Engineering Analysis with Probability Boxes: a Review on Computational Methods M
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Literature Review of Methods for Representing Uncertainty
Methods for representing uncertainty A literature review Enrico Zio and Nicola Pedroni n° 2013-03 THEME Risk analysis he Foundation for an Industrial Safety Culture (FonCSI) is a french public-interest research T foundation created in 2005. It aims to: undertake and fund research activities that contribute to improving safety in hazardous organizations (industrial firms of all sizes, in all industrial sectors); work towards better mutual understanding between high-risk industries and civil society, aiming for a durable compromise and an open debate that covers all the dimensions of risk; foster the acculturation of all stakeholders to the questions, tradeoffs and problems related to risk and safety. In order to attain these objectives, the FonCSI works to bring together researchers from different scientific disciplines with other stakeholders concerned by industrial safety and the management of technological risk: companies, local government, trade unions, NGOs. We also attempt to build bridges between disciplines and to promote interaction and cross-pollination between engineering, sciences and the humanities. The work presented in this document is the result of research funded by the FonCSI.The opinions presented are those of the authors. Foundation for an Industrial Safety Culture A public-interest research foundation http://www.FonCSI.org/ 6 allée Émile Monso — BP 34038 Telephone: +33 534 32 32 00 31029 Toulouse cedex 4 Twitter: @LaFonCSI France Email: [email protected] iii Résumé Titre Revue de littérature sur la représentation de l’incertitude dans l’analyse de risque Mots-clefs incertitude, analyse de risque, approches possibilistes, analyse d’intervalle Auteurs Enrico Zio et Nicola Pedroni Date de publication avril 2013 Ce document fournit une analyse critique de différentes méthodes de représentation et d’analyse d’incertitude dans les analyses de risques: l’analyse probabiliste classique, les probabilités imprécises (analyse d’intervalle), les fonctions de croyance de Dempster-Shafer et la théorie des possibilités. -
Uncertainty Modelling for Scarce and Imprecise Data in Engineering Applications
Uncertainty Modelling for Scarce and Imprecise Data in Engineering Applications Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy by Jonathan Cyrus Sadeghi January 2020 Uncertainty Modelling for Scarce and Imprecise Data in Engineering Applications Abstract In this thesis, models for uncertainty quantification in the case of scarce and imprecise data are described, and the computational efficiency of simulations with these models is improved. Specifically, probability boxes are used to describe imprecision in cumulative distribution functions. This may be the case when imprecise data is used to train a model, or the prior knowledge regarding a property of the system being studied is very weak. Performing simulations with probability boxes is often computationally expensive, because an optimisation program must be solved to obtain each sample in a Monte Carlo simulation. When the system model is known analytically, it is possible to significantly reduce the cost of the analysis. However, the system model is often a black box which can only be queried for a particular point value of the input. Each evaluation or query of the system model is often computationally expensive in itself. Currently, few efficient methods exist to perform computations with probability boxes, and the techniques which exist do not provide rigorous bounds on the obtained probability of failure. Interval Predictor Models are a technique to create an approximate representation of a function, where the uncertainty in the true function is described as an interval, with statistical guarantees on the coverage of the true function. This thesis proposes the use of Interval Predictor Models to create an approximate surrogate model for the true black box system model and hence obtain rigorous bounds on the probability of failure of a system. -
Imprecise Probability in Risk Analysis
IMPRECISE PROBABILITY IN RISK ANALYSIS D. Dubois IRIT-CNRS, Université Paul Sabatier 31062 TOULOUSE FRANCE Outline 1. Variability vs incomplete information 2. Blending set-valued and probabilistic representations : uncertainty theories 3. Possibility theory in the landscape 4. A methodology for risk-informed decision- making 5. Some applications Origins of uncertainty • The variability of observed repeatable natural phenomena : « randomness ». – Coins, dice…: what about the outcome of the next throw? • The lack of information: incompleteness – because of information is often lacking, knowledge about issues of interest is generally not perfect. • Conflicting testimonies or reports:inconsistency – The more sources, the more likely the inconsistency Example • Variability: daily quantity of rain in Toulouse – May change every day – It can be estimated through statistical observed data. – Beliefs or prediction based on this data • Incomplete information : Birth date of Brazil President – It is not a variable: it is a constant! – You can get the correct info somewhere, but it is not available. – Most people may have a rough idea (an interval), a few know precisely, some have no idea: information is subjective. – Statistics on birth dates of other presidents do not help much. • Inconsistent information : several sources of information conflict concerning the birth date (a book, a friend, a website). The roles of probability Probability theory is generally used for representing two aspects: 1. Variability: capturing (beliefs induced by) variability through repeated observations. 2. Incompleteness (info gaps): directly modeling beliefs via betting behavior observation. These two situations are not mutually exclusive. Using a single probability distribution to represent incomplete information is not entirely satisfactory: The betting behavior setting of Bayesian subjective probability enforces a representation of partial ignorance based on single probability distributions. -
Reliability of Computational Science
Reliability of Computational Science I. Babuska∗ F. Nobile† R. Tempone‡ November 2, 2006 Abstract Today’s computers allow us to simulate large, complex physical prob- lems. Many times the mathematical models describing such problems are based on a relatively small amount of available information such as exper- imental measurements. The question arises whether the computed data could be used as the basis for decision in critical engineering, economic, medicine applications. The representative list of engineering accidents oc- curred in the past years and their reasons illustrates the question. The paper describes a general framework for Verification and Validation which deals with this question. The framework is then applied to an illustrative engineering problem, in which the basis for decision is a specific quantity of interest, namely the probability that the quantity does not exceed a given value. The V&V framework is applied and explained in detail. The result of the analysis is the computation of the failure probability as well as a quantification of the confidence in the computation, depending on the amount of available experimental data. 1 Introduction Computational Science is a discipline concerned with the use of computers for the prediction of physical phenomena. These predictions are used as the basis for critical decisions in engineering and in other fields such as environment, heath, management, etc. Rapid development of computer hardware allows us to make predictions of more and more complex phenomena. The major problem arises: How reliable are these predictions? Could they be the basis for decisions, often very crucial and with large implications? The reliability problem has many aspects: mathematical, numerical, computational, ∗ICES, University of Texas at Austin, Texas, USA. -
Uncertainty Propagation of P-Boxes Using Sparse Polynomial Chaos Expansions R Schöbi, B Sudret
Uncertainty propagation of p-boxes using sparse polynomial chaos expansions R Schöbi, B Sudret To cite this version: R Schöbi, B Sudret. Uncertainty propagation of p-boxes using sparse polynomial chaos expansions. 2016. hal-01421112 HAL Id: hal-01421112 https://hal.archives-ouvertes.fr/hal-01421112 Preprint submitted on 21 Dec 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNCERTAINTY PROPAGATION OF P-BOXES USING SPARSE POLYNOMIAL CHAOS EXPANSIONS R. Schobi,¨ B. Sudret CHAIR OF RISK,SAFETY AND UNCERTAINTY QUANTIFICATION Risk, Safety & STEFANO-FRANSCINI-PLATZ 5 Uncertainty Quantification CH-8093 ZURICH¨ Data Sheet Journal: - Report Ref.: RSUQ-2016-009 Arxiv Ref.: http://arxiv.org/abs/1608.05565 - [stat.CO] DOI: - Date submitted: August 19, 2016 Date accepted: - Uncertainty propagation of p-boxes using sparse polynomial chaos expansions R. Sch¨obi1 and B. Sudret1 1Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Stefano-Franscini-Platz 5, 8093 Zurich, Switzerland Abstract In modern engineering, physical processes are modelled and analysed using advanced com- puter simulations, such as finite element models. Furthermore, concepts of reliability analysis and robust design are becoming popular, hence, making efficient quantification and propagation of uncertainties an important aspect. -
On the Calculation of the Bounds of Probability of Events Using Infinite
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector International Journal of Approximate Reasoning 43 (2006) 241–267 www.elsevier.com/locate/ijar On the calculation of the bounds of probability of events using infinite random sets Diego A. Alvarez * Arbeitsbereich fu¨r Technische Mathematik am Institut fu¨r Grundlagen der Bauingenieurwissenschaften, Leopold-Franzens Universita¨t, Technikerstrasse 13, A-6020 Innsbruck, EU, Austria Received 5 December 2005; received in revised form 9 March 2006; accepted 19 April 2006 Available online 22 May 2006 Abstract This paper presents an extension of the theory of finite random sets to infinite random sets, that is useful for estimating the bounds of probability of events, when there is both aleatory and epistemic uncertainty in the representation of the basic variables. In particular, the basic variables can be mod- elled as CDFs, probability boxes, possibility distributions or as families of intervals provided by experts. These four representations are special cases of an infinite random set. The method intro- duces a new geometrical representation of the space of basic variables, where many of the methods for the estimation of probabilities using Monte Carlo simulation can be employed. This method is an appropriate technique to model the bounds of the probability of failure of structural systems when there is parameter uncertainty in the representation of the basic variables. A benchmark example is used to demonstrate the advantages and differences of the proposed method compared with the finite approach. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Random sets; Dempster–Shafer evidence theory; Epistemic uncertainty; Aleatory uncertainty, Monte Carlo simulation * Tel.: +43 512 5076825; fax: +43 512 5072941. -
A Critical Discussion and Practical Recommendations on Some Issues
A critical discussion and practical recommendations on some issues relevant to the non-probabilistic treatment of uncertainty in engineering risk assessment Nicola Pedroni 1, Enrico Zio 2, Alberto Pasanisi 3, Mathieu Couplet 4 1Corresponding author. Chair “System Science and the Energy challenge”-Fondation Electricité de France (EdF) at the Laboratoire Genie Industriel (LGI), CentraleSupélec, Université Paris-Saclay, Grande voie des Vignes, 92290 Chatenay-Malabry, France. E-mail: [email protected] 2Chair “System Science and the Energy challenge”-Fondation Electricité de France (EdF) at the Laboratoire Genie Industriel (LGI), CentraleSupélec, Université Paris-Saclay, Grande voie des Vignes, 92290 Chatenay-Malabry, France. E-mail: [email protected] . Also: Energy Department, Politecnico di Milano, Via Ponzio, 34/3 – 20133 Milano, Italy. E-mail: [email protected] . 3EDF R&D - EIFER, Emmy-Noether-Str. 11. 76131 Karlsruhe, Germany. E-mail address: [email protected] . 4Electricité de France, R&D, 6 Quai Watier, 78400, Chatou, France. E-mail address: [email protected] . 1 ABSTRACT Models for the assessment of the risk of complex engineering systems are affected by uncertainties, due to the randomness of several phenomena involved and the incomplete knowledge about some of the characteristics of the system. The objective of the present paper is to provide operative guidelines to handle some conceptual and technical issues related to the treatment of uncertainty in risk assessment for engineering practice. In particular, the following issues are addressed: (1) quantitative modeling and representation of uncertainty coherently with the information available on the system of interest; (2) propagation of the uncertainty from the input(s) to the output(s) of the system model; (3) (Bayesian) updating as new information on the system becomes available; (4) modeling and representation of dependences among the input variables and parameters of the system model. -
Belief Functions: a Gentle Introduction Seoul National University
Belief functions: A gentle introduction Seoul National University Professor Fabio Cuzzolin School of Engineering, Computing and Mathematics Oxford Brookes University, Oxford, UK Seoul, Korea, 30/05/18 Professor Fabio Cuzzolin Belief functions: A gentle introduction Seoul, Korea, 30/05/18 1 / 125 Uncertainty Outline Belief functions Decision making 1 Uncertainty Semantics 5 Second-order uncertainty Dempster’s rule Theories of uncertainty Classical probability Multivariate analysis Imprecise probability 2 Beyond probability Misunderstandings Monotone capacities Probability intervals Set-valued observations 4 Reasoning with belief Propositional evidence Fuzzy and possibility theory functions Probability boxes Scarce data Statistical inference Rough sets Representing ignorance Combination Rare events Conditioning 6 Belief functions on reals Uncertain data Belief vs Bayesian reasoning Continuous belief functions 3 Belief theory Generalised Bayes Theorem Random sets A theory of evidence The total belief theorem 7 Conclusions Professor Fabio Cuzzolin Belief functions: A gentle introduction Seoul, Korea, 30/05/18 2 / 125 Uncertainty Second-order uncertainty Orders of uncertainty the difference between predictable and unpredictable variation is one of the fundamental issues in the philosophy of probability second order uncertainty: being uncertain about our very model of uncertainty has a consequence on human behaviour: people are averse to unpredictable variations (as in Ellsberg’s paradox) how good are Kolmogorov’s measure-theoretic probability, -
CMIS Modelling Optimisation Simulation
Uncertainty and uncertainty analysis methods Keith R. Hayes February 2011 Report Number: EP102467 Issues in quantitative and qualitative risk modeling with application to import risk assessment ACERA project (0705) Enquiries should be addressed to: Keith R Hayes CSIRO Mathematics, Informatics and Statistics GPO Box 1538 Hobart, Tasmania, Australia, 7001 Castray Esplanade, Hobart, Australia, 7000 Telephone : +61 3 6232 5260 Fax : +61 3 6232 5485 Email : [email protected] Distribution List Client (1) Publications Officer (X) Stream Leader (1) Authors (1) Copyright and Disclaimer c CSIRO To the extent permitted by law, all rights are reserved and no part of this publication covered by copyright may be reproduced or copied in any form or by any means except with the written permission of CSIRO. Important Notice CSIRO advises that the information contained in this publication comprises general statements based on scientific research. The reader is advised and needs to be aware that such information may be incomplete or unable to be used in any specific situation. No reliance or actions must therefore be made on that information without seeking prior expert professional, scientific and technical advice. To the extent permitted by law, CSIRO (including its employees and consul- tants) excludes all liability to any person for any consequences, including but not limited to all losses, damages, costs, expenses and any other compensation, arising directly or indirectly from using this publication (in part or in whole) and any information or material contained in it. Contents 1 Introduction..................................... 8 1.1 Project background...............................8 1.2 Report structure and outline...........................8 1.3 Fundamental issues.............................. -
Imprecise Probability for Multiparty Session Types in Process Algebra APREPRINT
IMPRECISE PROBABILITY FOR MULTIPARTY SESSION TYPES IN PROCESS ALGEBRA A PREPRINT Bogdan Aman Gabriel Ciobanu Romanian Academy, Institute of Computer Science Romanian Academy, Institute of Computer Science Alexandru Ioan Cuza University, Ia¸si, Romania Alexandru Ioan Cuza University, Ia¸si, Romania [email protected] [email protected] February 18, 2020 ABSTRACT In this paper we introduce imprecise probability for session types. More exactly, we use a proba- bilistic process calculus in which both nondeterministic external choice and probabilistic internal choice are considered. We propose the probabilistic multiparty session types able to codify the structure of the communications by using some imprecise probabilities given in terms of lower and upper probabilities. We prove that this new probabilistic typing system is sound, as well as several other results dealing with both classical and probabilistic properties. The approach is illustrated by a simple example inspired by survey polls. Keywords Imprecise Probability · Nondeterministic and Probabilistic Choices · Multiparty Session Types. 1 Introduction Aiming to represent the available knowledge more accurately, imprecise probability generalizes probability theory to allow for partial probability specifications applicable when a unique probability distribution is difficult to be identified. The idea that probability judgments may be imprecise is not novel. It would be enough to mention the Dempster- Shafer theory for reasoning with uncertainty by using upper and lower probabilities [1]. The idea appeared even earlier in 1921, when J.M. Keynes used explicit intervals for approaching probabilities [2] (B. Russell called this book “undoubtedly the most important work on probability that has appeared for a very long time”). -
Propagation of Uncertainties Modelled by Parametric P-Boxes Using Sparse Polynomial Chaos Expansions Roland Schöbi, Bruno Sudret
Propagation of Uncertainties Modelled by Parametric P-boxes Using Sparse Polynomial Chaos Expansions Roland Schöbi, Bruno Sudret To cite this version: Roland Schöbi, Bruno Sudret. Propagation of Uncertainties Modelled by Parametric P-boxes Using Sparse Polynomial Chaos Expansions. 12th Int. Conf. on Applications of Statistics and Probability in Civil Engineering (ICASP12), Jul 2015, Vancouver, Canada. hal-01247151 HAL Id: hal-01247151 https://hal.archives-ouvertes.fr/hal-01247151 Submitted on 21 Dec 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 Propagation of Uncertainties Modelled by Parametric P-boxes Using Sparse Polynomial Chaos Expansions Roland Schöbi Ph.D. student, Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Zurich, Switzerland Bruno Sudret Professor, Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Zurich, Switzerland ABSTRACT: Advanced simulations, such as finite element methods, are routinely used to model the be- haviour of physical systems and processes. At the same time, awareness is growing on concepts of struc- tural reliability and robust design. This makes efficient quantification and propagation of uncertainties in computation models a key challenge. -
Demystifying Dilation
Forthcoming in Erkenntnis. Penultimate version. Demystifying Dilation Arthur Paul Pedersen Gregory Wheeler Abstract Dilation occurs when an interval probability estimate of some event E is properly included in the interval probability estimate of E conditional on every event F of some partition, which means that one’s initial estimate of E becomes less precise no matter how an experiment turns out. Critics main- tain that dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. How- ever, two points are often overlooked: (i) knowing that E is stochastically independent of F (for all F in a partition of the underlying state space) is sufficient to avoid dilation, but (ii) stochastic independence is not the only independence concept at play within imprecise probability models. In this paper we give a simple characterization of dilation formulated in terms of deviation from stochastic independence, propose a measure of dilation, and distinguish between proper and improper dilation. Through this we revisit the most sensational examples of dilation, which play up independence be- tween dilator and dilatee, and find the sensationalism undermined by either fallacious reasoning with imprecise probabilities or improperly constructed imprecise probability models. 1 Good Grief! Unlike free advice, which can be a real bore to endure, accepting free information when it is available seems like a Good idea. In fact, it is: I. J. Good (1967) showed that under certain assumptions it pays you, in expectation, to acquire new infor- mation when it is free. This Good result reveals why it is rational, in the sense of maximizing expected utility, to use all freely available evidence when estimating a probability.