Types of Uncertainty

This statement captures well the spirit of science in the nineteenth century scientific knowledge should be expressed in precise numerical terms imprecision and other types of uncertainty do not belong to science. 

When uncertainty was viewed as unscientific, there was little motivation to seriously study it. This explains why uncertainty has traditionally been neglected by science. It was only with the emergence of statistical mechanics at the beginning of this century that uncertainty became recognized as useful, or even essential, in certain scientific inquiries. However, this recognition was strongly qualified uncertainty was conceived solely in terms of probability theory. It took more than a half century to liberate uncertainty from its narrow confines of probability theory and to study its various other (nonprobabilistic) manifestations as well as their utility in science and technology. 

The focus in this chapter is on the mathematics pertaining to fuzzy set theory and its role in science. However, other approaches to uncertainty are also briefly introduced. 

The classical mathematical theories by which certain types of uncertainty can be expressed are classical set theory and probability theory. In terms of set theory, uncertainty is expressed by any given set of possible alternatives in situations where only one of the alternatives may actually happen. For example, when an interval of values of a variable is predicted by a given mathematical model, the set of values in the interval represents a predictive uncertainty, when an unsettled historical question allows a set of possible answers rather than a unique one, the set represents a retrodictive uncertainty when medical diagnosis of a patient results in a set of possible diseases rather than a single disease, the set represents a diagnostic uncertainty when design requirements are specified in terms of sets of alternatives, the sets represent a prescriptive uncertainty. 

Uncertainty expressed in terms of sets of alternatives results from the nonspecificity inherent in each set. Large sets result in less specific predictions (retrodictions, prescriptions, etc.) than their smaller counterparts. One area of mathematics that deals with this kind of uncertainty is interval analysis.  

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In contrast with continuous plants, batch plants are intrinsically flexible. Designed properly, they meet changes in product demand and characteristics of processes in the plant. These changes originate from uncertainties in variables determining the performance of the batch plant. There are two different types of uncertainties in the design of batch plants (1) short-term, and (2) long-term variations. 

Semantic uncertainty is the type of uncertainty for which we shall need fuzzy logic. Expressed by phrases such as "acidic" or "much weaker," this is imprecision in the description of an event, state, or object rather than its measurement. Fuzzy logic offers a way to make credible deductions from uncertain statements. We shall illustrate this with a simple example. 

The risk assessment framework we have described for chemical toxicity is applicable to microbial risk assessment. Once the information is available on microbial hazards, which are for the most part acute (immediately observable) conditions resulting from acute (one-time) exposures, and their dose (pathogen count)-response characteristics, we should be ready to assess the risks associated with any dose of interest. Hazard information for the important pathogens is readily available but, as expected, their dose-response characteristics are much harder to come by. So with pathogen risk assessment we see the same types of uncertainties creeping into the framework as we have encountered for chemicals. 

Also under the IPCS harmonization project, a working group is preparing a harmonized set of principles for the treatment of uncertainty in exposure assessment. The document will review the types of uncertainty analyses used in exposure assessments, evaluate their effectiveness in giving decision-makers the types of information they need, and derive a set of principles for uncertainty analysis (WHO/IPCS 2006). 

There are many ways of classifying the various types of uncertainty and variability that are associated with these two terms. In the peer-reviewed literature, it is... 

A probabilistic risk assessment (PRA) deals with many types of uncertainties. In addition to the uncertainties associated with the model itself and model input, there is also the meta-uncertainty about whether the entire PRA process has been performed properly. Employment of sophisticated mathematical and statistical methods may easily convey the false impression of accuracy, especially when numerical results are presented with a high number of significant figures. But those who produce PR As, and those who evaluate them, should exert caution there are many possible pitfalls, traps, and potential swindles that can arise. Because of the potential for generating seemingly correct results that are far from the intended model of reality, it is imperative that the PRA practitioner carefully evaluates not only model input data but also the assumptions used in the PRA, the model itself, and the calculations inherent within the model. This chapter presents information on performing PRA in a manner that will minimize the introduction of errors associated with the PRA process. 

Next, the applications have to be validated and placed into standardized forms. Validation should consist of two steps. First, simulated data sets of aerosol properties should be generated from pre-selected source contributions as did Watson in his simulation studies of the chemical mass balance method. These data should be perturbed with the types of uncertainties expected under field conditions. The types of sources and their contributions predicted by the receptor model application should be compared with the known source model values and the extent of perturbation tolerable should be assessed. 

There are several terms used in measurement uncertainty that must be defined. An uncertainty arising from a particular source, expressed as a standard deviation, is known as the standard measurement uncertainty (u). When several of these are combined to give an overall uncertainty for a particular measurement result, the uncertainty is known as the combined standard measurement uncertainty (uc), and when this figure is multiplied by a coverage factor ( ) to give an interval containing a specified fraction of the distribution attributable to the measurand (e.g., 95%) it is called an expanded measurement uncertainty [U). I discuss these types of uncertainties later in the chapter. 

However, since exposure assessment is an interdisciplinary activity, this monograph is also expected to be a useful resource for a wider audience, considering that each group may use the information contained herein for different purposes. The monograph provides an overview of the types of uncertainty encountered by an exposure analyst and provides guidance on how uncertainty can be characterized, analysed and described in a risk assessment and communicated effectively to a range of stakeholders. 

Sample uncertainty is also referred to as statistical random sampling error. This type of uncertainty is often estimated assuming that data are sampled randomly and without replacement and that the data are random samples from an unknown population distribution. For example, when measuring body weights of different individuals, one might randomly sample a particular number of individuals and use the data to make an estimate of the interindividual variability in body weight for the entire population of similar individuals (e.g. for a similar age and sex cohort). 

This is an example exposure assessment that illustrates quantitative representations of uncertainty and variability at the higher tiers of an exposure assessment. This case-study is based on human exposures to a persistent, bioaccumulative and lipid-soluble compound through fish consumption. This compound is fictional and referred to here as PBLx, but it has properties that correspond to those of known persistent compounds. Specific goals of this case-study are to illustrate (1) the types of uncertainty and variability that arise in exposure assessments, (2) quantitative uncertainty assessment, (3) how distributions are established to represent variability and uncertainty, (4) differences among alternative variance propagation methods, (5) how to distinguish uncertainty from variability and (6) how to communicate the results of an uncertainty analysis. 

Transparent documentation of numerical values may depend on how the numbers were originally generated. Numerical values estimated from a population-based measurement study should describe the study design in detail, the basis for the population sample selected for measurement, how measurements were performed, how values were stored, how statistical summaries of the stored values were calculated, and so on. In other cases, a numerical value in an exposure assessment may be a value that is not associated with any particular data set but has been mandated by regulation or science policy or developed from theory. In this case, transparency requires a description of the source of the value, the types of situations or populations for which it was thought to be relevant and appropriate, any limitations of the value, and the degree and type of uncertainties associated with its use in a particular assessment. 

One illustration of this type of uncertainty comes from studies of paralogs of malate dehydrogenase. The mitochondrial paralog of malate dehydrogenase (mMDH) has long been... 

Assessment factors (AFs) are tools for dealing with the uncertainty that is implicit in any risk assessment. Three types of uncertainty should be considered (Figure 5.8) ... 

Risk descriptor VII, national dimensions , would be defined as whether or not target-setting or national permitting schemes should be applied for the scenarios that were described in Box 5.3 on. Examples of four types of uncertainty that can lead to conclusion (i) are provided in Table 5.5. 

In addition to covering all toxic chemicals, the legislation must cover all relevant persons. As discussed above, there is room for uncertainty in both the Australian and the French legislation with regard to the persons covered, which is resolved by reference to other legislation. This type of uncertainty could easily be clarified by the addition of a definitional section in the CWC implementing legislation itself. 

Uncertainty Represents a lack of knowledge about factors affecting exposure or risk and can lead to inaccurate or biased estimates of exposure. The types of uncertainty include scenario uncertainty, parameter uncertainty and model uncertainty (USEPA, 1997c). 

While designing complex systems, we can basically encounter two types of uncertainties. In the first, we know that the system will work but it is difficult to deter-... 

The second type of uncertainty in propositions of the given type results from information deficiency regarding the object x. While the predicate P is in this case defined precisely, information about x is insufficient to determine whether or not x satisfies P. The proposition is in this case either true or false, but its actual truth status cannot be determined. However, it is useful to assign a number in the unit interval [0,1] to the proposition to express the degree of evidence that the proposition is true. Assigning degrees of evidence to relevant propositions is a topic dealt with in measure theory. 

We can see in these debates once again that most opponents of fuzzy set theory from the area of probability theory attempt to compare probabilities with degrees of truth (or degrees of membership). However, these are not comparable. As explained in Section III in the context of possibility theory, degrees of truth result from linguistic uncertainty, while probabilities result from information deficiency. These two types of uncertainty may be combined, but their comparison is meaningless. 

Second, even if the number of knowns in Eq. (5) were greater than or equal to the number of unknowns, a new type of uncertainty would be observed. As stated previously, we set the values of the S vector to (0,1), i.e., the parameters derived from the spectrum are crisp. From the values of these parameters assigned to the vertices, the structure of the unknown molecule (given by the elements a of the adjacency matrbc) is to be deduced. From a graph-theoretical point of view these vertices may be considered as colored by different spectral parameters, e.g., chemical shifts from NMR spectra, or by frequencies in IR spectroscopy, etc. This can be done by assignment of these parameters to the corresponding atoms. The assignment procedure implies that the experimental values are compared with values assumed or evaluated from the structure. Insofar as these parameters are functions of the atom connectivity environment, they are widely used to elucidate structural connectivity. Several severe problems... 

Sensitivity analysis is a type of uncertainty analysis that is used to consider the impacts of uncertainty. In such analyses, one input is changed at a time to determine how the results of a model will change over the range of possible values of that single input. Multiple inputs can be varied simultaneously, using a sampling technique called Monte Carlo analysis, to obtain an overall distribution of the result. 

As noted earlier, the conceptual model is the product of the problem formulation phase, which, in turn, provides the foundation for the analysis phase and the development of the exposure and stressor-response profiles. If incorrect assumptions are made during conceptual model development regarding the potential effects of a stressor, the environments impacted, or the species residing within those systems, then the final risk assessment will be flawed. These types of uncertainties are perhaps the most difficult to identify, quantify, and reduce. 

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