Uncertainties of parameters

In this chapter we investigate the interaction between experimental design and information quality in two-factor systems. However, instead of looking again at the uncertainty of parameter estimates, we will focus attention on uncertainty in the response surface itself. Although the examples are somewhat specific (i.e., limited to two factors and to full second-order polynomial models), the concepts are general and can be extended to other dimensional factor spaces and to other models. 

Inappropriate methodology of determinations Uncertainties of parameters determined in separate measurements and used in calculating the final result, such as physicochemical constants... 

The local concentration sensitivity matrix shows the effect of equal perturbations of parameters. This means equal unit perturbation in the units of each parameter for the case of the original sensitivity matrix, and unit fractional (percentage) perturbation in the case of the normed sensitivity matrix. The local sensitivity matrix itself does not carry any information on the uncertainty of parameters. 

Possible reasons are the limited knowledge of the uncertainty of parameters, and the fact that global sensitivity methods are computationally very intensive. In the future, it is expected that both these limitations will be lifted and detailed uncertainty analyses will appear for combustion calculations. On the other hand, one of the main applications of sensitivity analysis has been to form a qualitative picture about which parameters should be known precisely in order to reproduce accurately a set of experimental observations. 

Reliability of the model requires that the model be assessed for the uncertainty of parameters and random effects. We are interested in the standard errors of estimated parameters and random effects in the model. The uncertainty should be small for parameters, uncertainty should be less than 25% of the relative standard error and for random effects, it should be less than 35% of the relative standard error (25). 

Due to the fact that the model consisting of objective function (5.17) and constraints contains random variables makes the max and the constraints have no complete mathematical sense, we need to transform the model accordingly. Considering the reliability of the actual decision-making environment and the uncertainty of parameters, this book wiU use the chance-constrained programming theory to... 

In most uncertainty studies published so far (see e.g. Brown et al. (1999), Turanyi et al. (2002), Zsely et al. (2005), Zador et al. (2005a, b, 2006a) and Zsely et al. (2008)), where the uncertainties of the rate coefficients were utilised, the uncertainty of k was considered to be equal to the xmcertainty of the pre-exponential factor A. This implies that the uncertainty of parameters E and n is zero, which is an unrealistic assumption. In a global sensitivity analysis study of a turbulent reacting atmospheric plume, Ziehn et al. (2009a) demonstrated the importance of uncertainties in EIR for the reaction N0 + 03 = N02 + 02. In this case for the prediction of mean plume centre line O3 concentratiOTis, the sensitivity to the assumed value for EIR was almost a factor of 20 higher than that of the A-factor, based on input parameter uncertainty factors provided by the evaluation of Androulakis (2004, 2004). However, in this case the parameters of the Anhenius expression for the chemical reactions considered were allowed to vary independently. In fact, the characterisation of the joint uncertainty of the Arrhenius parameters is important for the reahstic calculatiOTi of the uncertainty of chemical kinetic simulation results as will be discussed in the next section. 

As discussed in Sect. 5.6.2, a full evaluation of the input uncertainties to a model should, where relevant, provide information on the correlations between input parameters. This can be represented through the joint probability distribution of the parameters or through a covariance matrix Sp such as that shown in Eq. (5.68). The joint probability distribution of model parameters can be determined from experimental data using the Bayes method (Berger 1985). Kraft et al. (Smallbone et al. 2010 Mosbach et al. 2014), Braman et al. (2013) and Miki et al. (Panes et al. 2012 Miki et al. 2013) have calculated the p< of rate parameters from experimental data. The covariance matrix of the rate parameters was calculated from the back propagation of experimental errors to the uncertainty of parameters by Sheen et al. (Sheen et al. 2009,2013 Sheen and Wang 201 la, b) and by [Turanyi... 

In this chapter various methods applicable for sensitivity and uncertainty analyses were reviewed, and the usual definitions of uncertainty information, as given in chemical kinetic databases, were summarised. The uncertainty of chemical kinetic models, calculated from the uncertainty of parameters, was presented, for examples of simulations of a methane flame model. In this section the general features of the various uncertainty analysis methods are reviewed and some general conclusions are made. 

The set of uncertain input parameters is abbreviated by V Q,R), neglecting further properties. For example, 2 (M, [0,1]) is a set of fuzzy numbers. The variable p" V Q,R) contains all uncertain parameters. The uncertainty of parameters can be observed for actions and resistance. This means that, e.g., dead loads, earthquake accelerations, or material parameters cannot be described by a deterministic value. 

Unfortunately, many commonly used methods for parameter estimation give only estimates for the parameters and no measures of their uncertainty. This is usually accomplished by calculation of the dependent variable at each experimental point, summation of the squared differences between the calculated and measured values, and adjustment of parameters to minimize this sum. Such methods routinely ignore errors in the measured independent variables. For example, in vapor-liquid equilibrium data reduction, errors in the liquid-phase mole fraction and temperature measurements are often assumed to be absent. The total pressure is calculated as a function of the estimated parameters, the measured temperature, and the measured liquid-phase mole fraction. 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

The diagonal elements of this matrix approximate the variances of the corresponding parameters. The square roots of these variances are estimates of the standard errors in the parameters and, in effect, are a measure of the uncertainties of those parameters. 

If there is sufficient flexibility in the choice of model and if the number of parameters is large, it is possible to fit data to within the experimental uncertainties of the measurements. If such a fit is not obtained, there is either a shortcoming of the model, greater random measurement errors than expected, or some systematic error in the measurements. 

In many process-design calculations it is not necessary to fit the data to within the experimental uncertainty. Here, economics dictates that a minimum number of adjustable parameters be fitted to scarce data with the best accuracy possible. This compromise between "goodness of fit" and number of parameters requires some method of discriminating between models. One way is to compare the uncertainties in the calculated parameters. An alternative method consists of examination of the residuals for trends and excessive errors when plotted versus other system variables (Draper and Smith, 1966). A more useful quantity for comparison is obtained from the sum of the weighted squared residuals given by Equation (1). 

The epipolar constrains calculated using the estimated camera parameters restrict the search for corresponding image features in different images to a ID search. Taking the uncertainty of the epipolar constrains into account, in our approach, the search is restricted to a small area around the epipolar lines in the images. 

The uncertainties of the minimization parameters are calculated just as they were for the linear case except that now there are three of them... 

Model Development PreHminary modeling of the unit should be done during the familiarization stage. Interactions between database uncertainties and parameter estimates and between measurement errors and parameter estimates coiJd lead to erroneous parameter estimates. Attempting to develop parameter estimates when the model is systematically in error will lead to systematic error in the parameter estimates. Systematic errors in models arise from not properly accounting for the fundamentals and for the equipment boundaries. Consequently, the resultant model does not properly represent the unit and is unusable for design, control, and optimization. Cropley (1987) describes the erroneous parameter estimates obtained from a reactor study when the fundamental mechanism was not properly described within the model. 

It is clear that a large number of parameters influence the formation of the sheets in a tubule and that their relative importance is still unknown, as is also the cause of the occurrence of the defects responsible for the eventual polygonization of the sheets. Although the model presented here highlights the necessity of including, as one of the parameters, the uncertainty 8r or dd on the separation of successive cylindrical sheets, it is impossible to predict with absolute certainty the final characteristics of any of these sheets, symmetric or not, on the basis of the characteristics of the previous one. Nevertheless, a number of features of their structure, such as the presence or absence of helicity, and the presence of groups of sheets with nearly the same angle of pitch, can be explained and quantified. 

If a measurement is repeated only a few times, the estimate for the distribution variance calculated from this sample is uncertain and the tiornial distribution cannot be applied. In this case another distribution is used, f his distribution is Student s distribution or the /-distribution, and it has one more parameter the number of degrees of freedom, t>. The /-distribution takes into account, through the p parameter, the uncertainty of the variance. The values of the cumulative /-distribution function cannot be evaluated by elementary methods, and tabulated values or other calculation methods have to be used. 

It can be argued that the main advantage of least-squares analysis is not that it provides the best fit to the data, but rather that it provides estimates of the uncertainties of the parameters. Here we sketch the basis of the method by which variances of the parameters are obtained. This is an abbreviated treatment following Bennett and Franklin.We use the normal equations (2-73) as an example. Equation (2-73a) is solved for <2o-... 

We also want estimates of the uncertainties of the parameters Equations (2-99) and (2-101), repeated here, provide these. 

Zintl Hauke cited no limits of error, but Ketelaar cited an uncertainty of 0-005 A for a0 and an uncertainty of 0-003 in y and z. From comparisons among the intensities obtained for the five compounds that they investigated, Zintl Hauke concluded that the y and z parameters are substantially the same for all five. 

In experiments related to flow and heat transfer in micro-channels, some parameters, such as the flow rate and channel dimensions are difficult to measure accurately because they are very small. For a single-phase flow in micro-channels the uncertainty of ARe is (Guo and Li 2002,2003)... 

The approach to standardization used by Haaijman (53) and others (66,67), in which the fluorophor is incorporated within or bound to the surface of a plastic sphere, is more versatile than the use of inorganic ion>doped spheres, since the standard can be tailored exactly to the specifications required by the analyte species. However, this approach increases the uncertainty of the measurement because the photobleaching characteristics of both the standard and the sample must be considered. The ideal approach is to employ both types of standards. The glass microspheres can be used to calibrate instruments and set instrument operating parameters on a day-to-day basis, and the fluorophor-doped polymer materials can be used to determine the concentration-instrument response function. 

Example 57 The three files can be used to assess the risk structure for a given set of parameters and either four, five, or six repeat measurements that go into the mean. At the bottom, there is an indicator that shows whether the 95% confidence limits on the mean are both within the set limits ( YES ) or not ( NO ). Now, for an uncertainty in the drug/weight ratio of 1%, a weight variability of 2%, a measurement uncertainty of 0.4%, and fi 3.5% from the nearest specification limit, the ratio of OOS measurements associated with YES as opposed to those associated with NO was found to be 0 50 (n == 4), 11 39 (n = 5), respectively 24 26 (u = 6). This nicely illustrates that it is possible for a mean to be definitely inside some limit and to have individual measurements outside the same limit purely by chance. In a simulation on the basis of 1000 sets of n - 4 numbers e ND(0, 1), the Xmean. Sx, and CL(Xmean) were calculated, and the results were categorized according to the following criteria ... 

---