Uncertainty representation

Uncertainty analysis and sensitivity analysis are tools that provide insight on how model predictions are affected by data precision. One of the issues in uncertainty analysis that must be confronted is how to rank both individual inputs and groups of inputs according to their contribution to overall uncertainty. In particular, there is a need to distinguish between the relative contribution of true uncertainty versus variability (i.e. heterogeneity), as well as to distinguish model uncertainty from parameter uncertainty. This case-study illustrates methods of uncertainty representation and variance characterization. 

THE DEMPSTER-SHAFER APPROACH TO MODEL UNCERTAINTY REPRESENTATION... 

In this paper, we have investigated the use of different frameworks for uncertainty representation and propagation in fault tree analysis. The frameworks considered are the probabihstic (Bayesian) and possi-bilistic frameworks, as well as an integratedprobabihs-tic/possibilistic computational framework, referred to as a hybrid approach. The tailoring of the integrated computational framework to the fault tree setting is the main contribution of the paper. Interpretations for the results obtained within the different approaches are provided, as well as a discussion of the approaches in relation to a specific case. However, a direct comparison of the actual results obtained for the different approaches has not been made, as no efforts have been made to make the probabihty and possibility distributions used as input coherent. In future work, we intend to make this comparison based on coherent probability-possibility transforms presented in the literature and to extend the computational procedures in the hybrid approach to produce uncertainty statements about the top event of the fault tree. 

Settings V and VI are characterised by a moderate or high degree of belief in deviation from x, as well as moderate or high sensitivity of R(x ) with respect to x. Then there is a need to both characterise uncertainty related to X in a more comprehensive maimer, as well as to integrate this uncertainty characterisation with the risk index R(X) in a coherent manner. This necessitates a quantitative approach to uncertainty representation. 

In setting VI the background knowledge on which to base a quantitative uncertainty representation of X is moderate or weak. It may still be possible to establish a probability distribution F(x z , K) however, such a distribution will be based on more or less reasonable assumptions, and so the risk index [F(V) z , K resulting from an integration according to Equation (1) needs to be supplemented by an assumption deviation risk assessment of the assumption Z = z. This could be achieved by reporting of (interval) probabilities... 

We conducted two user studies to identify major usability issues and evaluate how our system works. Our qualitative study with biologists showed that while we have improved the uncertainty representation so that task performance and insight-building is high even with large trees, ways to improve satisfaction are needed. Also, while most users concur with relative uncertainty scores, there is not universal agreement on... 

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 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. 

The most informative method of expressing uncertainty in HCIIP or ultimate recovery (UR) is by use of the expectation curve, as introduced in Section 6.2. The high (H) medium (M) and low (L) values can be read from the expectation curve. A mathematical representation of the uncertainty n a parameter (e.g. STOMP) can be defined as... 

Conventional computers initially were not conceived to handle vague data. Human reasoning, however, uses vague information and uncertainty to come to a decision. In the mid-1960 this discrepancy led to the conception of fuzzy theory [14]. In fuzzy logic the strict scheme of Boolean logic, which has only two statements true and false), is extended to handle information about partial truth, i.e., truth values between "absolutely true" and absolutely false". It thus gives a mathematical representation of uncertainty and vagueness and provides a tool to treat them. 

A more useful representation of uncertainty is to consider the effect of indeterminate errors on the predicted slope and intercept. The standard deviation of the slope and intercept are given as... 

Descriptions of Physical Objects, Processes, or Abstract Concepts. Eor example, pumps can be described as devices that move fluids. They have input and output ports, need a source of energy, and may have mechanical components such as impellers or pistons. Similarly, the process of flow can be described as a coherent movement of a Hquid, gas, or coUections of soHd particles. Flow is characterized by direction and rate of movement (flow rate). An example of an abstract concept is chemical reaction, which can be described in terms of reactants and conditions. Descriptions such as these can be viewed as stmctured coUections of atomic facts about some common entity. In cases where the descriptions are known to be partial or incomplete, the representation scheme has to be able to express the associated uncertainty. 

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. 

Rules may represent either guidelines based on experience, or compact descriptions of events, processes, and behaviors with the details and assumptions omitted. In either case, there is a degree of uncertainty associated with the appHcation of the rule to a given situation. Rule-based systems allow for expHcit ways of representing and dealing with uncertainty. This includes the representation of the uncertainty of individual rules, as weU as the computation of the uncertainty of a final conclusion based on the uncertainty of individual rules, and uncertainty in the data. There are numerous approaches to uncertainty within the rule-based paradigm (2,35,36). One of these approaches is based on what are called certainty factors. In this approach, a certainty factor (CF) can be associated with variable—value pairs, and with individual rules. The certainty of conclusions is then computed based on the CF of the preconditions and the CF for the rule. For example, consider the foUowing example. 

It will be noted that because of the low self-diffusion coefficients the numerical values for representations of self-diffusion in silicon and germanium by Anhenius expressions are subject to considerable uncertainty. It does appear, however, that if this representation is used to average most of the experimental data the equations are for silicon... 

If Pmfv) and the plant uncertainty A(.v) are combined to give P(.v), then Figure 9.29 can be simplified as shown in Figure 9.30, also referred to as the two-port state-space representation. 

Notice that in this example, the speed of the packet is inversely proportional to the packet s spatial size. While there is certainly nothing unique about this particular representation, it is interesting to speculate, along with Minsky, whether it may be true that, just as the simultaneous information about position and momentum is fundamentally constrained by Heisenberg s uncertainty relation in the physical universe, so too, in a discrete CA universe, there might be a fundamental constraint between the volume of a given packet and the amount of information that can be encoded within it. 

FIGURE 1.22 A representation of the uncertainty principle, (a) The location ot the particle is ill defined and so the momentum of the particle (represented by the arrow) can be specified reasonably precisely, (b) The location of the particle is well defined, and so the momentum cannot be specified very precisely. 

In Eq. (2.4) the temperatures Tout and Tin, denote the bulk mean air temperatures at the outlet and inlet cross-sections, respectively. Their representation by point measurements introduces a bias error equal to the difference between the latter and the corresponding bulk means. In evaluating it, allowance should be made for the residual uncertainty involved in the bias errors from the probe calibration, etc. 

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