.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/sensitivity/cramer_von_mises/cvm_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_cramer_von_mises_cvm_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_cramer_von_mises_cvm_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}.


The function was also used in the Chatterjee indices section to demonstrate the 
computation of the Chatterjee indices. We can see clearly that the estimates are 
equivalent.

.. 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.CramerVonMisesSensitivity import CramerVonMisesSensitivity as cvm
    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.arange(1, num_vars+1, 1)

    model = PythonModel(
        model_script="local_sobol_func.py",
        model_object_name="evaluate",
        var_names=[r"$X_1$", "$X_2$"],
        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 Cramér-von Mises indices**

.. GENERATED FROM PYTHON SOURCE LINES 62-67

.. code-block:: default

    SA = cvm(runmodel_obj, dist_object)

    # Compute Sobol indices using rank statistics
    SA.run(n_samples=50_000, estimate_sobol_indices=True)


.. GENERATED FROM PYTHON SOURCE LINES 68-69

**Cramér-von Mises indices**

.. GENERATED FROM PYTHON SOURCE LINES 71-80

.. code-block:: default

    SA.first_order_CramerVonMises_indices

    # **Plot the CVM indices**
    fig1, ax1 = plot_sensitivity_index(
        SA.first_order_CramerVonMises_indices[:, 0],
        plot_title="Cramér-von Mises indices",
        color="C4",
    )


.. GENERATED FROM PYTHON SOURCE LINES 81-96

**Estimated Sobol indices**

Expected first order Sobol indices:

:math:`S_1` = 0.46067666

:math:`S_2` = 0.20474518

:math:`S_3` = 0.11516917

:math:`S_4` = 0.07370827

:math:`S_5` = 0.0511863

:math:`S_6` = 0.03760626

.. GENERATED FROM PYTHON SOURCE LINES 98-106

.. code-block:: default

    SA.total_order_sobol_indices

    # **Plot the first order Sobol indices**
    fig2, ax2 = plot_sensitivity_index(
        SA.total_order_sobol_indices[:, 0],
        plot_title="First order Sobol indices",
        color="C0",
    )


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

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


.. _sphx_glr_download_auto_examples_sensitivity_cramer_von_mises_cvm_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/cramer_von_mises/cvm_sobol_func.ipynb
        :alt: Launch binder
        :width: 150 px



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

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

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

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


.. only:: html

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

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