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More Details Other Editions 1. Friend Reviews. To see what your friends thought of this book, please sign up. To ask other readers questions about Measuring Uncertainty within the Theory of Evidence , please sign up. Be the first to ask a question about Measuring Uncertainty within the Theory of Evidence. Generally a standard probabilistic, possibly with a. Generally a mixed deterministic-probabilistic setting. Risk and uncertainty separated or not in a double or single-level probabilistic or mixed setting.

Sensor calibration heterogeneity. Dataset sizes. Acceptability of deterministic vs. Computational and mathematical complexity in handling rare probabilities.

## Measuring Uncertainty Within the Theory of Evidence

Scarcity of data to control extreme event distributions. Complexity to handle double probabilistic criteria. These should be seen as applying both to industrial or environmental fields within which there is already some experience in uncertainty treatment, as nuclear safety or waste management think of the Yucca Mountain nuclear waste facility, for which the performance assessment involves more considerable effort to represent uncertainty , or to fields where practice is more recent.

Climate change may be seen as a frontier case in that respect, for which each of the three challenges discussed hereafter, mobilizing information on uncertainty sources, numerically treating uncertainty, and building acceptable representations along a precautionary principle, will take acute dimensions. Needless to say, the relevance and significance of the entire uncertainty study relies upon the quality of such an input uncertainty model. Key difficulties arise with the highly-limited sampling information directly available on uncertain input variables in real-world industrial cases: real samples in water management or environmental studies, if any, fall often below critical sizes required for stable statistical estimation.

Indeed, flood monitoring generates data on maximal water elevations or velocities rather than on uncertain friction coefficient or riverbed topology. Run-off coefficients, and Strickler or Manning friction coefficients are typical examples in the hydrological or hydraulic domain of uncertain inputs for which no direct data is made available, although rainfall-flow, stage-discharge or stage line curves could provide indirect data to be calibrated against. This approach, which involves the inversion of a physical model to transform the indirect information, is intimately connected to classical data assimilation, parameter identification model calibration or updating techniques, although inverse uncertainty identification has some distinctive features: it regards the way uncertainty sources are conceptually acknowledged and mathematically modelled on unknown model parameters or model uncertainty.

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However such an estimation uncertainty happens to be purely epistemic or reducible which is not satisfactory in the case of intrinsically uncertain or variable physical systems, i. We have seen how, in a strong deterministic paradigm, the maximisation of the response for uncertain domains implies numerous optimisation calculations; in a simple probabilistic paradigm, even with accelerated methods, several dozens or hundreds of calculations are necessary at least.

In the former case, computational greediness is associated to the need of nesting a maximisation by intervals for the deterministic components with, for each point, a conditional probabilistic calculation for the probabilised variables. In the latter two cases, nesting optimization algorithms with propagation by sampling is required in the general non-linear, non-Gaussian case, generating a large computational cost. Aside from the internal optimisation of the code solvers themselves and their parallelisation, the numerical challenges posed by large-scale uncertainty treatment, depending on the propagation methods adopted, may indeed benefit from massively distributed computing.

Beyond simple Monte-Carlo, accelerated sampling e. Everything nevertheless depends on the quantity of interest fixed for the study, the paradigm chosen, for example "mixed deterministic-probabilistic", and the more or less probabilistic treatment devoted to the different sources of uncertainty, i. Step A in the framework.

More generally, a decision rule with non-linear probability that is to say, based on the subjective transformation of probabilities by the agents equally constitutes a generalised representation of the risk aversion involved Quiggin, In the case of probabilities judged to be imprecise, this transformation can be interpreted as an aversion to uncertainty on the distribution of probabilities.

While the calculation of the criterion c Z is more sophisticated, because the functional on the measure of Z is more complex, that does not structurally change the global framework defined above. The mode of treatment of uncertainties is in effect a subject for discussion, depending on the level of subjective uncertainty or cognitive uncertainty, when the consequences are long term or serious, for example when it is a question of the storage of wastes with a long life, or climate change. Accepting a probabilistic treatment when the uncertain occurrence will only occur once, and in a manner that is not observable ex ante for the deciders, is a delicate matter: for example, one might prefer a min-max approach that minimizes the damage associated with the worst case, which can be made to correspond, in our industrial framework, with the assignation of deterministic intervals for serious sources of uncertainty in a mixed deterministic-probabilistic approach: which is, nevertheless, a very burdensome approach.

This progression leads to the examination of methods and selection of the most relevant, those which closely associate the applied mathematics involved and the analysis of the physical system, with the emphasis on the problem to be resolved, which itself depends on the type of regulation concerned or the decision criterion chosen Step A , and not on the specific industrial or phenomenological domain.

Furthermore, while this global approach shows that other mathematical paradigms are possible, the mixed deterministic-probabilistic framework appears to play a central role in current industrial applications, giving rise to numerous problems of statistical modeling and of scientific computing. This suggests that, in preference to the implementation of a single research field, applications should be oriented towards capitalizing on all the various numerical and statistical algorithms available, made transparent in order to increase public accountability on the key uncertainty and risk issues, which is the whole object of the development of open source initiatives.

Beyond, several directions for research remain to be explored.

This is especially so when it is linked to the delicate but promising area of the evaluation of expert judgment, or the question of modeling dependences also in Step B. Over and above efforts to provide cross-cultural training, the question rapidly becomes one of the acceptability, precaution and legibility associated with regulatory criteria, a debate lying at the heart of the society's attitude to risk.

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These questions remain open, in particular as regards the delicate choices of differentiation and aggregation of uncertainties according to natures, joined with the global conceptual apprehension of risks when considering issues of such public significance as industrial safety, natural risk or environmental impact. At the very least, recent applied research and industrial experience leads us to think that by systematically encouraging people to question hypotheses in step B and to test numerous alternative calculations in step C , a study of uncertainties , even if its ambition appears presumptuous, will always be more reliable than a straightforward deterministic approach whose justification resides in the impossibility of giving complete confidence to quantitative uncertainty modeling.

Andrianov G. Apostolakis G. Science ; Apostolakis, G. The distinction between aleatory and epistemic uncertainties is important; an example from the inclusion of aging effects in the PSA.

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## The mathematical theory of evidence and measurement uncertainty - Semantic Scholar

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Contents - Previous document - Next document. Abstract Uncertainty is ubiquitous in modern decision-making supported by quantitative modeling. Outline Introduction. A debate on the natures of uncertainty and associated quantification methods. A Generic methodological framework in the industrial world. Industrial system models considered. The central concept of the quantity of interest or risk measure. Generic goals of uncertainty studies.

Generic conceptual framework and key steps in any uncertainty study. Metrology of environmental emissions.