Taylor importance factorsΒΆ

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 Taylor Expansion. Let \uX be the input random vector. We assume that the marginals of \uX are independent. Let Y = \model(\ux) with \model: \Rset^\inputDim \rightarrow \Rset be a function with a scalar output.

Refer to Taylor Expansion for details on the expressions of the first-order and second-order Taylor expansions and to Taylor Expansion Moments for details on the approximations of the mean and the variance of Y.

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

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

for i \in \{1, ..., d\} where \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:

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

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