Second-order sensitivity index

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. ... 

This second-order sensitivity index shows the fraction of the total variance of 7, which is reduced when two parameters xj and x/ are fixed. In other words, the second-order sensitivity index characterises the interaction of the corresponding parameters. The sensitivity indices can be calculated up to an arbitrary n-th order by keeping n parameters fixed, but the computational time requirement increases exponentially with the order. 

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,. 

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 SHG efficiency depends critically on the phase integral (Eq. 17), as already mentioned. Ideally, phase matching is achieved when the phase mismatch A/3 is zero. This mismatch, however, depends on several parameters, e.g., the waveguide thickness h and width W, the refractive index , and the temperature T. In real devices, each of these parameters p is subject to fluctuations, causing the efficiency to drop. In order to investigate the sensitivity of the phase matching condition and calculate acceptable tolerances, the mismatch is expanded into a Taylor series to second order around the phase matching point... 

An example of the second type of detector is the refractive index monitor which functions by recording the refractive changes in the eluant as the solute passes through the detector cell. Bulk property detectors, though more versatile, are generally several orders of magnitude less sensitive than specific property detectors and the choice for a particular application is often dictated by solute characteristics. 

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