Uncertainty in parameters

With the various period-doubling sequences seen above, however, the parameter ranges over which some of the higher-order periodicities exist can become very small. If we cannot control our conditions within these ranges, we may not be able to sustain these patterns. 

As we have already commented, mappings of the type discussed above are not in any way easily related to a given set of reaction rate equations. Such mappings have, however, been used for chemical systems in a slightly different way. A quadratic map has been used to help interpret the oscillatory behaviour observed in the Belousov-Zhabotinskii reaction in a CSTR. There, the variable x is not a concentration but the amplitude of a given oscillation. Thus the map correlates the amplitude of one peak in terms of the amplitude of the previous excursion. 

With this identification, the stable stationary-state behaviour (found for the cubic model with 1 A 4) corresponds to oscillations for which each amplitude is exactly the same as the previous one, i.e. to period-1 oscillatory behaviour. The first bifurcation (A = 4 above) would then give an oscillation with one large and one smaller peak, i.e. a period-2 waveform. The period doubling then continues in the same general way as described above. The B-Z reaction (chapter 14) shows a very convincing sequence, reproducing the Feigenbaum number within experimental error. 

Mapping techniques, and the associated bifurcation analyses, are also of great importance when applied with the Poincare map described in the appendix to chapter 5. These are used to establish local stability, and changes 

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Determination of confidence limits for non-linear models is much more complex. Linearization of non-linear models by Taylor expansion and application of linear theory to the truncated series is usually utilized. The approximate measure of uncertainty in parameter estimates are the confidence limits as defined above for linear models. They are not rigorously valid but they provide some idea about reliability of estimates. The joint confidence region for non-linear models is exactly given by Eqn. (B-34). Contrary to ellipsoidal contours for linear models it is generally banana-shaped. 

Reactive scheduling is an online procedure which modifies nominal schedules in reaction to the occurrence of an unexpected event. Reactive scheduling is traditionally used to handle short-term uncertainties in parameters as, e.g., processing times, or equipment failures. The underlying-models themselves usually do not incorporate information on the uncertainty. 

The amount of uncertainty in parameter estimates obtained for the hyperbolic models is particularly large. It has been pointed out, for example, that parameter estimates obtained for hyperbolic models are usually highly correlated and of low precision (B16). Also, the number of parameters contained in such models can be too great for the range of the experimental data (W3). Quantitative measures of the precision of parameter estimates are thus particularly important for the hyperbolic models. (Cl). 

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... 

Bayesian statistics are applicable to analyzing uncertainty in all phases of a risk assessment. Bayesian or probabilistic induction provides a quantitative way to estimate the plausibility of a proposed causality model (Howson and Urbach 1989), including the causal (conceptual) models central to chemical risk assessment (Newman and Evans 2002). Bayesian inductive methods quantify the plausibility of a conceptual model based on existing data and can accommodate a process of data augmentation (or pooling) until sufficient belief (or disbelief) has been accumulated about the proposed cause-effect model. Once a plausible conceptual model is defined, Bayesian methods can quantify uncertainties in parameter estimation or model predictions (predictive inferences). Relevant methods can be found in numerous textbooks, e.g., Carlin and Louis (2000) and Gelman et al. (1997). 

These partial derivatives provide a lot of information (ref. 10). They show how parameter perturbations (e.g., uncertainties in parameter values) affect the solution. Identifying the unimportant parameters the analysis may help to simplify the model. Sensitivities are also needed by efficient parameter estimation procedures of the Gauss - Newton type. Since the solution y(t,p) is rarely available in analytic form, calculation of the coefficients Sj(t,p) is not easy. The simplest method is to perturb the parameter pj, solve the differential equation with the modified parameter set and estimate the partial derivatives by divided differences. This "brute force" approach is not only time consuming (i.e., one has to solve np+1 sets of ny differential equations), but may be rather unreliable due to the roundoff errors. A much better approach is solving the sensitivity equations... 

Hence, the estimated values are within a factor of three (chlorothion) and two (pentachlorobenzene), respectively, of the experimental BAFt values. But recall that when dealing with living media, due to the rather large uncertainties in parameter estimation, any predicted values have to be considered good to within factors of... 

Scepanovic O, Bechtel KL, Haka AS, Shih WC, Koo TW, Feld MS. Determination of uncertainty in parameters extracted from single spectroscopic measurements. Journal of Biomedical Optics 2007, 12, 064012. 

A process design is flexible if it can tolerate uncertainties in parameters and can handle disturbances. A flexibility index is a measure of the amount of uncertainty that can be tolerated with the desired process operation remaining feasible. A schematic of the flexibility index, 5, is shown in Fig. 8. Here, the two degrees of freedom represent uncertain parameters or disturbance variables, which have assumed upper and lower bounds. The feasible operating region lies within the cross-hatched area. The flexibility index is the fraction of the parameter range that still results... 

Referring again to Fig. 2 one can see that the effect of mass transport on the rate of benzene hydrogenation is somewhat higher then predicted by model specially at shorter poisoning time. The reason for such behaviour can be found in the inaccuracy involved in the measurements and in the contribution of experimental uncertainty in parameter determination. 

The global sensitivity analysis methods address the problem of the precise calculation of the uncertainty of the model output as a result of uncertainties in parameters. These methods can handle any large uncertainty in the input parameters. More refined techniques include the determination of the extent of the uncertainty in output as a result of the uncertainty of each parameter. 

The major source of uncertainty in parameter determination by EXAFS analysis arises from the correlation that exists between the coordination number (Nj) and the Debye-Waller factor (crj) for each shell. This correlation occurs through the amplitude of the backscattered wave [Eq. (4)] and results in the uncertainty in Nj being 25%. The primary manifestation of is in the relative phases of the outgoing and backscattered waves and, although Rj is strongly correlated to... 

TTie mean of the SPE is now calculated with its SE and the mean SPE (MSPE) with its Cl is constructed. Again this Cl should include 0 and the standard deviation of the SPE should include 1. The above methods may be overly conservative as uncertainty in parameters is not taken into account, resulting in an appropriate model being rejected or declared to have substantive error (28). 

Figures 6.1 and 6.2 show that the number of cross-over points directly available from two experiments is far sparser than the number of points available in a standard TSR experiment and therefore the subsequent calculations will lead to greater uncertainty in parameter estimates. It is possible, however, to employ an interpolation scheme to manufacture new virtual crossing points. This would seem to allow an unlimited in-...

Lofts, S., and E. Tipping. 2011. Assessing WHAM/Model VII against field measurements of free metal ion concentrations Model performance and the role of uncertainty in parameters and inpnts. Environmental Chemistry 8, no. 5 501-516. 

Some issues arise. Risk assessment should consider all available information. Thus, a desirable characteristic of a conceptual framework would be to handle multiple sources of evidence. A nice feature offered by Monte Carlo simulation is that it is possible to account for correlations between input parameters. In its simplest form, these correlations are assigned assuming a pair-wise linear association between parameters. However, there are other ways to capture the correlation between parameters based on the same sources of evidence. Some sources of uncertainty are missed when parameters are predictions from other models. For example the link to the source of evidence is lost when not considering uncertainty in parameters that have been predicted by statistical regression models. 

Figure 1. a) Monte Carlo simulations of population size for alternative A. The dash line indicates a suggested threshold of 45 individuals, b) Probability of quasiextinction under different threshold for alternative A, B and C with median parameters of MCMC, and c) Probability of quasi-extinction (p, ) for alternative A, B and C for a given threshold N = 45) with uncertainty in parameters shown as boxplots. 

In the classical statistical framework uncertainty in parameters (which are perceived as having a true value) is expressed as relative frequencies resulting from drawing random samples from an underlying population. Some argue for a classical statistical approach since it is seen as objective, and objectivity is a valued characteristic for something based on science. However, risk assessment is seldom objective. It can for example be difficult to motivate a mixture of parameters with uncertainty derived by a classical statistical approach and parameters with uncertainty based on expert judgment. 

In order to assess the structural reliability and robustness with respect to deterioration, it is required to select appropriate models which describe the deterioration process. The parameters associated with these models have to be estimated through statistical inference, which introduces uncertainties in parameter estimates. The reliability index which takes into account these uncertainties is the predictive reliability index (Der Kiureghian 2008). As the structural reliability indices which... 

In the evaluation of the model, sensitivity analysis, i.e. the change in model output due to uncertainties in parameter values, is important. 

The effect of RTR variability on the variance of collapse is directly obtained by performing IDAs for a set of ground motions on a deterministic system. The variance of collapse capacity of nonmaterial deteriorating systems is also affected by uncertainty in parameters, such as the yield moment and post-yield hardening ratio, but their effect on the variance of collapse capacity is small compared to that originated by RTR variability. 

Meinrath, G., Ekberg, C, Landgren, A, and Liljenzin, J.-O. (2000) Assessment of uncertainty in parameter evaluation and prediction. Talanta, 51, 231-246,... 

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