Sensitivity analysis is the task of evaluating the sensitivity of a model output Y to input variables (X1,…,Xp). Quite often, it is assumed that this output is related to the input through a known function f :Y= f(X1,…,Xp).
Sobol indices are generalizing the coefficient of the coefficient of determination in regression. The ith first order indice is the proportion of the output variance that is “due” to the ith input variable. Numerous details and references are found e.g. in this paper. In particular, the mentionned paper defines an estimation procedure that we use here.
This estimation procedure relies on the use of montecarlo estimates where computation speed and precision of the estimate are highly connected. Let us show how to compute these indices efficiently in c++ while keeping the code simple.
To fix idea we use a “test” function f that was introduced in the First papers of Sobol around sensitivity indices estimation, see e.g. his 1993 paper “Sensitivity analysis for non-linear mathematical models”. This function is called “the Sobol Function”.
Here is a cpp implementation of sobol function and of the Sobol indices estimation procedure: