First order, 99 Sensitization

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

The reaction set presented in Table XV also was subjected to a full sensitivity analysis in order to determine the rank order of the reactions in the mechanism. For this, normalized first-order sensitivity gradients (Sy) were calculated along the flame using the following definition ... 

Gao, D W. R. Stockwell, and J. B. Milford, First-Order Sensitivity and Uncertainty Analysis for a Regional-Scale Gas-Phase Chemical Mechanism, J. Geophys. Res., 100, 23153-23166 (1995). 

This parameter controls the option of calculating the first-order sensitivity functions. 

Seefeld, S. and W.R. Stockwell First-order sensitivity analysis of models with time-dependent parameters An application to PAN and ozone, Atmoi. Environ. 33 (1999) 2941-2953. 

Moving Beyond First-Order Sensitivity Analysis... 

The formulas involving thermodynamic properties, however, are somewhat different because the formula for calculating a thermodynamic quantity differs from that for calculating an ensemble-averaged quantity. For example, the first-order sensitivity coefficient relating the Helmholtz free energy A of a biomolecular system to a potential parameter is expressed in the form... 

Because first-order sensitivity coefficients are easier to calculate than higher order sensitivity coefficients, it is likely that the former may be used more frequently in guiding molecular design. However, first-order sensitivity theory can provide reliable predictions only when the sensitivities of the properties of interest are approximately linear with respect to the model parameters. This linear response limit is satisfied when the perturbations of model parameters are small. For certain applications, such as in protein engineering where one amino acid is mutated into another, the linear response approximation may fail to reliably predict the change in the properties of a protein resulting from a point mutation. It is therefore useful to examine in more detail how well first-order sensitivity theory performs in guiding such predictions. 

The two-dimensional square lattice protein folding model discussed earlier provides a simple basis for probing this issue. The model has the advantage of allowing one to carry out many exact calculations to check the predictions from first-order sensitivity theory. Unlike molecular dynamics or Monte Carlo simulations, there are no statistical errors or convergence problems associated with the calculations of the properties, and their parametric derivatives, of a model polypeptide on a two-dimensional square lattice. 

It is clear from the data in Table 3 that first-order sensitivity theory works best when e. and Ae, are both small. When e. and Ae, are of the order of 2 k, the predictive reliability decreased to 75%. Therefore, first-order sensitivity theory does not always give correct predictions. However, since first-order sensitivity coefficients can usually be calculated more easily than higher order sensitivity coefficients in (bio)molecular simulations, first-order sensitivity coefficients can be used as a preliminary screening tool for suggesting a small number of modifications to a (bio)molecule that may lead to the desired biological effect. More sophisticated (but usually more expensive) calculations and/or suitable experimental studies can then be carried out to sort out from this small number of suggestions those that are more likely to achieve the desired biological effects. If experimentation is easier, the predictions can be tested in the laboratory. 

An obvious extension of first-order sensitivity theory is to develop higher order theories utilizing higher order sensitivity coefficients. For example, some... 

Table 3. Predicted Results of Mutations for a Two-Dimensional Square Lattice Model of Protein Folding Using First-Order Sensitivity Theory"...

Although first-order sensitivity theory is not always reliable in predicting the properties of a structurally modified (bio)molecule, it may be useful as a preliminary classification tool for suggesting a small number of modifications... 

ABSTRACT We present an algorithm named EASI that estimates first order sensitivity indices from given data, hence allowing its use as a post-processing module for pre-computed model evaluations. Ideas for the estimation of higher order sensitivity indices and the computation of regression curves are also discussed. 

Gatelli, D., S. Kucherenko, M. Ratio, and S. Tarantola (2009). Calculating first-order sensitivity measures A benchmark of some recent methodologies. Reliability Engineer-ing System Safety 94,1212 1219. 

More accurate evaluation of sensitivity indices can be achieved by variance-based sensitivity analysis (Sobol indices). First order sensitivity indices are defined as ... 

Figure 7. First order sensitivity indices for variable m,...

Sensitivity Analysis. To a great extent, the risk associated with the variability of a given parameter is dependent on the effect that a change in that parameter has on the profitability criterion of interest. For the sake of this discussion, the NPV will be used as the measure of profitability. However, this measure could just as easily be the DCFROR, DPEP, or any other profitability criterion discussed in Section 10.2. If it is assumed that the NPV is affected by n parameters (x, X2, X3,. .., x ), then the first-order sensitivity to parameter x is given in mathematical terms by the following quantity ... 

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,. 

Gao, D., Stockwell, W.R., Milford, J.B. First-order sensitivity and uncertainty analysis for a regional-scale gas-phase chemical mechanism. J. Geophys. Res. Atm. 100, 23153-23166... 

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