Über die Autorin bzw. den Autor

Dorota Kurowicka, Delft University of Technology, The Netherlands.

Roger M. Cooke, Delft University of Technology, The Netherlands.

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Mathematical models are used to simulate complex real-world phenomena in many areas of science and technology. Large complex models typically require inputs whose values are not known with certainty. Uncertainty analysis aims to quantify the overall uncertainty within a model, in order to support problem owners in model-based decision-making. In recent years there has been an explosion of interest in uncertainty analysis. Uncertainty and dependence elicitation, dependence modelling, model inference, efficient sampling, screening and sensitivity analysis, and probabilistic inversion are among the active research areas. This text provides both the mathematical foundations and practical applications in this rapidly expanding area, including:

• An up-to-date, comprehensive overview of the foundations and applications of uncertainty analysis.
• All the key topics, including uncertainty elicitation, dependence modelling, sensitivity analysis and probabilistic inversion.
• Numerous worked examples and applications.
• Workbook problems, enabling use for teaching.
• Software support for the examples, using UNICORN – a Windows-based uncertainty modelling package developed by the authors.
• A website featuring a version of the UNICORN software tailored specifically for the book, as well as computer programs and data sets to support the examples.

Uncertainty Analysis with High Dimensional Dependence Modelling offers a comprehensive exploration of a new emerging field. It will prove an invaluable text for researches, practitioners and graduate students in areas ranging from statistics and engineering to reliability and environmetrics.

Aus dem Klappentext

Mathematical models are used to simulate complex real-world phenomena in many areas of science and technology. Large complex models typically require inputs whose values are not known with certainty. Uncertainty analysis aims to quantify the overall uncertainty within a model, in order to support problem owners in model-based decision-making. In recent years there has been an explosion of interest in uncertainty analysis. Uncertainty and dependence elicitation, dependence modelling, model inference, efficient sampling, screening and sensitivity analysis, and probabilistic inversion are among the active research areas. This text provides both the mathematical foundations and practical applications in this rapidly expanding area, including:

•  An up-to-date, comprehensive overview of the foundations and applications of uncertainty analysis.
• All the key topics, including uncertainty elicitation, dependence modelling, sensitivity analysis and probabilistic inversion.
• Numerous worked examples and applications.
• Workbook problems, enabling use for teaching.
• Software support for the examples, using UNICORN – a Windows-based uncertainty modelling package developed by the authors.
• A website featuring a version of the UNICORN software tailored specifically for the book, as well as computer programs and data sets to support the examples.

Uncertainty Analysis with High Dimensional Dependence Modelling offers a comprehensive exploration of a new emerging field. It will prove an invaluable text for researches, practitioners and graduate students in areas ranging from statistics and engineering to reliability and environmetrics.

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Uncertainty Analysis with High Dimensional Dependence Modelling

John Wiley & Sons

Chapter One

1.1 Wags and Bogsats

'... whether truef or not [it] is at least probable; and he who tells nothing exceeding the bounds of probability has a right to demand that they should believe him who cannot contradict him'. Samuel Johnson, author of the first English dictionary, wrote this in 1735. He is referring to the Jesuit priest Jeronimo Lobo's account of the unicorns he saw during his visit to Abyssinia in the 17th century (Shepard (1930) p. 200).

Johnson could have been the apologist for much of what passed as decision support in the period after World War II, when think tanks, forecasters and expert judgment burst upon the scientific stage. Most salient in this genre is the book The Year 2000 (Kahn and Wiener (1967)) in which the authors published 25 'even money bets' predicting features of the year 2000, including interplanetary engineering and conversion of humans to fluid breathers. Essentially, these are statements without pedigree or warrant, whose credibility rests on shifting the burden of proof. Their cavalier attitude toward uncertainty in quantitative decision support is representative of the period. Readers interested in how many of these even money bets the authors have won, and in other examples from this period, are referred to (Cooke (1991), Chapter 1).

Quantitative models pervade all aspects of decision making, from failure probabilities of unlaunched rockets, risks of nuclear reactors and effects of pollutants on health and the environment to consequences of economic policies. Such quantitative models generally require values for parameters that cannot be measured or assessed with certainty. Engineers and scientists sometimes cover their modesty with churlish acronyms designating the source of ungrounded assessments. 'Wags' (wild-ass guesses) and 'bogsats' (bunch of guys sitting around a table) are two examples found in published documentation.

Decision makers, especially those in the public arena, increasingly recognize that input to quantitative models is uncertain and demand that this uncertainty be quantified and propagated through the models.

Initially, it was the modellers themselves who provided assessments of uncertainty and did the propagating. Not surprisingly, this activity was considered secondary to the main activity of computing 'nominal values' or 'best estimates' to be used for forecasting and planning and received cursory attention.

Figure 1.1 shows the result of such an in-house uncertainty analysis performed by the National Radiological Protection Board (NRPB) and The Kernforschungszentrum Karlsruhe (KFK) in the late 1980s (Crick et al. (1988); Fischer et al. (1990)). The models in question predict the dispersion of radioactive material in the atmosphere following an accident in a nuclear reactor. The figure shows predicted lateral dispersion under stable conditions, and also shows wider and narrower plumes, which the modellers are 90% certain will enclose an actual plume under the stated conditions.

It soon became evident that if things were uncertain, then experts might disagree, and using one expert-modeller's estimates of uncertainty might not be sufficient. Structured expert judgment has since become an accepted method for quantifying models with uncertain input. 'Structured' means that the experts are identifiable, the assessments are traceable and the computations are transparent. To appreciate the difference between structured and unstructured expert judgment, Figure 1.2 shows the results of a structured expert judgment quantification of the same uncertainty pictured in Figure 1.1 (Cooke (1997b)). Evidently, the picture of uncertainty emerging from these two figures is quite different.

One of the reasons for the difference between these figures is the following: The lateral spread of a plume as a function of down wind distance x is modelled, per stability class, as

[sigma](x) = A[x.sup.B].

Both the constants A and B are uncertain as attested by spreads in published values of these coefficients. However, these uncertainties cannot be independent. Obviously if A takes a large value, then B will tend to take smaller values. Recognizing the implausibility of assigning A and B as independent uncertainty distributions, and the difficulty of assessing a joint distribution on A and B, the modellers elected to consider B as a constant; that is, as known with certainty.

The differences between these two figures reflect a change in perception regarding the goal of quantitative modelling. With the first picture, the main effort has gone into constructing a quantitative deterministic model to which uncertainty quantification and propagation are added on. In the second picture, the model is essentially about capturing uncertainty. Quantitative models are useful insofar as they help us resolve and reduce uncertainty. Three major differences in the practice of quantitative decision support follow from this shift of perception.

First of all, the representation of uncertainty via expert judgment, or some other method is seen as a scientific activity subject to methodological rules every bit as rigorous as those governing the use of measurement or experimental data.

Second, it is recognized that an essential part of uncertainty analysis is the analysis of dependence. Indeed, if all uncertainties are independent, then their propagation is mathematically trivial (though perhaps computationally challenging). Sampling and propagating independent uncertainties can easily be trusted to the modellers themselves. However, when uncertainties are dependent, things become much more subtle, and we enter a domain for which the modellers' training has not prepared them.

Finally, the domains of communication with the problem owner, model evaluation, and so on, undergo significant transformations once we recognize that the main purpose of models is to capture uncertainty.

1.2 Uncertainty analysis and decision support: a recent example

A recent example serves to illustrate many of the issues that arise in quantifying uncertainty for decision support. The example concerns transport of Campylobacter infection in chicken processing lines. The intention here is not to understand Campylobacter infection, but to introduce topics covered in the following chapters. For details on Campylobacter, see Cooke et al. (Appearing); Van der Fels-Klerx et al. (2005); Nauta et al. (2004).

Campylobacter contamination of chicken meat may be responsible for up to 40% of Campylobacter-associated gastroenteritis and for a similar proportion of deaths. A recent effort to rank various control options for Campylobacter contamination has led to the development of a mathematical model of a processing line for chicken meat (these chickens are termed 'broilers').

A typical broiler processing line involves a number of phases as shown in Figure 1.3. Each phase is characterized by transfers of Campylobacter colony forming units from the chicken surface to the environment, from the environment back to the surface and from the faeces to the surface (until evisceration), and the destruction of the...