.. _taylor_importance_factors:

Taylor importance factors
-------------------------

The importance factors derived from a Taylor expansion are
defined to rank the sensitivity of the output to
the inputs for central dispersion analysis.

We consider the Taylor expansion of a function. We use the notations introduced in
:ref:`Taylor Expansion <taylor_expansion>`. Let :math:`\uX` be the input random vector. We assume that
the marginals of :math:`\uX` are independent. Let :math:`Y = \model(\ux)` with
:math:`\model: \Rset^\inputDim \rightarrow \Rset` be a function with a scalar output.

Refer to :ref:`Taylor Expansion <taylor_expansion>` for details on the expressions
of the first-order and second-order Taylor expansions and to
:ref:`Taylor Expansion Moments <taylor_expansion_moments>` for
details on the approximations of the mean and the variance of :math:`Y`.

The importance factor of :math:`X_i` is defined by (see [daveiga2022]_ eq. 2.6 page 38) :

.. math::

    \eta_i = \frac{ \left(\frac{\partial \model}{\partial x_i}(\vect{\mu})\right)^2 \sigma_i^2}{\Var Y}.

for :math:`i \in \{1, ..., d\}` where :math:`\sigma_i = \sqrt{\Var{X_i}}` is the standard
deviation of the i-th input.
If the model is linear (i.e. if the model is equal to its first-order Taylor expansion),
then the importance factors sum to one:

.. math::

    \eta_1 + \eta_2 + \ldots + \eta_\inputDim = 1

These importance factors are also called *importance factors derived from perturbation methods*.

.. topic:: API:

    - See :class:`~openturns.TaylorExpansionMoments`

.. topic:: Examples:

    - See :doc:`/auto_reliability/central_dispersion/plot_estimate_moments_taylor`
