# 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.