Derivative based global sensitivity measures and their link with global sensitivity indices

A model function f(x1,…,xn) defined in the unit hypercube Hn with Lebesque measure dx = dx1…dxn is considered. If the function is square integrable, global sensitivity indices provide adequate estimates for the influence of individual factors xi or groups of such factors. Alternative estimators that require less computer time can also be used. If the function f is differentiable, functionals depending on ∂f/∂xi have been suggested as estimators for the influence of xi. The Morris importance measure modified by Campolongo, Cariboni and Saltelli μ  * is an approximation of the functional μi=∫Hn|∂f/∂xi|dxμi=∫Hn∂f/∂xidx.In this paper a similar functional is studiedνi=∫Hn∂f∂xi2dxEvidently, μi≤νi, and νi≤Cμiνi≤Cμi if |∂f/∂xi|≤C∂f/∂xi≤C. A link between νi and the sensitivity index Sitot is established:Sitot≤νiπ2Dwhere D is the total variance of f(x1,…,xn). Thus small νi imply small Sitot, and unessential factors xi (that is xi corresponding to a very small Sitot) can be detected analyzing computed values ν1,…,νn. However, ranking influential factors xi using these values can give false conclusions.Generalized Sitot and νi can be applied in situations where the factors x1,…,xn are independent random variables. If xi is a normal random variable with variance σi2, then Sitot≤νiσi2/D.