First-order sensitivity index

Ilja M. Sobol, a mathematician, worked out the coherent concept of sensitivity analysis, which enables the analysis of the influence of arbitrary subgroups of input factors (doubles, triples, etc.) on a monitored output (Sobol 1993). Sobol s first order sensitivity index may be written as ... 

If the integral in Eq. (5.25) is calculated with a fixed value of a single parameter Xj, then the variance caused by all other parameters except for Xj denoted by V(J, xj) is obtained. If this V(Ji xj) value is calculated for many values of Xj, selected according to its pdf, then the expected value E(y(Xi xj)) can be calculated. This requires the integration of V(Yi Xj) over the pt oi Xj (see Saltelli (2002) for details). The value V(Yi)-E(V(Yi Xj)) is equal to the reduced variance of F, caused by fixing the value of Xj and is equal to V E(Yi xj)). Dividing this conditional variance by the unconditional variance, the first-order sensitivity index for parameter Xj can be calculated ... 

This measure shows the fraction of the total variance of 7, which is reduced when the value of Xj is held at a fixed value and is therefore a measure of the influence of uncertainty in Xj. The first-order sensitivity index is between 0 and 1, although sometimes this is multiplied by 100 yielding 5)(,)%. The calculation of the integrals in Eq. (5.25) is nontrivial and the use of a Monte Carlo sampling method is described in Saltelli (2002) requiring N (2 m+1) model runs where N is the sample size chosen for the Monte Carlo estimates. 

Liidtke et al. (2007) developed a new version of the method above that they call information-theoretic sensitivity analysis. Here the model is considered as a communication channel , which is a transmitter of information between inputs and outputs. Instead of analysing the variance of the ouQjut distribution, they measured output xmcertainty in terms of Shannon s entropy. The first-order sensitivity index, the higher-order sensitivity indices and the total sensitivity index aU have information-theoretic coxmterparts. 

Again, the first-order sensitivity index 5, shows the exclusive effect of parameter x, on the model result. The second-order sensitivity index shows the interaction of parameters x, and x,. 

S, is a measure of the first order (for e.g. additive) effect (the so-called main effect) of A) on the model output Y. The second order sensitivity index Sy is the interaction term (2) between factors A, X which captures that part of the response of Yto A), Xj that cannot be written as a superposition of the separate effects due to A. and A. ... 

As we already noted the choice of the SA methods was left to the participants. In this section we concentrate on the methods used in the second phase of the benchmark where the SA techniques were restricted to variance-based sensitivity analysis methods. The sensitivity index of first order effect is given by... 

The activities were found to correlate with x (the first order valence molecular connectivity index) of P2/P2 substituents attached to the two nitrogens (N1 8c N3) and some indicator parameters. It was suggested [86] that the less polar and more hydrophobic substituent would be beneficial for improving the activities. The authors also found that the translation of the enzyme inhibition activity to antiviral potency is more sensitive for compoimds having dissimilar P2 and P2 substituents. 

Two example applications are described in detail, which address such issues as the convergence of the reliability index for the two methods of random field discretization, the effect of correlation length on the convergence, the influence of correlation length on reliability, reliability sensitivity with respect to distribution parameters, and multiplicity of failure modes. These examples clearly demonstrate the applicability and usefulness of the first>order reliability approach in conjunction with the finite element method for reliability analysis of complex structures. 

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