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Interpreting Sobol sensitivity analysis indices

I’ve read a lot about the Sobol method for sensitivity analysis, and I’ll admit I don’t have a firm grasp of the mathematics, so I’d like to pose what might be a trivial question here. If I analyze the linear model

Y=X1+X2+X3+X4

where Xi∼N(0,1) and are independent.

Then the Sobol first-order and total order indices are naturally 0.25 for each independent variable or factor. If the linear model is modified to the following:

Y=X1+2*X2+X3+X4

the first-order and total order indices are about 0.14 for X1, X3, and X4, and 0.58 for X2. I don’t understand these results. I would expect the values for X1, X3, and X4 to be 0.2 and the value for X2 to be 0.4. I wonder if anyone can provide a good explanation for the results.

Dear @mlaurens29

You have to come back to the definition of Sobol’ indices as partial variances (normalized by the total variance).

In your first example, the (so-called total) variance of Y is obviously 4 because the X_i's are independent. In the second case,it is equal to:

\text{Var}[Y] = \text{Var}[X_1] + \text{Var}[2\,X_2] + \text{Var}[X_3] + \text{Var}[X_4] = 1+4+1+1 = 7.

The first-order Sobol’ index S_i is defined by:

S_i = \frac{D_i}{\text{Var}[Y]},

where D_i is the partial variance defined by:

D_i = \text{Var}_{X_i}[E[Y | X_i]].
  • In your first example, all D_i's are identical and equal to \text{Var}_{X_i}[X_i] = 1.
  • in the second example \text{Var}_{X_i}[X_i] = 1 for i=1,\, 3, \, 4 and \text{Var}_{X_2}[2\,X_2] =4.

As a consequence, the corresponding Sobol’ indices are:

\begin{split} S_i &= \frac{1}{7} \quad \text{for } i=1,3,4, \\ S_2 &= \frac{4}{7} . \end{split}

I hope this explanation will help.
Best regards
Bruno