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Added references to bibliography for CVM
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Prateek Bhustali committed May 23, 2022
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6 changes: 2 additions & 4 deletions docs/code/sensitivity/cramer_von_mises/README.rst
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Expand Up @@ -4,10 +4,8 @@ These examples serve as a guide for using the Cramér-von Mises sensitivity modu

1. **Exponential function**

For the Exponential model, analytical Cramér-von Mises indices are available [1]_.
For the Exponential model, analytical Cramér-von Mises indices are available :cite:`CVM`.

2. **Sobol function**

The Cramér-von Mises indices are computed using the Pick and Freeze approach [1]_. These model evaluations can be used to estimate the Sobol indices as well. We demonstrate this using the Sobol function.

.. [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>`_)
The Cramér-von Mises indices are computed using the Pick and Freeze approach :cite:`CVM`. These model evaluations can be used to estimate the Sobol indices as well. We demonstrate this using the Sobol function.
8 changes: 2 additions & 6 deletions docs/source/sensitivity/cramer_von_mises.rst
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Cramér-von Mises indices
----------------------------------------

A sensitivity index based on the Cramér-von Mises distance. In contrast to the variance based Sobol indices, it takes into account the whole distribution of the model output and is therefore considered as a moment-free method [1]_. Furthermore the index can be naturally extended to multivariate model outputs (not implemented yet in UQPy).
A sensitivity index based on the Cramér-von Mises distance. In contrast to the variance based Sobol indices, it takes into account the whole distribution of the model output and is therefore considered as a moment-free method :cite:`CVM`. 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:

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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 [2]_.)

.. [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>`_)
.. [2] Gamboa, F., Gremaud, P., Klein, T., & Lagnoux, A. (2020). Global Sensitivity Analysis: a new generation of mighty estimators based on rank statistics. arXiv [math.ST]. (`Link <http://arxiv.org/abs/2003.01772>`_)
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 :cite:`gamboa2020global`.)

Cramér-von Mises Class
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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