_indices[:, 0],
        plot_title="First order Sobol indices",
        color="C0",
    )


.. GENERATED FROM PYTHON SOURCE LINES 96-112

**Total order Sobol indices**

Expected total order Sobol indices:

:math:`S_{T_1}` = 6.90085892e-01

:math:`S_{T_2}` = 3.56173364e-01

:math:`S_{T_3}` = 5.63335422e-02

:math:`S_{T_4}` = 9.17057664e-03

:math:`S_{T_5}` = 9.20083854e-05

:math:`S_{T_6}` = 9.20083854e-05


.. GENERATED FROM PYTHON SOURCE LINES 114-116

.. code-block:: default

    SA.total_order_indices


.. GENERATED FROM PYTHON SOURCE LINES 117-118

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

.. GENERATED FROM PYTHON SOURCE LINES 118-126

.. code-block:: default

    fig2, ax2 = 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",
    )


.. GENERATED FROM PYTHON SOURCE LINES 127-160

**Second order Sobol indices**

Expected second order Sobol indices:

:math:`S_{T_{12}}` = 0.0869305

:math:`S_{T_{13}}` = 0.0122246

:math:`S_{T_{14}}` = 0.00195594

:math:`S_{T_{15}}` = 0.00001956

:math:`S_{T_{16}}` = 0.00001956

:math:`S_{T_{23}}` = 0.00543316

:math:`S_{T_{24}}` = 0.00086931

:math:`S_{T_{25}}` = 0.00000869

:math:`S_{T_{26}}` = 0.00000869

:math:`S_{T_{34}}` = 0.00012225

:math:`S_{T_{35}}` = 0.00000122

:math:`S_{T_{36}}` = 0.00000122

:math:`S_{T_{45}}` = 0.00000020

:math:`S_{T_{46}}` = 0.00000020

:math:`S_{T_{56}}` = 2.0e-9

.. GENERATED FROM PYTHON SOURCE LINES 162-164

.. code-block:: default

    SA.second_order_indices


.. GENERATED FROM PYTHON SOURCE LINES 165-166

**Plot the second order sensitivity indices**

.. GENERATED FROM PYTHON SOURCE LINES 166-170

.. code-block:: default

    fig3, ax3 = plot_second_order_indices(
        SA.second_order_indices[:, 0],
        num_vars=num_vars,
    )


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

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


.. _sphx_glr_download_auto_examples_sensitivity_sobol_sobol_func.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_func.ipynb
        :alt: Launch binder
        :width: 150 px



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

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

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

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


.. only:: html

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

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