Forms of uncertainty

Application of this equation to the probability distributions given in Table 40.6 shows that H for the less precise method is larger than for the more precise method. Uniform distributions represent the highest form of uncertainty and disorder. Therefore, they have the largest entropy. 

Prett and Garcia (1988) pose the validation problem as a discrete time linear optimal control problem under uncertainty. The uncertainty is defined by simple bounds, giving a polyhedral set of uncertain parameters V. For this problem, certain forms of uncertainty, e.g., in gains only, together with a quadratic performance index can be shown to satisfy the convexity requirements for the worst-case parameters to lie at vertices of V. This allows the algorithm of Gross-mann et al, based on examination only of vertices of V, to be applied (see Section II.A.l). The mathematical formulation is... 

To provide the most realistic representation of risk, all forms of uncertainty arc considered. Rather than assuming the existence of some representative condition prior to the accident scenario, a study models the full range of conditions and other uncertainties that can affect the scenario. Results include uncertainties in the frequency and consequences of each scenario. The upper uncertainty bound shown for the QRA risk estimates is a measure of the analysts confidence in the results. There is a 95 percent chance that the risk is less than the upper bound. 

Equivocality reduction It is generally accepted that high-quality information may reduce the imperfection or equivocality that might otherwise be present. This equivocality generally takes the form of uncertainty, imprecision, inconsistency, or incompleteness. It is very important to note that it is neither necessary nor desirable to obtain decision information that is unequivocal or totally perfect. Information need only be sufficiently unequivocal or unambiguous for the task at hand. To make it better may well be a waste of resources ... 

What follows mainly focuses on the detailed scheduling of the jobs. Given a collection of jobs that have to be processed in a given machine environment, the problem is to schedule the jobs, subject to given constraints, in such a way that one or more performance criteria are optimized. Various forms of uncertainties, such as random job-processing times, machines subject to breakdown, and rush orders, may have to be dealt with. 

It may seem that these distinctions have been lost on the way, at least in the business of commercial risk analysis and management. For the purpose of this paper, the implication is that we have lost one potential way of addressing the wildness in wait , for which there was no room in the hygienist utopia of Castels. However, this form of uncertainty, as an expression of possible wildness , can be rehabilitated by means of acknowledging the distinctions of order, as depicted by the left side of Cynefin (Fig. 1). 

The classical notion of risk in decision theory is primarily modeled using utility theory. Utility theory assumes that people are rational and should choose the option that maximizes the expected utility, which is the product of probability and payoff. Utility theory also assumes that all risk probabilities and payoff are known to a point estimate but does not allow ambiguity, or a variant form of uncertainty. In reality, however, uncertainty does occur when risk probabilities or payoff is missing or unknown. The subjective expected utility (SEU) model of utility theory proposed by Savage [26] argues that people s subjective preferences and beliefs, rather than objective probabilities, are used in the evaluation of an uncertain prospect for decision making. The SEU model is based on a set of seven axioms designed for consistent and rational behavior. 

Conflict is a form of uncertainty associated with classical probability theory. Probability theory expresses evidence in the form of a probability distribution, which distributes evidential assignments over multiple values. The probability assigned to any given element, however, is evidence that conflicts with the evidence assigned to other elements contained in the distribution (Klir and Smith 2001). In other words, evidence defined for some element x,- in the form of a probability distribution conflicts with evidence that supports other elements, the probabilities assigned to other elements, such as Xj and xj,. Similarly, the probability in support of the elements Xj and x/c indicates less support for the element x,. These elements are in competition with one another so to speak, and hence, the form of uncertainty characterized by classical probability theory is commonly referred to as conflict (Klir and Smith 2001). 

The range of uncertainty in the UR may be too large to commit to a particular development plan, and field appraisal may be required to reduce the uncertainty and allow a more suitable development plan to be formed. Unless the range of uncertainty is quantified using statistical techniques and representations, the need for appraisal cannot be determined. Statistical methods are used to express ranges of values of STOMP, GIIP, UR, and reserves. 

Propagation of uncertainty allows us to estimate the uncertainty in a calculated result from the uncertainties of the measurements used to calculate the result. In the equations presented in this section the result is represented by the symbol R and the measurements by the symbols A, B, and C. The corresponding uncertainties are sr, sa, sb) and sq. The uncertainties for A, B, and C can be reported in several ways, including calculated standard deviations or estimated ranges, as long as the same form is used for all measurements. 

Units are cm throughout. Measurements are of band heads, formed by the rotational stmcture, not band origins. Figures in parentheses are differences variations in differences (e.g. between the first two columns) are a result of uncertainties in experimental measurements. 

Relationships Between Objects, Processes, and Events. Relationships can be causal, eg, if there is water in the reactor feed, then an explosion can take place. Relationships can also be stmctural, eg, a distiUation tower is a vessel containing trays that have sieves in them or relationships can be taxonomic, eg, a boiler is a type of heat exchanger. Knowledge in the form of relationships connects facts and descriptions that are already represented in some way in a system. Relational knowledge is also subject to uncertainty, especiaUy in the case of causal relationships. The representation scheme has to be able to express this uncertainty in some way. 

Sometimes the expected consequences of an accident alone may provide you with sufficient information for decision-making purposes. Conventionally, the form of these estimates will be dictated by the purpose (concern) of the study (safety, economics, etc.). Absolute consequence estimates are best estimates of the impacts of an accident and, like frequency estimates, may have considerable uncertainty. Table 4 contains examples of typical consequence estimates obtained from QRA. These examples point to the difficulty in comparing various safety and economic results on a common basis—there is no common denominator. 

The ASME, Performance Test Code on Test Uncertainty Instruments and Apparatus PTC 19.1 specifies procedures for evaluation of uncertainties in individual test measurements, arising form both random errors and... 

There is a great deal of uncertainty as to the mechanism of PVC degradation but certain facts have emerged. Firstly dehydrochlorination occurs at an early stage in the degradation process. There is some infrared evidence that as hydrogen chloride is removed polyene structures are formed (Figure 12.18). 

Velocity The precise control of velocity and the study of effects of velocity on corrosion are extremely difficult, especially when high velocities are involved. A major problem is to prevent, or to take into account properly, the tendency of a liquid to follow the motion of a specimen moved through it, e.g. by rotation at high velocity. This can be controlled to some extent by proper baffling, but uncertainties as to the true velocity remain —as they do also when the test liquid is made to pass at some calculated velocity over a stationary test-piece or through a test-piece in the form of a tube or pipe ... 

Another problem arises from the presence of higher terms in the multipole expansion of the electrostatic interaction. While theoretical formulas exist for these also, they are even more approximate than those for the dipole-dipole term. Also, there is the uncertainty about the exact form of the repulsive interaction. Quite arbitrarily we shall group the higher multipole terms with the true repulsive interaction and assume that the empirical repulsive term accounts for both. The principal merit of this assumption is simplicity the theoretical and experimental coefficients of the R Q term are compared without adjustment. Since the higher multipole terms are known to be attractive and have been estimated to amount to about 20 per cent of the total attractive potential at the minimum, a rough correction for their possible effect can be made if it is believed that this is a preferable assumption. 

We do not underestimate the difficulties inherent in this task. The heterogeneity of highly composite books, in spite of the present vogue which spawns than, always impairs their usefulness and certainly detracts from their teachability. Nor is there a way of avoiding this difficulty. One confronts here the principle of complementarity for editorial surveillance. Written in the form of an uncertainty relation, that principle reads... 

The values of i calculated from (8) and (8) do not agree very closely, and it would appear, as Weinstein (loc. cit. 1068) remarks, that Although the calculations undoubtedly establish the legitimacy of the system of equations, the great uncertainty in the numerical determination of the decisive magnitudes forms a practical defect which will only be removed by observations over very wide intervals of the variables. Any discrepancy between the results of actual observations of equilibria, and those calculated by means of Nernst s chemical constants, need not, in the present state of uncertainty of the latter, cause any great alarm. Nernst himself apparently regards the constant < >, obtained from vapour-pressure measurements, as the most certain, and the others as more or less tentative. 

Detonation pressure may be computed theoretically or measured exptly. Both approaches are beset with formidable obstacles. Theoretical computations depend strongly on the choice of the equation of state (EOS) for the detonation products. Many forms of the EOS have been proposed (see Vol 4, D269—98). So.far none has proved to be unequivocally acceptable. Probably the EOS most commonly, used for pressure calcns are the polytropic EOS (Vol 4, D290-91) and the BKW EOS (Vol 4, D272-74 Ref 1). A modern variant of the Lennard Jones-Devonshire EOS, called JCZ-3, is now gaining some popularity (Refs -11. 14). Since there is uncertainty about the correct form of the detonation product EOS there is obviously uncertainty in the pressures computed via the various types of EOS ... 

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