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

• Authors: I. M. Sobol';S. Kucherenko
• Venue: Mathematics and Computers in Simulation
• Year: 2009

A model function f(x"1,...,x"n) defined in the unit hypercube H^n 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"i 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/@?x"i have been suggested as estimators for the influence of x"i. The Morris importance measure modified by Campolongo, Cariboni and Saltelli @m* is an approximation of the functional @m"i=@!"H"^"n@?f/@?x"idx. In this paper a similar functional is studied@n"i=@!"H"^"n@?f@?x"i^2dxEvidently, @m"i@?@n"i, and @n"i@?C@m"i if @?f/@?x"i@?C. A link between @n"i and the sensitivity index S"i^t^o^t is established:S"i^t^o^t@?@n"i@p^2Dwhere D is the total variance of f(x"1,...,x"n). Thus small @n"i imply small S"i^t^o^t, and unessential factors x"i (that is x"i corresponding to a very small S"i^t^o^t) can be detected analyzing computed values @n"1,...,@n"n. However, ranking influential factors x"i using these values can give false conclusions. Generalized S"i^t^o^t and @n"i can be applied in situations where the factors x"1,...,x"n are independent random variables. If x"i is a normal random variable with variance @s"i^2, then S"i^t^o^t@?@n"i@s"i^2/D.