Possibility and probability: application examples and comparison of two different approaches to uncertainty evaluation

This paper proposes two interesting applications of the approach to uncertainty evaluation and representation in terms of Random-Fuzzy Variables. One covers the expression of the calibration uncertainty of gauge blocks, and one considers unknown temperature variations, with respect to temperature at calibration time, in expressing a voltmeter uncertainty. Both considered examples show that the proposed approach is more effective than the traditional GUM approach

Dimensional flow and fuzziness in quantum gravity: emergence of stochastic spacetime

We show that the uncertainty in distance and time measurements found by the heuristic combination of quantum mechanics and general relativity is reproduced in a purely classical and flat multi-fractal spacetime whose geometry changes with the probed scale (dimensional flow) and has non-zero imaginary dimension, corresponding to a discrete scale invariance at short distances. Thus, dimensional flow can manifest itself as an intrinsic measurement uncertainty and, conversely, measurement-uncertainty estimates are generally valid because they rely on this universal property of quantum geometries. These general results affect multi-fractional theories, a recent proposal related to quantum gravity, in two ways: they can fix two parameters previously left free (in particular, the value of the spacetime dimension at short scales) and point towards a reinterpretation of the ultraviolet structure of geometry as a stochastic foam or fuzziness. This is also confirmed by a correspondence we establish between Nottale scale relativity and the stochastic geometry of multi-fractional models.Comment: 25 pages. v2: minor typos corrected, references adde

A Box Particle Filter for Stochastic and Set-theoretic Measurements with Association Uncertainty

This work develops a novel estimation approach for nonlinear dynamic stochastic systems by combining the sequential Monte Carlo method with interval analysis. Unlike the common pointwise measurements, the proposed solution is for problems with interval measurements with association uncertainty. The optimal theoretical solution can be formulated in the framework of random set theory as the Bernoulli filter for interval measurements. The straightforward particle filter implementation of the Bernoulli filter typically requires a huge number of particles since the posterior probability density function occupies a significant portion of the state space. In order to reduce the number of particles, without necessarily sacrificing estimation accuracy, the paper investigates an implementation based on box particles. A box particle occupies a small and controllable rectangular region of non-zero volume in the target state space. The numerical results demonstrate that the filter performs remarkably well: both target state and target presence are estimated reliably using a very small number of box particles

A Fuzzy Set Theory Based Methodology for Analysis of Uncertainties in Stage-Discharge Measurements and Rating Curve

River stage and discharge records are essential for hydrological and hydraulic analyses. While stage is measured directly, discharge value is calculated from measurements of flow velocity, depth and channel cross-section dimensions. The measurements are affected by random and systematic measurement errors and other inaccuracies, such as approximation of velocity distribution and channel geometry with a finite number of measurements. Such errors lead to the uncertainty in both, the stage and the discharge values, which propagates into the rating curve established from the measurements. The relationship between stage and discharge is not strictly single valued, but takes a looped form due to unsteady flow in rivers. In the first part of this research, we use a fuzzy set theory based methodology for consideration of different sources of uncertainty in the stage and discharge measurements and their aggregation into a combined uncertainty. The uncertainty in individual measurements of stage and discharge is represented using triangular fuzzy numbers and their spread is determined according to the ISO – 748 guidelines. The extension principle based fuzzy arithmetic is used for the aggregation of various uncertainties into overall stage discharge measurement uncertainty. In the second part of the research we use fuzzy nonlinear regression for the analysis of the uncertainty in the single valued stage – discharge relationship. The methodology is based upon fuzzy extension principle. All input and output variables as well as the coefficients of the stage - discharge relationship are considered as fuzzy numbers. Two different criteria; the minimum spread and the least absolute deviation are used for the evaluation of output fuzziness. The results of the fuzzy regression analysis lead to a definition of lower and upper uncertainty bounds of the stage – discharge relationship and representation of discharge value as a fuzzy number. The third part of this research considers uncertainties in a looped rating curve with an application of the Jones formula. The Jones formula is based on approximate form of unsteady flow equation, which leads to an additional uncertainty. In order to take into account of the uncertainties due to the use of approximate formula and measurement of discharge values, the parameters of the Jones formula are considered fuzzy numbers. This leads to a fuzzified form of Jones formula. Its spread is determined by a multi-objective genetic algorithm. We used a criterion to minimize the spread of the fuzzified Jones formula so that the measurements points are bounded by the lower and upper bound curves. The study therefore considers individual sources of uncertainty from measurements to the single valued and looped rating curves. The study also shows that the fuzzy set theory provides an appropriate methodology for the analysis of the uncertainties in a nonprobabilistic framework.https://ir.lib.uwo.ca/wrrr/1023/thumbnail.jp

Galton's Error and the Under-Representation of Systematic Risk

Our methodology of 'complete identification,' using simple algebraic geometry, throws new light on the continued commitment of Galton's Error in finance and the resulting misinformation of investors. Mutual funds conventionally advertise their relative systematic market risk, or 'betas,' to potential investors based on incomplete measurement by unidirectional bivariate projections: they commit Galton's Error by under-representing their systematic risk. Consequently, far too many mutual funds are marketed as 'defensive' and too few as 'aggressive.' Using our new methodology we found that, out of a total of 3,217 mutual funds, 2,047 funds (63.7%) claimed to be defensive based on the current industry standard methodology, but only 608 (18.9%) actually are. This under-representation of systematic risk leads to inefficiencies in the capital allocation process, since biased betas lead to mis-pricing of mutual funds. Our complete bivariate projection produces a correct representation of the epistemic uncertainty inherent in the bivariate measurement of relative market risk. Our conclusions have also serious consequences for the proper 'bench-marking' and recent regulatory proposals for the mutual funds industry.

A proposed framework for characterising uncertainty and variability in rock mechanics and rock engineering

This thesis develops a novel understanding of the fundamental issues in characterising and propagating unpredictability in rock engineering design. This unpredictability stems from the inherent complexity and heterogeneity of fractured rock masses as engineering media. It establishes the importance of: a) recognising that unpredictability results from epistemic uncertainty (i.e. resulting from a lack of knowledge) and aleatory variability (i.e. due to inherent randomness), and; b) the means by which uncertainty and variability associated with the parameters that characterise fractured rock masses are propagated through the modelling and design process. Through a critical review of the literature, this thesis shows that in geotechnical engineering – rock mechanics and rock engineering in particular – there is a lack of recognition in the existence of epistemic uncertainty and aleatory variability, and hence inappropriate design methods are often used. To overcome this, a novel taxonomy is developed and presented that facilitates characterisation of epistemic uncertainty and aleatory variability in the context of rock mechanics and rock engineering. Using this taxonomy, a new framework is developed that gives a protocol for correctly propagating uncertainty and variability through engineering calculations. The effectiveness of the taxonomy and the framework are demonstrated through their application to simple challenge problems commonly found in rock engineering. This new taxonomy and framework will provide engineers engaged in preparing rock engineering designs an objective means of characterising unpredictability in parameters commonly used to define properties of fractured rock masses. These new tools will also provide engineers with a means of clearly understanding the true nature of unpredictability inherent in rock mechanics and rock engineering, and thus direct selection of an appropriate unpredictability model to propagate unpredictability faithfully through engineering calculations. Thus, the taxonomy and framework developed in this thesis provide practical tools to improve the safety of rock engineering designs through an improved understanding of the unpredictability concepts.Open Acces

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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Clouds, p-boxes, fuzzy sets, and other uncertainty representations in higher dimensions

Uncertainty modeling in real-life applications comprises some serious problems such as the curse of dimensionality and a lack of sufficient amount of statistical data. In this paper we give a survey of methods for uncertainty handling and elaborate the latest progress towards real-life applications with respect to the problems that come with it. We compare different methods and highlight their relationships. We introduce intuitively the concept of potential clouds, our latest approach which successfully copes with both higher dimensions and incomplete information