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Update sensitivity_sobol_from_pce.rst
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Fixed typos.
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mbaudin47 authored Feb 5, 2025
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Expand Up @@ -37,7 +37,7 @@ to the index :math:`P`.
Depending on the way the coefficients are computed, the set of multi-indices
is the consequence of the choice of the polynomial degree, the enumeration rule,
and, if necessary, the selection method (e.g. the LARS selection method).
Let :math:`\metaModel(\standardRV)` be the polynomial chaos expansion:
Let :math:`\tilde{h}(\standardRV)` be the polynomial chaos expansion:

.. math::
\tilde{h}(\standardRV)
Expand All @@ -51,15 +51,15 @@ functional basis.
The variance of the polynomial chaos expansion is:

.. math::
\Var{\metaModel(\standardRV)} = \sum_{\idx \in \mathcal{J}_P} a_\idx^2 \|\psi_\idx\|^2.
\Var{\metaModel(\standardRV)} = \sum_{\idx \in \mathcal{J}_P, \; \idx \neq \vect{0}} a_\idx^2 \|\psi_\idx\|^2.
In the previous expression, let us emphasise that the variance is a sum
of squares, excepted the :math:`a_0` coefficient.
If the polynomial basis is orthonormal, the expression is particularly
simple (see [legratiet2017]_ eq. 38.43 page 1301):

.. math::
\Var{\metaModel(\standardRV)} = \sum_{\idx \in \mathcal{J}_P} a_\idx^2.
\Var{\metaModel(\standardRV)} = \sum_{\idx \in \mathcal{J}_P, \; \idx \neq \vect{0}} a_\idx^2.
The part of variance of the multi-index :math:`\idx` is:

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