Sensitivity analysis using Sobol indices¶

This method deals with analyzing the influence the random vector $\vect{X} = \left( X^1,\ldots,X^{n_X} \right)$ has on a random variable $Y^k$ which is being studied for uncertainty. Here we attempt to evaluate the part of variance of $Y^k$ due to the different components $X^i$ .

The estimators for the mean of $m_{Y^j}$ and the standard deviation $\sigma$ of $Y^k$ can be obtained from a first sample, as Sobol indices estimation requires two samples of the input variables : $(X^1,\ldots,X^{n_X})$ , that is two sets of N vectors of dimension $n_X$ $(x_{11}^{(1)},\ldots,x_{1n_X})^{(1)},\ldots,(x_{N^1}^{(1)},\ldots,x_{Nn_X}^{(1)})$ and $(x_{11}^{(2)},\ldots,x_{1n_X})^{(2)},\ldots,(x_{N^1}^{(2)},\ldots,x_{Nn_X}^{(2)})$

The estimation of sensitivity indices for first order consists in estimating the quantity

$V_i = \Var{\Expect{ Y^k \vert X_i}} = \Expect{ \Expect{Y^k \vert X_i}^2} - \Expect{\Expect{ Y^k \vert X_i }} ^2 = U_i - \Expect{Y^k} ^2$

Sobol proposes to estimate the quantity $U_i = \Expect{\Expect{ Y^k \vert X_i}^2}$ by swapping every variables in the two samples except the variable $X_i$ between the two calls of the function:

$\hat U_i = \frac{1}{N}\sum_{k=1}^N{ Y^k\left( x_{k1}^{(1)},\dots, x_{k(i-1)}^{(1)},x_{ki}^{(1)},x_{k(i+1)}^{(1)},\dots,x_{kn_X}^{(1)}\right) \times Y^k\left( x_{k1}^{(2)},\dots,x_{k(i-1)}^{(2)},x_{ki}^{(1)},x_{k(i+1)}^{(2)},\dots,x_{kn_X}^{(2)}\right)}$

Then the $n_X$ first order indices are estimated by

$\hat S_i = \frac{\hat V_i}{\hat \sigma^2} = \frac{\hat U_i - m_{Y^k}^2}{\hat \sigma^2}$

For the second order, the two variables $X_i$ and $X_j$ are not swapped to estimate $U_{ij}$ , and so on for higher orders, assuming that order $< n_X$ . Then the $\binom {n_X}{2}$ second order indices are estimated by