Assuming a model with a scalar output Y as a function of a given set of m random input parameters X_i

(2–1)

the first order sensitivity index is defined as

(2–2)

where V(Y) is the unconditional variance of the model output and V(Y|X_i) is the variance of Y caused by a variation of X_i only.

Since first order sensitivity indices measure only the decoupled influence of each variable an extension for higher order coupling terms is necessary. Therefore total effect sensitivity indices have been introduced

(2–3)

where V(Y|X_∼i) is the variance of Y caused by all model inputs without X_i .

In order to estimate the first order and total sensitivity indices, a matrix combination approach is very common (Saltelli et al. 2008). This approach calculates the conditional variance for each variable with a new sampling set. In order to obtain a certain accuracy, this procedure requires often more than 1000 samples for each estimated conditional variance. Thus, for models with a large number of variables and time consuming solver calls, this approach can not be applied efficiently.