Uncertainty combining

The certainty factor approach has been among the more popular rule-based approaches to uncertainty. However, although it is easy to apply given the individual CFs, acquiring the raw CFs from the experts is often quite difficult. Further, although the formulas for CF combination are mathematically appealing, they often have no relation to the ways in which experts combine evidence to arrive at conclusions. Some of the task-specific approaches discussed later address uncertainty combination in a more intuitive way (35). 

Well done if you got this one right. If you picked the wrong answer, do not be too disappointed because, as you can see, it is very easy to get equations mixed-up . Go over the section again and try making up some of your own exercises to test the rules of uncertainty combination. You are bound to improve with practice. 

Although it is relatively straightforward to estimate the uncertainty on an individual measurement, many of the quantities we wish to calculate involve the combination of these individual measurements in some way. The question then arises as to how the uncertainties combine when the quantities to which they refer are combined. 

A simplified explanation of the information in Table 3 is that the analytical component represents the uncertainty revealed by repeating the measurement on the same specimen (ex situ) or in the same position (in situ). Sampling uncertainty arises mainly because of heterogeneity (or Tumpiness ) of radionuclides in the field. The combined uncertainty is just the analytical and sampling uncertainties combined by the sums of their squares. ... 

A great deal of effort has been invested in the scientific research realm in the attempt to reduce such uncertainty, combining technological assessments with economic evaluations and environmental sustainability studies. Also policy and regulation issues have been investigated. In the following sections, we will provide an account of all these studies, in an attempt to assess the development status of biofuels as a fundamental step toward a sustainable transition. 

When providing input for the STOMP calculation a range of values of porosity (and all of the other input parameters) should be provided, based on the measured data and estimates of how the parameters may vary away from the control points. The uncertainty associated with each parameter may be expressed in terms of a probability density function, and these may be combined to create a probability density function for STOMP. 

The parametric method is an established statistical technique used for combining variables containing uncertainties, and has been advocated for use within the oil and gas industry as an alternative to Monte Carlo simulation. The main advantages of the method are its simplicity and its ability to identify the sensitivity of the result to the input variables. This allows a ranking of the variables in terms of their impact on the uncertainty of the result, and hence indicates where effort should be directed to better understand or manage the key variables in order to intervene to mitigate downside and/or take advantage of upside in the outcome. 

There is some uncertainty connected with testing techniques, errors of characteristic measurements, and influence of fectors that carmot be taken into account for building up a model. As these factors cannot be evaluated a priori and their combination can bring unpredictable influence on the testing results it is possible to represent them as additional noise action [4], Such an approach allows to describe the material and testing as a united model — dynamic mathematical model. 

If the signal features already have been chosen, another important problem is how to optimally combine these features in order to obtain the best estimate of the material property. The physical reasoning will give us ideas of how to combine the features but there will be no guarantee that we are using the chosen features in an optimal way. One reason for this is that we have to take into account the uncertainties that always are present in measurement data. 

The uncertainties in choice of potential function and in how to approximate the surface distortion contribution combine to make the calculated surface energies of ionic crystals rather uncertain. Some results are given in Table VII-2, but comparison between the various references cited will yield major discrepancies. Experimental verification is difficult (see Section VII-5). Qualitatively, one expects the surface energy of a solid to be distinctly higher than the surface tension of the liquid and, for example, the value of 212 ergs/cm for (100)... 

Again the uncertainty about the proportion of an observed result which is due to nitration and the proportion which is due to nitrosation exists. Thus, in expt. 11 phenol was being nitrated above the encounter rate and the observed isomer distribution could arise from a combination of nitration by whatever is the usual electrophile with nitration by a new, less reactive electrophile, or with nitrosation, or all three processes could be at work. 

It is easy to see that combining uncertainties in this way overestimates the total uncertainty. Adding the uncertainty for the first delivery to that of the second delivery assumes that both volumes are either greater than 9.992 mL or less than 9.992 mL. At the other extreme, we might assume that the two deliveries will always be on opposite sides of the pipet s mean volume. In this case we subtract the uncertainties for the two deliveries,... 

Many chemical calculations involve a combination of adding and subtracting, and multiply and dividing. As shown in the following example, the propagation of uncertainty is easily calculated by treating each operation separately using equations 4.6 and 4.7 as needed. 

A solution can be diluted by a factor of 200 using readily available pipets (f-mL to fOO-mL) and volumetric flasks (fO-mL to fOOO-mL) in either one, two, or three steps. Limiting yourself to glassware listed in Table 4.2, determine the proper combination of glassware to accomplish each dilution, and rank them in order of their most probable uncertainties. 

Minimills and other EAF plants ate expanding iato flat-roUed steel products which, by some estimates, requite 50—75% low residual scrap or alternative raw material. Up to 16 million t of new capacity are expected to be added ia the United States between 1994 and 2000 (18). Developments ia other parts of the world also impact scrap use and supply. Possible scrap deficiencies of several million tons have been projected for EAFs ia East Asia and ia parts of Europe. This puts additional strains on the total scrap supply, particularly low residual scrap (19,20). The question of adequate supply of low residual scrap is always a controversial one. Some analysts see serious global shortages ia the first decade of the twenty-first century others are convinced that the scrap iadustry has the capabiUty to produce scrap ia the quantities and quaUty to meet foreseeable demand. This uncertainty ia combination with high scrap prices has led to iacreased use of scrap alternatives where the latter is price competitive with premium scrap. Use of pig iroa has iacreased ia EAF plants and mote capacity is being iastaHed for DRI and HBI outside the United States. 

In dealing with future uncertainties. Royal Dutch/SheU pioneered Scenario planning (54,55). Alternative assumptions for future developments can be combined under this approach in various ways to give a number of consistent possible outcomes (56) and provide a basis for both actions and reactions. The approach has rewarded Shell handsomely. 

The above FF controller can be implemented using analog elements or more commonly by a digital computer. Figure 8-33 compares typical responses for PID FB control, steady-state FF control (.s = 0), dynamic FF control, and combined FF/FB control. In practice, the engineer can tune K, and Tl in the field to improve the performance oTthe FF controller. The feedforward controller can also be simplified to provide steady-state feedforward control. This is done by setting. s = 0 in Gj. s). This might be appropriate if there is uncertainty in the dynamic models for Gl and Gp. 

Multicomponent analysis by non-selective methods is based on the measurement of total analytical signal (AS) of mixture of components at several intensive parameters and on the constmction of combined equations and the solving of it. The difference of partial sensitivity of components determined in common defines uncertainty. 

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. 

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