Added documentation Cramér-von Mises sensitivity

docs/code/sensitivity/cramer_von_mises/README.rst

@@ -0,0 +1,3 @@
+Cramér-von Mises Sensitivity indices
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+

docs/source/sensitivity/cramer_von_mises.rst

@@ -0,0 +1,61 @@
+Cramér-von Mises indices
+----------------------------------------
+
+A sensitivity index based on the Cramér-von Mises distance. In contrast to variance based Sobol indices it takes into account the whole distribution of the model output and is therefore considered as a moment-free method [5]_. Furthermore the index can be naturally extended to multivariate model outputs (not implemented yet in UQPy). 
+
+Consider a model :math:`Y=f(X): \mathbb{R}^d \rightarrow \mathbb{R}^k` with :math:`d` inputs :math:`X_{(1)}, X_{(2)}, \ldots, X_{(d)}` and :math:`k` outputs :math:`Y_{(1)}, Y_{(2)}, \ldots, Y_{(k)}`. We define the cumulative distribution function :math:`F(t)` of :math:`Y` as:
+
+.. math::
+
+   F(t)=\mathbb{P}(Z \leqslant t)=\mathbb{E}\left[\mathbb{1}_{\{Z \leqslant t\}}\right] \text { for } t=\left(t_{1}, \ldots, t_{k}\right) \in \mathbb{R}^{k}
+
+and the conditional distribution function :math:`F(t)` of :math:`Y` as:
+
+.. math::
+
+   F^{v}(t)=\mathbb{P}\left(Z \leqslant t \mid X_{v}\right)=\mathbb{E}\left[\mathbb{1}_{\{Z \leqslant t\}} \mid X_{v}\right] \text { for } t=\left(t_{1}, \ldots, t_{k}\right) \in \mathbb{R}^{k}
+
+where, :math:`\{Z \leqslant t\} \text { means that } \left\{Z_{1} \leqslant t_{1}, \ldots, Z_{k} \leqslant t_{k}\right\}`.
+
+The first order Cramér-von Mises index :math:`S_{2, C V M}^{i}` (for input :math:`v = {i}`) is defined as:
+
+.. math::
+
+   S_{2, C V M}^{i}:=\frac{\int_{\mathbb{R}^{k}} \mathbb{E}\left[\left(F(t)-F^{i}(t)\right)^{2}\right] d F(t)}{\int_{\mathbb{R}^{k}} F(t)(1-F(t)) d F(t)}
+
+and the total Cramér-von Mises index :math:`S_{2, C V M}^{T o t, i}` (for input :math:`v = {i}`) is defined as:
+
+.. math::
+
+   S_{2, C V M}^{T o t, i}:=1-S_{2, C V M}^{\sim i}=1-\frac{\int_{\mathbb{R}^{k}} \mathbb{E}\left[\left(F(t)-F^{\sim i}(t)\right)^{2}\right] d F(t)}{\int_{\mathbb{R}^{k}} F(t)(1-F(t)) d F(t)}
+
+The above first and total order indices are estimated using the Pick-and-Freeze approach. This requires :math:`N(d+2)` model evaluations, where :math:`N` is the number of samples. (For implementation details, see also [6]_.)
+
+Cramér-von Mises Class
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+The :class:`Cramér-von Mises` class is imported using the following command:
+
+>>> from UQpy.sensitivity.cramer_von_mises import CramerVonMises
+
+Methods
+"""""""
+.. autoclass:: UQpy.sensitivity.CramervonMises
+   :members: run
+
+Attributes
+""""""""""
+.. autoattribute:: UQpy.sensitivity.CramervonMises.CVM_i
+.. autoattribute:: UQpy.sensitivity.CramervonMises.CI_CVM_i
+.. autoattribute:: UQpy.sensitivity.CramervonMises.sobol_i
+.. autoattribute:: UQpy.sensitivity.CramervonMises.sobol_total_i
+.. autoattribute:: UQpy.sensitivity.CramervonMises.n_samples
+.. autoattribute:: UQpy.sensitivity.CramervonMises.num_vars
+
+
+Examples
+""""""""""
+
+.. toctree::
+
+   Cramér-von Mises Examples <../auto_examples/sensitivity/cramer_von_mises/index>

docs/source/sensitivity/index.rst

@@ -6,6 +6,7 @@ This module contains functionality for all the sampling methods supported in :py
 The module currently contains the following classes:
 
 - :py:class:`.Sobol`: Class to compute Sobol sensitivity indices.
+- :py:class:`.CramervonMises`: Class to compute Cramér-von Mises sensitivity indices.
 - :py:class:`.MorrisSensitivity`: Class to perform Morris.
 - :py:class:`.PceSensitivity`: Class to compute the sensitivity indices using the :class:`.PolynomialChaosExpansion` method.
 
@@ -20,3 +21,4 @@ Sensitivity analysis comprises techniques focused on determining how the variati
     Morris Sensitivity <morris>
     Polynomial Chaos Sensitivity <pce>
     Sobol Sensitivity <sobol>
+    Cramér-von Mises Sensitivity <cramer_von_mises>