I use Sobol’ sensitivity indices for my model and obtain that I have zero first-order Sobol’ indices for all input parameters but high second-order Sobol’ indices which constitute most of the Total Sobol’s indices. As much as I read, the first order Sobol’ indices constitute most of the Total Sobol’s indices.
My question is: Is it usual that in some models the first-order indices can be zero, and the impact of the interaction of input parameters can be higher than the individual effect of each parameter? If yes, How do I have to interpret my results? Because I know that my input parameters are independent. So, there is no single effect but some indirect relation between my input parameters on output results?
What you describe is indeed rather unusual, but not impossible. In most engineering problems, we observe that first-order indices are a big chunk of the total ones. However, I know about some environmental models where almost only interactions were observed indeed. So in that case it means that only certain combinations of parameters are “important”, which can be interpreted as follows: the parameters with which you describe your problem may not be the relevant ones for the"real" governing equations / physics of your problem.
Another example I have in mind was related to scattering of electromagnetic waves on buildings, where the windows were parameterized by width w and height h, whereas the important quantity was the surface w \times h.
Question: how did you compute the Sobol’ indices? using Monte Carlo simulation, or using a surrogate model like polynomial chaos expansion or Kriging? If you used the latter, make sure that the surrogate modelling error is small enough (e.g., leave-one-out error smaller than 0.01).
Best regards
Bruno
I compute the Sobol’ indices with the polynomial chaos expansion.
I will try to obtain a leave-one-out error smaller than 0.01 by increasing the sampling number.