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