INDEX sensitivity analysis

A global sensitivity analysis was performed for the lead concentration in the arterial blood model (Fig. 7) over the simulation period for each parameter. Parameters considered for the sensitivity analysis are listed in Table 8 (Annex 1). The magnitude of sensitivity is shown by relative sensitivity index. It was observed that the most influential parameter is the porosity of the sediment of the river (phi sed)... 

Examples of mathematical methods include nominal range sensitivity analysis (Cullen Frey, 1999) and differential sensitivity analysis (Hwang et al., 1997 Isukapalli et al., 2000). Examples of statistical sensitivity analysis methods include sample (Pearson) and rank (Spearman) correlation analysis (Edwards, 1976), sample and rank regression analysis (Iman Conover, 1979), analysis of variance (Neter et al., 1996), classification and regression tree (Breiman et al., 1984), response surface method (Khuri Cornell, 1987), Fourier amplitude sensitivity test (FAST) (Saltelli et al., 2000), mutual information index (Jelinek, 1970) and Sobol s indices (Sobol, 1993). Examples of graphical sensitivity analysis methods include scatter plots (Kleijnen Helton, 1999) and conditional sensitivity analysis (Frey et al., 2003). Further discussion of these methods is provided in Frey Patil (2002) and Frey et al. (2003, 2004). 

Monitoring and control of CMP slurry properties is essential for effective and uniform CMP processes. Bench-top blend sensitivity analysis helps in identification of the most sensitive blend monitoring and control parameter. Two- and three-component blends of CMP slurries can be created and monitored based on the measurements of density, wt% solids, refractive index, pH, and oxidizer level. Typical silica oxide slurry blend ratio is controlled using density as a control parameter, whereas tungsten and copper CMP slurries usually need an autotitrator for periodic monitoring of the oxidizer level. 

The refractive index sensitivity affects gradient analysis and should be checked by measuring the absorbance when the cell is filled with methanol (n = 1.329) and cyclohexane (n = 1.427) at 270 nm. 

More importantly, as written, the partial derivative implies that a single rate constant is to be varied, holding all the others constant, and indeed this is the way it is implemented in many sensitivity analysis routines. The index y runs out to 2Areactions because each reaction has two rate constants, one for the forward direction, and one for the reverse. However, in order for the model to remain consistent with the laws of thermodynamics, the rate constant for the reverse of reaction y must vary simultaneously with the forward reaction, since the two rate constants must maintain a detailed-balance ratio related to the AGreaction, Eq. (2). This can be assured by specifying that the partial derivative is taken only for the forward reaction, while holding the thermochemistry fixed. Note that this also cuts the number of partial derivatives to be computed in half. These sensitivities should then be supplemented with sensitivities to the individual species thermochemistry as in Eq. (25) overall the number of partial derivatives to be computed per model prediction M, is... 

With the complexity of modern pharmacokinetic-pharmacodynamic models, analytical derivation of sensitivity indexes is rarely possible because rarely can these models be expressed as an equation. More often these models are written as a matrix of derivatives and the solution to finding the sensitivity index for these models requires a software package that can do symbolic differentiation of the Jacobian matrix. Hence, the current methodology for sensitivity analysis of complex models is empirical and done by systematically varying the model parameters one at a time and observing how the model outputs change. While easy to do, this approach cannot handle the case where there are interactions between model parameters. For example, two... 

If the causal model did not include the identifiability assumptions, then it is unknown if /q equals the desired causal effect ipo- A variety of sensitivity analyses could be employed, which involve posing a new causal model that still allows identification of the desired causal effect but that represents a deviation from the original causal model. Such a causal model is indexed by a sensitivity parameter a that is assumed known, and for each a-specific causal model, one redevelops the identifiability result and estimator with corresponding statistical inference (Robins et al., 1999 Scharfstein et al., 1999 Rotnitzky et al., 2001). In a recent article by Dfaz and van der Laan (2013), we develop a much simpler sensitivity analysis that simply defines the sensitivity parameter as the bias il o - o or an upper bound thereof, and for each plausible value of this bias, it reports the estimator and possibly conservative confidence interval for the causal effect rpS. The latter method relies on fewer assumptions, and does not involve any extra work. 

Schmidtlein, M. Deutsch, R. Piegorsch,W. Cutter Susan. 2008. A sensitivity artalysis of the social vulnerability index. Risk Analysis, Vol.28, n.°4, 1099-1114. 

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... 

In plasmon resonance gas sensors, which also usually operate at room temperatures, we have another situation. In these devices, in order to shift the operating point of the surface plasmon resonance toward an aqueous environment, a thin high-refractive index dielectric overlayer can be employed. The use of overlayers with higher refractive indexes allow for thinner overlayers and potentially better sensor sensitivity. Analysis of results obtained indicates that tantalum pentoxide, which has high refractive index (see Table 2.18) and good environmental stability, may be used for this purpose (Dostalek et al. 2001 Boozer 2003). 

Reliability of the pile is analyzed considering the pile length / as a study parameter. A sensitivity analysis investigates effects of the coefficients of variation of C , and Q on the reliability index p. 

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 ... 

Many factors influence the accuracy of experimental data and each experimental run could be described by a different set of parameters n, K, Ty. Since the flow models strongly depend on input data and their evaluation, a sensitivity analysis was used to find effect of value of flow behaviour index n on accuracy of laminar and turbulent flow models. Dependency of slurry/water pressure gradient ratio i / io on mean slurry velocity V of the measured slurries for both tested turbulent models and yield power-law model is shown in Fig. 4, where also a role of parameter n is illustrated. The value of n given by best fitting of laminar data by Eq. (4) represents quite well laminar region. For turbulent data it is not valid (see dashed line). The best fitting value n for turbulent data varies not only with kind of solid material, but depends also on concentration. 

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. 

Saltelli, A., Annoni, P., Azzini, I., Campolongo, F., Ratto, M., Tarantola, S. Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Comp. Phys. Commun. 181, 259-270 (2010)... 

The Management Index (e.g.. Risk Index x Sensitivity Index) provides further ranking for those risks with equal Risk Indexes. Given its scope, sensitivity analysis may not necessarily constitute an integrated step of the risk analysis process. 

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