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 Sobolâ€™ 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 checkâ€”say, by bootstrapâ€”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 Sobolâ€™ indices and their references can be found on the Sensitivity User Manual (see Table 13).

Hope this helps!

Just to complement Damarâ€™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, Sobolâ€™ indices are computed analytically at any order. In contrast, the accurate estimation of **each** Sobolâ€™ index by Monte Carlo simulation typically requires 10^3 - 10^4 runs *per index*.

So there is no reason not to try PCEâ€™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