DOCSLIB.ORG

Uncertainty Modelling for Scarce and Imprecise Data in Engineering Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. Techniques are described to create Interval Predictor Models which are tailored to model the performance of a system for reliability analysis. This thesis also describes analytical techniques which can be used for probabilistic safety analysis, in the case that the system model is not a black box. This is advantageous as it enables engineers to perform calculations without spending time programming complex Monte Carlo simulations. A technique is presented to efficiently create Interval Predictor Models for datasets of arbitrary complexity and size, which may contain imprecise data, and we call these models Interval Neural Networks. Case studies or numerical examples are presented to demonstrate the performance of the proposed techniques, including some common benchmarks and a finite element model. The interval predictor models used in this thesis were implemented in the open source uncertainty quantification software OpenCossan, and are now freely i Jonathan Cyrus Sadeghi available.∗ ∗https://github.com/cossan-working-group/OpenCossan/tree/development/+opencossan/ +metamodels/@IntervalPredictorModel ii Acknowledgements I am grateful for the academic support of my supervisors Dr. Edoardo Patelli and Dr. Marco de Angelis. Edoardo Patelli has introduced me to the field of uncertainty quantification, whilst always giving me the opportunity to grow as a researcher. In particular, he has allowed me to contribute to the software development of OpenCossan, which has provided me with an invaluable educational experience. I am very grateful to him for having taken me on board. Marco de Angelis has provided excellent guidance, even before he joined the supervisory team for my PhD. His explanations of concepts in imprecise probability have had a profound impact on my understanding of the field, and these conversations have shaped my opinion of how reliable engineering can be achieved. In addition, he has frequently provided practical assistance to assist me with the OpenCossan software. I gratefully acknowledge the financial support of Wood Plc (formerly Amec Foster Wheeler) and the Engineering and Physical Sciences Research Council (EPSRC Grant reference: EP/L015390/1), which enabled me to undertake my studies. In addition, the Next Generation Nuclear centre for doctoral training provided many opportunities to develop my knowledge of the nuclear industry. The Institute for Risk and Uncertainty at the University of Liverpool was also an important part of doctoral studies, where I benefited from frequent seminars and events. The welcoming atmosphere of the Institute provided an enjoyable environment in which to conduct my studies. During my time there, I met many collaborators and friends, who are too numerous to mention on this page and hence I apologise to anyone who I have not mentioned. The contribution of Prof. Nawal Prinja (Wood Plc) to the work in Chapter 5 is gratefully acknowledged. Specifically, Prof. Prinja was responsible for the description of the structural experiments and devising the structural model to which he applied the First Order Reliability Method. Conversations with Prof. Prinja also inspired many of the developments in Chapter 6. His decades of experience in the nuclear industry have been a great inspiration. The contribution of Dr. Matthias Faes to Chapter 7, is gratefully acknowledged. Specifically, Dr. Faes contributed the benchmarks to which the Interval Predictor Model methodology was compared in the case study, in addition to a large amount of the text for the original paper, on which this chapter is based (not including novel contributions related to Interval Predictor Models by the present author). iii Jonathan Cyrus Sadeghi I owe a debt of gratitude to Phoebe, my partner and closest friend, who has been beside me to continually raise my spirits during my PhD. Finally, I wish to thank my family, and in particular my parents, for their invaluable support over the years. They have always pushed me forward at difficult times throughout my studies. iv Declaration I hereby confirm that the results presented in this dissertation are from my own work and that I have not presented anyone else's work and that full and appropriate acknowledgements have been given where references have been made to the work of others. This dissertation contains 202 pages, 59 figures, 23 tables and 38362 words. Jonathan Sadeghi v Jonathan Cyrus Sadeghi vi Contents Abstract i Acknowledgements iii Declaration v Contents x List of Figures xv List of Tables xvii List of Published Work xviii Nomenclature xx 1 Introduction 1 1.1 Motivation . .1 1.1.1 Uncertainty in engineering simulation . .1 1.1.2 Reliability engineering . .2 1.1.3 Models of uncertainty . .3 1.2 Problem definition and objectives . .4 1.3 Structure of this thesis . .5 2 Models of Uncertainty 9 2.1 Overview of uncertainty models . .9 2.1.1 Probabilistic models of uncertainty . 10 2.1.2 Set-based models of uncertainty . 14 2.1.3 Imprecise probabilistic models of uncertainty . 17 2.2 Creating models in practice . 20 2.2.1 Choosing a model . 20 2.2.2 Training models from data . 21 vii Jonathan Cyrus Sadeghi 2.2.3 Creating uncertainty models without data . 25 2.2.4 Validating a trained model . 27 2.3 Chapter summary . 28 3 Machine Learning of Regression Models 30 3.1 Parametric regression models . 30 3.1.1 Bayesian parameter learning . 33 3.1.2 Validation . 36 3.1.3 The bias-variance tradeoff . 37 3.2 Non-parametric models . 37 3.3 Learning bounds on a model . 38 3.3.1 Training interval predictor models . 39 3.3.2 Validating models with the scenario approach . 44 3.3.3 Software for interval predictor models . 48 3.4 Chapter summary . 48 4 Reliability Analysis 49 4.1 Reliability analysis with random variables . 49 4.1.1 Problem definition . 49 4.1.2 Methods to compute the failure probability . 52 4.2 Convex set models for reliability . 58 4.2.1 Problem definition . 58 4.3 Reliability analysis with probability boxes . 59 4.3.1 Problem definition . 59 4.3.2 Methods to compute the failure probability . 60 4.4 Chapter summary . 62 5 Structural Reliability of Pre-stressed Concrete Containments under Dis- tributional Uncertainty 63 5.1 Introduction . 63 5.2 Structural model . 64 5.3 Analysis . 68 5.4 Discussion . 70 5.5 Chapter summary . 79 6 Analytic Imprecise Probabilistic Safety Analysis 81 6.1 Introduction . 81 6.2 Probabilistic safety analysis . 82 6.3 Probability bounds analysis . 85 6.3.1 Fragility curve . 85 6.3.2 Product of log-normally distributed random variables . 89 viii Uncertainty Modelling for Scarce and Imprecise Data in Engineering Applications 6.3.3 Failure probability . 90 6.3.4 Summary of failure probability expressions . 93 6.3.5 Imprecise FORM . 98 6.4 Numerical examples . 99 6.4.1 Reliability analysis of a simple concrete containment . 99 6.4.2 Containment with additive component strengths . 101 6.5 Chapter summary . 103 7 Interval Predictor Models for Reliability Analysis 104 7.1 Introduction . 104 7.2 Interval failure probability . 105 7.3 Uncertain surrogate model predictions . 106 7.3.1 Kriging . 107 7.3.2 Interval predictor models . 108 7.4 Case study . 110 7.4.1 Advanced Monte Carlo sampling . 111 7.4.2 Surrogate model based estimation . 115 7.4.3 When is creating an IPM surrogate worthwhile? . 117 7.5 Conclusions . 119 8 Interval Predictor Models for Propagation of Probability Boxes 120 8.1 Introduction . 120 8.2 Proposed approaches: obtaining bounds on the failure probability . 121 8.2.1 Approach 1: metamodels for na¨ıve double loop approach . 122 8.2.2 Approach 2: IPMs trained on propagated focal elements . 123 8.2.3 Approach 3: metamodels for non-na¨ıve approach . 124 8.2.4 Confidence bounds on failure probability . 126 8.3 Numerical examples . 127 8.3.1 Cantilever beam . 127 8.3.2 Dynamic response of a non-linear oscillator . 129 8.3.3 Small satellite . 137 8.4 Chapter summary . 139 9 Interval Neural Networks 143 9.1 Introduction .
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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,
    [Show full text]
  • 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..............................
    [Show full text]
  • 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.
    [Show full text]
  • Eastern Box Turtles (Terrapene C. Carolina) in the Oak Openings Region of Northwestern Ohio
    SPATIAL ECOLOGY OF EASTERN BOX TURTLES (TERRAPENE C. CAROLINA) IN THE OAK OPENINGS REGION OF NORTHWESTERN OHIO Matthew D. Cross A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2016 Committee: Karen V. Root, Advisor Salim Elwazani Graduate Faculty Representative Enrique Gomezdelcampo Jeffery G. Miner Shannon L. Pelini © 2016 Matthew D. Cross All Rights Reserved iii ABSTRACT Karen V. Root, Advisor Eastern Box Turtles (Terrapene c. carolina) have experienced range-wide declines as the result of extensive habitat loss, fragmentation, and alteration. The Oak Openings Region of northwestern Ohio is a biodiversity hotspot that exists in a highly fragmented landscape and provides a unique case study from which to examine the effects of anthropogenic disturbance on Eastern Box Turtles. In an effort to inform management and conservation efforts in the Oak Openings Region, I initiated a radio-telemetry project with the following objectives: 1) examine the spatial ecology of Eastern Box Turtles over several years to understand how they interact with their habitat in an area as unique as the Oak Openings Region, 2) develop predictive models depicting the temporal distributions of Eastern Box Turtles, 3) examine the impacts of one of the most common management tools in the Oak Openings Region, prescribed fire, on Eastern Box Turtles, and 4) evaluate pattern-recognition software as a low-cost alternative of identifying individual Eastern Box Turtles. Turtles at my study site exhibited larger home ranges than previously reported for this species as well as hierarchical habitat selection at multiple scales.
    [Show full text]
  • Propagation of Imprecise Probabilities Through
    PROPAGATION OF IMPRECISE PROBABILITIES THROUGH BLACK-BOX MODELS A Thesis Presented to The Academic Faculty by Morgan Bruns In Partial Fulfillment of the Requirements for the Degree Master of Science in the School of Mechanical Engineering Georgia Institute of Technology May 2006 PROPAGATION OF IMPRECISE PROBABILITIES THROUGH BLACK-BOX MODELS Approved by: Dr. Christiaan J.J. Paredis, Advisor School of Mechanical Engineering Georgia Institute of Technology Dr. Bert Bras School of Mechanical Engineering Georgia Institute of Technology Dr. Leon F. McGinnis School of Industrial and Systems Engineering Georgia Institute of Technology Date Approved: April 7, 2006 ACKNOWLEDGEMENTS This thesis would not have been possible without the assistance of many of my friends and colleagues. I would like to thank my advisor, Dr. Chris Paredis, for his support, guidance, and insight. I would also like to thank the students of the Systems Realization Laboratory for the intellectually nurturing environment they have provided. In particular, I would like to thank Jason Aughenbaugh, Scott Duncan, Jay Ling, Rich Malak, and Steve Rekuc for their criticism and advice regarding my research. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS iii LIST OF TABLES vi LIST OF FIGURES vii LIST OF ABBREVIATIONS viii LIST OF SYMBOLS ix SUMMARY xi CHAPTER 1 : INTRODUCTION 1 1.1 Design decision making 1 1.2 Imprecision in design 2 1.3 Challenges for decision making formalisms 4 1.4 Research questions, hypotheses, and thesis outline 5 CHAPTER 2 : DECISION MAKING UNDER IMPRECISE UNCERTAINTY
    [Show full text]
  • SIPTA Summer School: Engineering
    SIPTA Summer School: Engineering Day Scott Ferson, Institute for Risk and Uncertainty, University of Liverpool Society for Imprecise Probabilities: Theory and Applications 24-28 July 2018, Oviedo, Spain © 2018 Scott Ferson Engineering • Publication advice joke • Practical applications • Get things done, even if not perfectly • Random variables, not just events • Logic, arithmetic, FE and ODE models • Continuous, infinite • Quick & dirty is better than elegant & abstract • Sloppy with notation, but big on computability Installing R • Locate the latest version by Googling "R" or going to http://www.r-project.org/ • Download and install the base software – For instance, get version 3.5.0 for Windows from https://cloud.r-project.org/bin/windows/base/R-3.5.0-win.exe • Install R by executing this self-extracting file and invoke the program by double-clicking on the blue R icon it leaves on your desktop Leaving R • To leave R click File / Exit from the main menu or enter the command quit() • R may ask whether you want to save your work • If you do, your variables and functions will be available the next time you invoke R Install the pba.r probability box library • If you left R, invoke it again • Enter rm(list = ls()) to clear its memory • Click File / Source R code on the main menu • Locate and Open the file pba.r • You’ll get the message ":pba> library loaded" Die-hard RStudio users • If, against advice, you must use RStudio… • File / Open file pbox.r • Source (Ctrl+Shift+S) pbox.r • Enter the instruction RStudio = TRUE pba.r probability bounds
    [Show full text]
  • Verified Computation with Probabilities
    Verified Computation with Probabilities Scott Ferson and Jack Siegrist Applied Biomathematics Abstract. Because machine calculations are prone to errors that can sometimes accumulate disastrously, computer scientists use special strategies called verified computation to ensure output is reliable. Such strategies are needed for computing with probability distributions. In probabilistic calculations, analysts have routinely assumed (i) probabil- ities and probability distributions are precisely specified, (ii) most or all variables are independent or otherwise have well-known dependence, and (iii) model structure is known perfectly. These assumptions are usually made for mathematical convenience, rather than with empirical justi- fication, even in sophisticated applications. Probability bounds analysis computes bounds guaranteed to enclose probabilities and probability dis- tributions even when these assumptions are relaxed or removed. In many cases, results are best-possible bounds, i.e., tightening them requires ad- ditional empirical information. This paper presents an overview of prob- ability bounds analysis as a computationally practical implementation of the theory of imprecise probabilities that represents verified computation of probabilities and distributions. Keywords: probability bounds analysis, probability box, p-box, veri- fied computation, imprecise probabilities, interval analysis, probabilistic arithmetic. 1 Introduction Many high-profile disasters are attributable to numerical errors in computer calculations. The self-destruction of the Ariane 5 rocket on its maiden test flight was caused by integer overflow (ESA 1996). The crash of the Mars Climate Orbiter during orbital insertion resulted from a units incompatibility (Isbell et al. 1999). The Sleipner A offshore drilling platform sank because of an inaccurate finite element approximation (Selby et al. 1997). The Aegis cruiser USS Yorktown was dead in the water for several hours because of a propagated divide-by-zero error (Slabodkin 1998).
    [Show full text]