Sobol second order arangement in the "AllOrders" list

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1

Dear UQworld community,

I am studying the sensitivity of a model’s outputs with Sobol indices computed with PCE meta-models. I computed the second order Sobol indices and am wandering how these second order indices are arranged in the list “AllOrders”.
My model has 22 parameters, and the size of the second list in the “AllOrders” list is 231, which is ((22*21)/2) so I guess it’s like the upper triangular part of a matrix.
Does this list contain all the second order Sobol of the first parameter, then the second order Sobol of the second, except for the one in interaction with the first parameter, ect…?

Regards,
Guillaume

2

Dear Guillaume

Thanks for your question. You can have a look at the example uq_Example_Sensitivity_02_SobolIndices. After creating a sensitivity analysis, say using a polynomial chaos expansion, the results are stored in mySobolAnalysisPCE.Results.
In this struc you have indeed:

  • the AllOrders field that contain the first and second order indices in cell arrays,
  • the VarIdx field, that contain the related enumeration.

More precisely, mySobolAnalysisPCE.Results.VarIdx{2} contains all pairs of integers indexing the second-order indices. In this example (Borehole function with 8 inputs), there are \binom{8}{2}=28 pairs. The corresponding value is stored in mySobolAnalysisPCE.Results.AllIndices{2}.

Example:

mySobolAnalysisPCE.Results.VarIdx{2}(3,:)
ans =
     1     4

mySobolAnalysisPCE.Results.AllOrders{2}(3,:)
ans =
    0.0091

This means that the second-order Sobol’ index S_{1,4} = 0.0091.

Good luck!
Best regards
Bruno

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