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

摘要

A model function f(x_1……x_n) defined in the unit hypercube H" with Lebesque measure dx = dx_1...dx_n is considered. If the function is square integrable, global sensitivity indices provide adequate estimates for the influence of individual factors x, or groups of such factors. Alternative estimators that require less computer time can also be used. If the function/is differentiable, functional depending on af/ax_i, have been suggested as estimators for the influence of x_i . The Morris importance measure modified by Campolongo, Cariboni and Saltelli μ* is an approximation of the functional μ_i = ∫_H ~n|af/ax_i|dx.rnIn this paper a similar functional is studiedrnv_i=∫_H ~n(af/ax_i)~2dxrnEvidently, μ_i ≤ v~(1/2) and v_i,≤ Cμ_ i if|af/ax_i| ≤ C. A link between v_i and the sensitivity index S~(101) _i is established:rnS~(101) _i≤v_i/π~2Drnwhere D is the total variance of f(x_1……x_n). Thus small v_i imply small S~(101) _i, and unessential factors x_ i (that is x_ i corresponding to arnvery small S~(101) _i) can be detected analyzing computed values v_i……v_n. However, ranking influential factors x_ i using these values canrngive false conclusions.rnGeneralized S~(101) _i and v_i can be applied in situations where the factors x_1……x_n are independent random variables. If v_i is a normal random variable with varianceσ~2 _i , then S~(101) _i) ≤ v_iσ~2 _i/D.