Order of second Sobol indices

Hello all,
I have a question about the order of second-order Sobol Indices. For the first-order index, we know the maximum value is unity and from there we can have an estimation of the impact of each parameter on the output. However, for the second-order indices, as far as I know, we do not have any maximum values. How can we interpret these values then? which number shows “High” interaction between parameters? Do we have a rule of thumb for that?
In my simulations, the maximum second-order index between parameters is in the order of 0.01. I have values down to the order of 0.0001 in my second order indices as well. How can I interpret these? Which parameters considerable interactions?
Thanks in advance.

Any helps is greatly appreciated.

Hi @Aep93

The Sobol’ indices depend on the Sobol’ decomposition (see the sensitivity analysis manual) that decomposes the model response into terms that depend only on a subset of the input parameters. The Sobol’ index S_{\mathcal{I}} is just the variance of the term in this decomposition that depends on the input parameters \{X_i\}_{i\in\mathcal{I}}.

Naturally, summing up all Sobol’ indices yields 1 (not considering total Sobol’ indices). The first order indices are just the subset of all Sobol’ indices for single parameters like S_1, S_2, etc.. Therefore, their sum needs to be smaller than 1 and is exactly 1, only if all higher order indices (S_{1,2}, S_{4,7,3}) are 0.

In many engineering models, higher-order interactions are negligible (sparsity of effects principle) and the more parameters an index considers, the smaller it typically becomes. This is why, you can often judge parameter importance by interpreting the first-order indices, when in fact you should be looking at the total Sobol’ indices for this matter.

To answer your question:

If the higher-order indices are very small (i.e., <1\%), I would call them negligible, although this can depend on your application. You could also have a look at whether certain interaction effects are more important than some first-order effects, e.g. S_1<S_{2,3} which would indicate a “high” interaction in my opinion.

Hope this helps