.. 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_exponential.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_exponential.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_exponential.py:


Exponential function
==============================================

The exponential function was used in [1]_ to demonstrate the 
Cramér-von Mises indices.

.. math::
    f(x) := \exp(x_1 + 2x_2), \quad x_1, x_2 \sim \mathcal{N}(0, 1)

.. [1] Gamboa, F., Klein, T., & Lagnoux, A. (2018). Sensitivity Analysis Based on Cramér-von Mises Distance. SIAM/ASA Journal on Uncertainty Quantification, 6(2), 522-548. doi:10.1137/15M1025621. (`Link <https://doi.org/10.1137/15M1025621>`_)

.. GENERATED FROM PYTHON SOURCE LINES 19-28

.. code-block:: default

    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.CramerVonMisesSensitivity import CramerVonMisesSensitivity as cvm
    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 30-44

.. code-block:: default


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

    runmodel_obj = RunModel(model=model)

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


.. GENERATED FROM PYTHON SOURCE LINES 45-46

**Compute Cramér-von Mises indices**

.. GENERATED FROM PYTHON SOURCE LINES 48-53

.. code-block:: default

    SA = cvm(runmodel_obj, dist_object)

    # Compute CVM indices using the pick and freeze algorithm
    SA.run(n_samples=20_000, estimate_sobol_indices=True)


.. GENERATED FROM PYTHON SOURCE LINES 54-61

**Cramér-von Mises indices**

Expected value of the sensitivity indices:

:math:`S^1_{CVM} = \frac{6}{\pi} \operatorname{arctan}(2) - 2 \approx 0.1145`

:math:`S^2_{CVM} = \frac{6}{\pi} \operatorname{arctan}(\sqrt{19}) - 2 \approx 0.5693`

.. GENERATED FROM PYTHON SOURCE LINES 63-72

.. 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 73-80

**Estimated first order Sobol indices**

Expected first order Sobol indices:

:math:`S_1` = 0.0118

:math:`S_2` = 0.3738

.. GENERATED FROM PYTHON SOURCE LINES 82-91

.. code-block:: default

    SA.first_order_sobol_indices

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


.. GENERATED FROM PYTHON SOURCE LINES 92-93

**Estimated total order Sobol indices**

.. GENERATED FROM PYTHON SOURCE LINES 95-105

.. code-block:: default

    SA.total_order_sobol_indices

    # **Plot the first and total order sensitivity indices**
    fig3, ax3 = plot_index_comparison(
        SA.first_order_sobol_indices[:, 0],
        SA.total_order_sobol_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_cramer_von_mises_cvm_exponential.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_exponential.ipynb
        :alt: Launch binder
        :width: 150 px



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

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

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

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


.. only:: html

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

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