Parameter uncertainties

Uncertainty Represents a lack of knowledge about factors affecting exposure or risk and can lead to inaccurate or biased estimates of exposure. The types of uncertainty include scenario uncertainty, parameter uncertainty and model uncertainty (USEPA, 1997c). 

The code does not include parameters for uncertainty. Parameter ZDVSL noted in chapter text is termed BZDV in the NONMEM code. 

Uncertainty Parameter associated with the result of a measurement, which characterizes the dispersion of the values that could reasonably be attributed to the measurand. 

Generally, the uncertainty in model predictions may derive from parameter or data uncertainties as well as from model uncertainties. Parameter or data uncertainty is defined as the lack of knowledge on parameter values or input data. Model uncertainty denotes uncertainty on the concept (accuracy, structure, completeness, suitability) of the model, on the mathematical model with its simplifications and approximations, and on the numerical model with its discretizations, coding errors, etc. (Gallegos Bonano 1993). 

In the IAEA data analysis, various statistical techniques (IAEA 1998a) are used to derive separate estimates of the operator s and inspector s uncertainty parameters based on the collection of historical operator-inspector differences. The results of these evaluations are performance values, typical for a specific facility and for each stratum (material type) and measurement method combination. The actually observed verification measurement performance is then used for the planning (sample size calculations), the conduct (establishing reject limits), and the evaluation (material balance) of inspections in a given facility. 

There are many different ways to treat mathematically uncertainly, but the most common approach used is the probability analysis. It consists in assuming that each uncertain parameter is treated as a random variable characterised by standard probability distribution. This means that structural problems must be solved by knowing the multi-dimensional Joint Probability Density Function of all involved parameters. Nevertheless, this approach may offer serious analytical and numerical difficulties. It must also be noticed that it presents some conceptual limitations the complete uncertainty parameters stochastic characterization presents a fundamental limitation related to the difficulty/impossibility of a complete statistical analysis. The approach cannot be considered economical or practical in many real situations, characterized by the absence of sufficient statistical data. In such cases, a commonly used simplification is assuming that all variables have independent normal or lognormal probability distributions, as an application of the limit central theorem which anyway does not overcome the previous problem. On the other hand the approach is quite usual in real situations where it is only possible to estimate the mean and variance of each uncertainty parameter it being not possible to have more information about their real probabilistic distribution. The case is treated assuming that all uncertainty parameters, collected in the vector d, are characterised by a nominal mean value iJ-dj and a correlation =. In this specific... 

The fundamental point developing a MOOP is that of evaluating the mean and variance of a conventional objective function. A widely nsed solntion for this problem, referred to the generic stochastic structural response R d) which depends on the uncertainty parameter vector d, is the linear approximation of the DPM that furnishes the mean value and variance ... 

The above result shows that the optimal solution, obtained by minimizing the expected value of the OF (the mean), is quite sensitive to the fluctuation of the uncertainty parameters, as demonstrated by the corresponding high values of variance. 

To perform the robust optimum design, the OF mean and standard deviation are numerically evaluated with a new procedure based on a Lyapunov type equation. Robustness is formulated as a multiobjective optimization problem, in which both the mean and the standard deviation of the deterministic OF are minimized. The results show a significant improvement in performance control and OF real values dispersion limitation if compared with standard conventional solutions. Some interesting conclusions can be reached with reference to the results obtained for the adopted examples. With reference to TMD efficiency in vibration reduction, the real structural performance obtained by using conventional optimization has a reduced efficiency compared to those obtained when system uncertainty parameters is properly considered. With reference to the obtained robust solutions, it can be noted that they can control and limit final OF dispersion by limiting its standard deviation. Moreover, this goal is achieved by finding optimal solutions in terms of DV that induce an increase in OF mean value. 

There are two types of lead-time uncertainty parameters, representing the domestic order placing lead time f(t) and domestic order delay lead time l/(t) in the domestic SC and two types of lead-time uncertainties in the ISC representing the international order placing lead time 1 (1) and international order delay lead time These lead-time parameters may influence the customer orders that the manufacturer actually receives during one period. Note both types of lead time are dynamic and stochastic variables that may be different over time periods. 

In summary, this chapter has presented the mathematical models for the DSC and ISC. There are four sub-models customer order production raw material ordering and transportation and finished goods satisfying customer demand model. The differences between DSC and ISC are identified from several aspects, for example, uncertainty parameters, finished goods on-hand inventory level update and SCP calculation, which exist in the customer order sub-model. 

The use of polynomial chaos expansions for the generation of response surfaces is based on the spectral uncertainty method introduced for combustion models in Reagan et al. (2003, 2004, 2005) and Najm et al. (2009) which was extended to an RSM in, e.g.. Sheen et al. (2009). Here an uncertainty factor m, is first assigned to each input variable. Note that this uncertainty factor m, is related to uncertainty parameter/to be discussed in Sect. 5.6.1 by Mj= 10. Taking the example of rate coefficients, they are then normalised into factorial variables x as follows ... 

Equation (5.54) means that the rate coefficient is uncertain according to a multiplication factor u = 10. Typical values of the uncertainty parameter/are 0.3, 0.5 and 0.7, which means that the extreme values differ from the recommended value by multiplication factors of 2.00, 3.16 and 5.01, respectively (see Table 5.1). This xmcertainty parameter/has been defined for a range of gas-phase systems by a series of researchers including Wamatz (1984), Tsang and Hampson (1986), Tsang (1992), Baulch et al. (1992, 1994, 2(X)5) and Konnov (2008). The specification of /allows the calculation of uncertainty ranges which may be used within the context of the global sensitivity methods described in the previous section. 

Uncertainty parameter/ Multiplication factor u Multiplication factor corresponding to 1Multiplication factor corresponding to 2[Pg.103]

Table 5.1 shows the conversion of the uncertainty parameter/to other representations of the uncertainty of the rate coefficient The second colunrn shows, e.g., that an/value of 0.3 means that the rate coefficient is uncertain according to a factor of 2, that is, up to 200 % and down to 50 % of the recommended value is also possible. The table also shows that/= 0.1 and/= 0.3 (frequently adopted values of the uncertainty parameter used for the characterisation of well-known rate coefficients) approximately correspond to 8 % and 26 % uncertainty of the rate coefficient at the lo level but multiplication factors of 1.26 and 2.00 at the 3o level, respectively. 

The JPL databases define the temperature dependence of uncertainty parameter /jpl(T) using the following equation ... 

The uncertainty parameter defined this way is also a piecewise linear functiiMi of and this uncertainty has a minimum at temperature To = 298 K. The upper and lower limits belonging to the standard deviation (lrecommended value of the rate coefficient by the parameter... 

---