Total Sobol indices sum to more than 1

Hello UQlab community.

I have been carrying out sobol analysis for global sensitivity analysis of a simple analytical model.
My understanding is the total sobol indices should sum to one and represent the contribution of the input parameter’s variance to the total variance of the property of interest.

I have noticed my total sobol indices often sum to more than 1. What could be causing this? Note my input parameters are not correlated at all.

Kind regards,

Matt :slight_smile:

Hi @Matthew_Harrison,

Welcome to UQWorld! :smiley:

Actually, the total Sobol’ indices will, in general, sum to more than 1. I think you might have mistaken the total Sobol’ indices for the first-order Sobol’ indices (whose sum cannot be larger than 1). The total Sobol’ index of an input parameter is the sum of the first-order Sobol’ index and all the higher-order Sobol’ indices involving that parameter.

So, if you sum the total indices, you would double count many indices. It is only for a special case that the sum of the total indices is 1. That is, if all the higher-order indices are 0; in other words, the sum of the first-order indices is 1. Note that this property is not about whether your inputs are correlated or not, but whether they are interacting with respect to the output.

Regarding this topic, the explanation from the Wikipedia article might be helpful:

I hope this helps!

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Hi @damarginal

Amazing, thank you for the quick reply!
That all makes sense now, great explanation.