Sensitivity analysis using Sobol indices¶
This method deals with analyzing the influence the random vector
has on a random variable
which is being studied for uncertainty. Here we attempt to evaluate the
part of variance of
due to the different components
.
The estimators for the mean of and the standard deviation
of
can be obtained from a first sample, as Sobol
indices estimation requires two samples of the input variables :
,
that is two sets of N vectors of dimension
and
The estimation of sensitivity indices for first order consists in estimating the quantity
Sobol proposes to estimate the quantity
by swapping every variables in the two samples except the variable
between the two calls of the function:
Then the first order indices are estimated by
For the second order, the two variables and
are not swapped to estimate
,
and so on for higher orders, assuming that order
.
Then the
second order indices are estimated by
For the total order indices
, we only swap the variable
between the two samples.