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


Ishigami function
==============================================

The ishigami function is a non-linear, non-monotonic function that is commonly used to 
benchmark uncertainty and senstivity analysis methods.

.. math::
    f(x_1, x_2, x_3) = sin(x_1) + a \cdot sin^2(x_2) + b \cdot x_3^4 sin(x_1)

.. math::
    x_1, x_2, x_3 \sim \mathcal{U}(-\pi, \pi), \quad a, b\in \mathbb{R}

First order Sobol indices
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. math::
    S_1 = \frac{V_1}{\mathbb{V}[Y]}, \quad S_2 = \frac{V_2}{\mathbb{V}[Y]}, \quad S_3 = \frac{V_3}{\mathbb{V}[Y]} = 0,

.. math::
    V_1 = 0.5 (1 + \frac{b\pi^4}{5})^2, \quad V_2 = \frac{a^2}{8}, \quad V_3 = 0

.. math::
    \mathbb{V}[Y] = \frac{a^2}{8} + \frac{b\pi^4}{5} + \frac{b^2\pi^8}{18} + \frac{1}{2}

Total order Sobol indices
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. math::
    S_{T_1} = \frac{V_{T1}}{\mathbb{V}[Y]}, \quad S_{T_2} = \frac{V_{T2}}{\mathbb{V}[Y]}, \quad S_{T_3} = \frac{V_{T3}}{\mathbb{V}[Y]}

.. math::
    V_{T_1} = 0.5 (1 + \frac{b\pi^4}{5})^2 + \frac{8b^2\pi^8}{225}, \quad V_{T_2}= \frac{a^2}{8}, \quad V_{T_3} = \frac{8b^2\pi^8}{225}

.. math::
    \mathbb{V}[Y] = \frac{a^2}{8} + \frac{b\pi^4}{5} + \frac{b^2\pi^8}{18} + \frac{1}{2}

.. GENERATED FROM PYTHON SOURCE LINES 42-53

.. 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 54-55

**Define the model and input distributions**

.. GENERATED FROM PYTHON SOURCE LINES 57-58

Create Model object

.. GENERATED FROM PYTHON SOURCE LINES 58-71

.. code-block:: default

    model = PythonModel(
        model_script="local_ishigami.py",
        model_object_name="evaluate",
        var_names=[r"$X_1$", "$X_2$", "$X_3$"],
        delete_files=True,
        params=[7, 0.1],
    )

    runmodel_obj = RunModel(model=model)

    # Define distribution object
    dist_object = JointIndependent([Uniform(-np.pi, 2 * np.pi)] * 3)


.. GENERATED FROM PYTHON SOURCE LINES 72-73

**Compute Sobol indices**

.. GENERATED FROM PYTHON SOURCE LINES 75-79

.. code-block:: default

    SA = SobolSensitivity(runmodel_obj, dist_object)

    SA.run(n_samples=100_000, n_bootstrap_samples=100)


.. GENERATED FROM PYTHON SOURCE LINES 80-89

**First order Sobol indices**

Expected first order Sobol indices:

:math:`S_1` = 0.3139

:math:`S_2` = 0.4424

:math:`S_3` = 0.0

.. GENERATED FROM PYTHON SOURCE LINES 91-93

.. code-block:: default

    SA.first_order_indices


.. GENERATED FROM PYTHON SOURCE LINES 94-103

**Total order Sobol indices**

Expected total order Sobol indices:

:math:`S_{T_1}` = 0.55758886

:math:`S_{T_2}` = 0.44241114

:math:`S_{T_3}` =  0.24368366

.. GENERATED FROM PYTHON SOURCE LINES 105-107

.. code-block:: default

    SA.total_order_indices


.. GENERATED FROM PYTHON SOURCE LINES 108-109

**Confidence intervals for first order Sobol indices**

.. GENERATED FROM PYTHON SOURCE LINES 111-113

.. code-block:: default

    SA.first_order_confidence_interval


.. GENERATED FROM PYTHON SOURCE LINES 114-115

**Confidence intervals for total order Sobol indices**

.. GENERATED FROM PYTHON SOURCE LINES 117-119

.. code-block:: default

    SA.total_order_confidence_interval


.. GENERATED FROM PYTHON SOURCE LINES 120-121

**Plot the first order sensitivity indices**

.. GENERATED FROM PYTHON SOURCE LINES 121-129

.. code-block:: default

    fig1, ax1 = plot_sensitivity_index(
        SA.first_order_indices[:, 0],
        confidence_interval=SA.first_order_confidence_interval,
        plot_title="First order Sobol indices",
        variable_names=["$X_1$", "$X_2$", "$X_3$"],
        color="C0",
    )


.. GENERATED FROM PYTHON SOURCE LINES 130-131

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

.. GENERATED FROM PYTHON SOURCE LINES 131-141

.. code-block:: default

    fig2, ax2 = plot_index_comparison(
        SA.first_order_indices[:, 0],
        SA.total_order_indices[:, 0],
        confidence_interval_1=SA.first_order_confidence_interval,
        confidence_interval_2=SA.total_order_confidence_interval,
        label_1="First order Sobol indices",
        label_2="Total order Sobol indices",
        plot_title="First and Total order Sobol indices",
        variable_names=["$X_1$", "$X_2$", "$X_3$"],
    )


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

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


.. _sphx_glr_download_auto_examples_sensitivity_sobol_sobol_ishigami.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_ishigami.ipynb
        :alt: Launch binder
        :width: 150 px



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

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

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

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


.. only:: html

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

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