Hello all,
Considering a model that has n number of inputs. How many times should I evaluate the model to be able to find first and total Sobol indices (if I do not want to use PCE)?
Can you please introduce me some references in this regard?
Thanks in advance
Hi @Aep93,
The number of sample points for a direct MC simulation to estimate the first and total Sobol’ indices would depend on the estimator and the scheme used (it is an MC estimator of a multi-dimensional integral). For instance the estimators and scheme mentioned below (UQLab uses these or similar to them),
requires N_{total} = N_s \times (n + 2) model evaluations, where N_s and n are the number of sample points and the number of inputs (your notation), respectively. The actual value of N_s would vary from model to model but I would expect this would be in the order of thousands and more (PS: convergence check—say, by bootstrap—would be advisable).
I also find the explanation in Section 3 of:
to be a pretty accessible summary.
Finally, the actual available estimators for the Sobol’ indices and their references can be found on the Sensitivity User Manual (see Table 13).
Hope this helps!
Just to complement Damar’s reply: in most cases you can get a PCE of reasonable accuracy for PCE analysis using a few hundred runs of your model (if you have n in the range of 5-30). With this PCE, Sobol’ indices are computed analytically at any order. In contrast, the accurate estimation of each Sobol’ index by Monte Carlo simulation typically requires 10^3 - 10^4 runs per index.
So there is no reason not to try PCE’s if you look for efficiency 
Our experience shows that a PCE can safely be used for sensitivity analysis as soon as its leave-one-out error is less than 1% (see the field myPCE.Error).
Best regards
Bruno
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