Uncertainty Modeling

Butler [27] formulated a LFR for smart structures based on measurement errors during the identification process. Based on this uncertainty model, mixed H l /Hoo controllers [29] were designed incorporating actuator saturation. 

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Unlike non-radiometric methods of analysis, uncertainty modelling in NAA is facilitated by the existence of counting statistics, although in principle an additional source of uncertainty, because this parameter is instantly available from each measurement. If the method is in a state of statistical control, and the counting statistics are small, the major source of variability additional to analytical uncertainty can be attributed to sample inhomogeneity (Becker 1993). In other words, in Equation (2.1) ... 

Table 9.1 Uncertainties model of the uncertain demands for four periods.

Gupta/Maranas (2003) as one example for a demand uncertainty model present a demand and supply network planning model to minimize costs. Production decisions are made here and now and demand uncertainty is balanced with inventories independently incorporating penalties for safety stock and demand violations. Uncertain demand quantity is modeled as normally distributed random variables with mean and standard deviation. The philosophy to have one production plan separated from demand uncertainty can be transferred to the considered problem. Penalty costs for unsatisfied demand and normally distributed demand based on historical data... 

Holloway, C. "Decision Making Under Uncertainty Models and Choices" Prentice-Hall Englewood Cliffs, N.J., 1979. 

Skogestad and Morari recommend the use of uncertainty models for the design of robust controllers. The idea is easy to visualize for an SISO system. Suppose we have a process with the following openloop transfer function ... 

It is desirable to demonstrate that the proposed stochastic formulations provide robust results. According to Mulvey, Vanderbei, and Zenios (1995), a robust solution remains close to optimality for all scenarios of the input data while a robust model remains almost feasible for all the data of the scenarios. In refinery planning, model robustness or model feasibility is as essential as solution optimality. For example, in mitigating demand uncertainty, model feasibility is represented by an optimal solution that has almost no shortfalls or surpluses in production. A trade-off exists... 

Reproductive risk descriptors are intended to address variability of risk within the population and the overall adverse impact on the population. In particular, differences between high-end and central tendency estimates reflect variability in the population but not the scientific uncertainty inherent in the risk estimates. There is uncertainty in all estimates of risk, including reproductive risk. These uncertainties can result from measurement uncertainties, modelling uncertainties and assumptions made due to incomplete data. Risk assessments should address the impact of each of these uncertainties on confidence in the estimated reproductive risk values. 

Because the objective of an exposure assessment is to characterize both the magnitude and the reliability of exposure scenarios, planning for an uncertainty analysis is a key element of an exposure assessment. The aims of the uncertainty analysis in this context are to individually and jointly characterize and quantify the exposure prediction uncertainties resulting from each step of the analysis. In performing an uncertainty analysis, typically the main sources of uncertainties are first characterized qualitatively and then quantified using a tiered approach (see chapter 4). In general, exposure uncertainty analyses attempt to differentiate between key sources of uncertainties scenario uncertainties, model uncertainties and parameter uncertainties (for definitions, see section 3.2). 

The three main classes of sources of uncertainty (section 3.2) are scenario uncertainty , model uncertainty (in both conceptual model formulation and mathematical model formulation) and parameter uncertainty (both epistemic and aleatory). The uncertainty of the conceptual model source concentrates on the relationship between the selected model and the scenario under consideration. 

Jablonowski M (1998) Fuzzy risk analysis in civil engineering. In Ayyub BM, ed. Uncertainty modeling and analysis in civil engineering. Boca Raton, FL, CRC Press, pp. 137-148. 

Shylakhter AI (1994) Uncertainty estimates in scientific models Lessons from trends in physical measurements, population and energy projections. In Ayyub BM, Gupta MM, eds. Uncertainty modelling and analysis Theory and applications. Amsterdam, Elsevier Science B.V., pp. 477 196... 

Price ND, Shmulevich I. Biochemical and statistical network models for systems biology. Curr. Opin. Biotechnol. 2007. Shmulevich I, et al. Probabilistic Boolean Networks a rule-based uncertainty model for gene regulatory networks. Bioinformatics. 2002 18 261-274... 

Step 1. Derive an irreducible (i.e., with natural uncertainty) model from a probable set of data. 

Given that laboratories accredited to ISO 17025 are encouraged to participate in proficiency testing (PT) programs when available, historical data from PT participation can be used to estimate an individual laboratory s uncertainty. The approaches to using PT data to calculate uncertainties vary Horwitz models," modifications of the ISO 5725-2 approach," and propagation of uncertainty models with variations depending on reliability of PT participants ... 

Shmulevich, I., Dougherty, E. R., Kim, S., and Zhang, W. (2002). ProbabiUstic Boolean networks A mle-based uncertainty model for gene regulatory networks. Bioinformatics, 18 261—274. 

Sampling is a very important operation in any decision-making process based on measurement. The samples must be truly relevant and represent the population under consideration. Knowledge of the response and the uncertainty model are needed to be able to select a sampling protocol that will give the desired information about the process. The sampling plan should carefully describe how samples (number, type, etc.) are to be selected, and the... 

While their in-season demand levels are correlated, both retailers have to order inventory in advance in the start of the season and thus face their own independent single-period demand uncertainty models (newsboy models). Let Xj and X2 refer to the discount factor for early orders, thus the retail prices are (for Retailer 1) andpX2 (for Retailer 2) for early orders. 

As mentioned, the uncertainty in Wf ) could come from stochastic uncertainty, model uncertainty or data and parameter uncertainty. In our model, stochastic uncertainly is related to the random behavior of z(t). Model uncertainty will more or less always be present, as g() only is a simplification of the leaUly. Data and parameter uncertainty is in this context related to both the parameters in g() as well as the parameters in the probability distribution of z(t). 

The MMSE estimator of Ay under the uncertainty model can be shown to be... 

An extension of this model leads to the assumption that speech vectors may be in different states at different time instants. The speech presence uncertainty model assumes two states representing speech presence and speech absence. In another model of Drucker [3], five states were proposed representing fricative, stop, vowel, ghde, and nasal speech sounds. A speech absence state could be added to that model as a sixth state. This model requires an estimator for each state just as in Eq. (19.99). 

Van den Nieuwenhof, B. 2003. Stochastic finite elements for elastodynamics random field and shape uncertainty modelling using direct and modal perturbation-based approaches, Ph.D. thesis, Universite catholique de Louvain, Louvain-La-Neuve, Belgium. 

Campbell, M.E. Grocott, S.C.O. Parametric uncertainty model for control design and analysis. IEEE Trans. Control Systems Technol., 7, no. 1 (1999), pp. 85-96... 

Boulet, B. Francis, B.A. Hughes, PC. Hong, T. Uncertainty modeling and experiments in Hoo control of large flexible space structures. IEEE Trans, on Control Systems Technol., 5, no. 5 (1997), pp. 504-519... 

You F, Wassick JM, Grossmann IE (2009) Risk management for a global supply chain planning under uncertainty models and algorithms. Am Inst Chem Eng J 55(4) 931-946. http //people. scs.carleton.ca/ arpwhite/courses/95590Y/lectures/pso.pdf (approached 11 March 2015)... 

If a real world problan is sought that is inherently messy, where mathematical functions are difficult to apply, uncertainty modeling processes wonld be an excellent example. Engineering processes, at or near the land surface depend on topography, vegetation, and soil moisture, rainfall patterns and intensity, potential evapo-transpiration, air tanperatures, solar radiation, winds, and dew points. Each of the variables changes either in space or in time, and many change in both space and time. Nonetheless, it is necessary to calculate such processes in this real world... 

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