Local uncertainty analysis

The local sensitivity matrix S shows the effect of a unit change of parameter values on the model results. This can provide useful information on the relative influence of parameters close to their nominal values and it may also be useful to estimate how uncertainty in these parameter values can propagate to predictive uncertainty in model outputs. The normalised sensitivity matrix S shows the effect of a unit relative (e.g. 1 %) change of the parameters. If we assume that the uncertainty of the parameters is known and can be characterised by the covariance matrix then local uncertainty analysis is based on the application of the Gaussian error propagation rule  

This means that the covariance matrix y of the model solution vector Y can easily be calculated from the local sensitivity matrix S. 

For a single model result 7, depending on two parameters and X2 the corresponding equation is 

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Local Uncertainty Analysis of Reaction Kinetic Models... 

Atherton et al. (1975) provided an early example of the application of local uncertainty analysis in the chemical engineering literature. They calculated the variance of the output of dynamic models from the variance of the parameters cP ixj) and the local sensitivity coefficients dYJdxf. 

Using local uncertainty analysis, the variance of the model solution F, can be calculated from the variance <7 (ln A y ) of uncorrelated parameters (Turanyi et al. 2002 Zador et al. 2005b) ... 

The standard deviations of the model results were calculated using local uncertainty analysis as well as via Monte Carlo analysis with Latin hypercube sampling. For the Monte Carlo analysis, the parameters were assumed to be independent random variables with normal distributions. More precisely, truncated normal... 

The uncertainty analysis that is a part of formal EcoRA methodology is designed to ensure adequate estimation of ecological effects based on a state-of-the-art scientific basis. Moreover, if applied on a local scale for site-specific assessments, with the use of empirical input data as biogeochemical parameters, the CLL approach is likely to provide results with a higher degree of confidence than the formal EcoRA model. 

Local sensitivity analysis is of limited value when the chemical system is non-linear. In this case global methods, which vary the parameters over the range of their possible values, are preferable. Two global uncertainty methods have been used in this work, a screening method, the so-called Morris One-At-A-Time (MOAT) analysis and a Monte Carlo analysis with Latin Hypercube Sampling (Saltelli et al., 2000 Zador et al., submitted, 20041). The analyses were performed by varying rate parameters, branching ratios and constrained concentrations within their uncertainty interval,... 

The rate of production analysis was complemented by a local sensitivity analysis and by a global Morris screening analysis. These analyses demonstrate the necessity of accurate measurements of j(0 D) and [HCHO] and reduced uncertainty in the quantum yields for H from HCHO photolysis. 

The sensitivity analysis of a system of chemical reactions consist of the problem of determining the effect of uncertainties in parameters and initial conditions on the solution of a set of ordinary differential equations [22, 23], Sensitivity analysis procedures may be classified as deterministic or stochastic in nature. The interpretation of system sensitivities in terms of first-order elementary sensitivity coefficients is called a local sensitivity analysis and typifies the deterministic approach to sensitivity analysis. Here, the first-order elementary sensitivity coefficient is defined as the gradient... 

If, instead of the nonlocal Fourier analysis, one uses the local wavelet analysis to represent a quantum particle, the uncertainty relationships may change in form. On the other hand, this process has the advantage of containing the usual uncertainty relations when the size of the basic gaussian wavelet increases indefinitely. 

From Fig. 18 it is seen that the new the uncertainty relations derived with the local wavelet analysis exhibit the form... 

Local sensitivity information has numerous applications in uncertainty analysis, parameter estimation, experimental design, mechanism investigation and mechanism reduction. Uncertainty analysis, a quantitative study of the effect of parameter uncertainties on the solution of models, is... 

Application of formal uncertainty analysis to combustion systems has been very rare so far, and restricted to the utilization of local sensitivities. 

Here are the local sensitivity coefficients (Turanyi 1990). This approach has been used, for example, for the uncertainty analysis of the RADM2 tropospheric chemical mechanism (Gao et al. 1995). 

Zador, J., Zsely, I.G., Turanyi, T., Ratto, M., Tarantola, S., Saltelli, A. Local and global uncertainty analyses of a methane flame model. J. Phys. Chem. A 109, 9795-9807 (2(X)5b) Zador, J., Turanyi, T., Wirtz, K., Pilling, M.J. Uncertainty analysis backed investigation of chamber radical sources in the European Photoreactor (EUPHORE). J. Atmos. Chem. 55, 147-166 (2006a)... 

Zador, J., Zsely, I.G., Turanyi, T. Local and global uncertainty analysis of complex chemical kinetic systems. Reliab. Eng. Syst. Saf. 91, 1232-1240 (2006b)... 

In some cases Bragg s Law has been used in place of Eq. (6.35), to analyze the peak in the SAXS curve [67]. While this does provide information about the local order of the micelles, information, such as micelle size, is not provided. In other cases, the form factor for an ellipsoid has been used instead to resolve micelle size [68,69]. In these cases, not all the micelle information was resolved from the SAXS data. However, for large micelles a significantly wide q-range is required to resolve the micelle size and individual chain dimensions using the appropriate form factor [28]. With a limited available g-range, this is just one example of how sometimes the simplest model is best. As stated previously, an uncertainty analysis on the fit parameters can also be useful. 

The usual uncertainty relations are a direct mathematical consequence of the nonlocal Fourier analysis therefore, because of this fact, they have necessarily nonlocal physical nature. In this picture, in order to have a particle with a well-defined velocity, it is necessary that the particle somehow occupy equally all space and time, meaning that the particle is potentially everywhere without beginning nor end. If, on the contrary, the particle is perfectly localized, all infinite harmonic plane waves interfere in such way that the interference is constructive in only one single region that is mathematically represented by a Dirac delta function. This implies that it is necessary to use all waves with velocities varying from minus infinity to plus infinity. Therefore it follows that a well-localized particle has all possible velocities. 

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