We will consider our all purpose model
for the two slightly different cases we have presented previously:
Case 1: Additive
Case 2: Non-additive (interaction)
For the additive model represented by Case 1:
a) Compute the first order Sobol indices analytically:
b) Show that $ \sum_i \left( S_{Z_i}^{\sigma} \right )^2 = 1$ for Case 1.
c) Show that $S_{T_i} = S_{Z_i}$ for the additive model, i.e. that the variance of the model output is completely described by the first order indices.
a) Compute analytically the first order Sobol indices for the model defined by Case 2.
b) Compute the second order indices for the interaction model
c) Show that interaction indices are zero for order 3 and a above for Case 2.
d) Compute the total sensitivity indices $S_{T_i}$ for Case 2.