Data Uncertainty and important Measures

231 Pages·2018·3.172 MB·english

by  Christophe Simon, Philippe Weber, Mohamed Sallak

Systems Dependability Assessment Set
coordinated by
Jean-François Aubry
Volume 3
Data Uncertainty and
Important Measures
Christophe Simon
Philippe Weber
Mohamed Sallak
First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
ISTE Ltd John Wiley & Sons, Inc.
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London SW19 4EU Hoboken, NJ 07030
UK USA
www.iste.co.uk www.wiley.com
© ISTE Ltd 2018
Library of Congress Control Number: 2017958413

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-84821-993-9
Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter 1. Why and Where Uncertainties . . . . . . . . . . . . . 1
1.1.Sourcesandformsofuncertainty . . . . . . . . . . . . . . . . . 1
1.2.Typesofuncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.Sourcesofuncertainty . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Chapter 2. Models and Language of Uncertainty . . . . . . . . . 9
2.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.Probabilitytheory . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1.Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2.Fundamentalnotions . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.Belieffunctionstheory . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1.Representationofbeliefs . . . . . . . . . . . . . . . . . . . . 16
2.3.2.Combinationrules . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.3.Extensionandmarginalization . . . . . . . . . . . . . . . . . 20
2.3.4.Pignistictransformation . . . . . . . . . . . . . . . . . . . . . 20
2.3.5.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.Fuzzysettheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.1.Basicdefinitions . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.2.Operationsonfuzzysets . . . . . . . . . . . . . . . . . . . . 22
2.4.3.Fuzzyrelations . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.Fuzzyarithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.1.Fuzzynumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.2.Fuzzyprobabilities . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.3.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6.Possibilitytheory . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6.1.Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6.2.Possibilityandnecessitymeasures . . . . . . . . . . . . . . . 30
2.6.3.Operationsonpossibilityand
necessitymeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.7.Randomsettheory . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.7.1.Basicdefinitions . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7.2.Expectationofrandomsets . . . . . . . . . . . . . . . . . . . 34
2.7.3.Randomintervals . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7.4.Confidenceinterval . . . . . . . . . . . . . . . . . . . . . . . 35
2.7.5.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.8.Confidencestructuresorc-boxes . . . . . . . . . . . . . . . . . . 36
2.8.1.Basicnotions . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.8.2.Confidencedistributions . . . . . . . . . . . . . . . . . . . . 37
2.8.3.P-boxesandC-boxes . . . . . . . . . . . . . . . . . . . . . . 38
2.8.4.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.9.Impreciseprobabilitytheory . . . . . . . . . . . . . . . . . . . . 40
2.9.1.Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.9.2.Basicproperties . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.9.3.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.10.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Chapter 3. Risk Graphs and Risk Matrices: Application
of Fuzzy Sets and Belief Reasoning . . . . . . . . . . . . . . . . . 47
3.1.SILallocationscheme . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.1.Safetyinstrumentedsystems(SIS) . . . . . . . . . . . . . . . 48
3.1.2.ConformitytostandardsANSI/ISA
S84.01-1996andIEC61508 . . . . . . . . . . . . . . . . . . . . . . 49
3.1.3.Taxonomyofrisk/SILassessmentmethods . . . . . . . . . . 50
3.1.4.Riskassessment . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1.5.SILallocationprocess . . . . . . . . . . . . . . . . . . . . . . 52
3.1.6.Theuseofexperts’opinions . . . . . . . . . . . . . . . . . . 53
3.2.SILallocationbasedonpossibilitytheory. . . . . . . . . . . . . 54
3.2.1.Elicitingtheexperts’opinions . . . . . . . . . . . . . . . . . 54
3.2.2.Ratingscalesforparameters . . . . . . . . . . . . . . . . . . 55
3.2.3.Subjectiveelicitationoftheriskparameters . . . . . . . . . 56
3.2.4.Calibrationofexperts’opinions . . . . . . . . . . . . . . . . 59
3.2.5.Aggregationoftheopinions . . . . . . . . . . . . . . . . . . 61
3.3.Fuzzyriskgraph . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.1.Inputfuzzypartitionandfuzzification . . . . . . . . . . . . . 65
3.3.2.Risk/SILgraphlogicbyfuzzyinference
system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.3.Outputfuzzypartitionanddefuzzification . . . . . . . . . . 67
3.3.4.Illustrationcase . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.Risk/SILgraph: belieffunctionsreasoning . . . . . . . . . . . . 72
3.4.1.Elicitationofexpertopinionsinthebelief
functionstheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4.2.Aggregationofexpertopinions . . . . . . . . . . . . . . . . 73
3.5.Evidentialriskgraph . . . . . . . . . . . . . . . . . . . . . . . . 75
3.6.Numericalillustration . . . . . . . . . . . . . . . . . . . . . . . . 77
3.6.1.Clusteringofexperts’opinions . . . . . . . . . . . . . . . . . 77
3.6.2.Aggregationofpreferences . . . . . . . . . . . . . . . . . . . 78
3.6.3.Evidentialrisk/SILgraph . . . . . . . . . . . . . . . . . . . . 79
3.7.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Chapter 4. Dependability Assessment Considering
Interval-valued Probabilities . . . . . . . . . . . . . . . . . . . . . . 83
4.1.Intervalarithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.1.1.Interval-valuedparameters . . . . . . . . . . . . . . . . . . . 84
4.1.2.Interval-valuedreliability . . . . . . . . . . . . . . . . . . . . 85
4.1.3.Assessingtheimpreciseaverageprobability
offailureondemand . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2.Constraintarithmetic . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3.Fuzzyarithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3.1.Applicationexample . . . . . . . . . . . . . . . . . . . . . . 95
4.3.2.MonteCarlosamplingapproach . . . . . . . . . . . . . . . . 97
4.4.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4.1.Markovchains . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4.2.MultiphaseMarkovchains . . . . . . . . . . . . . . . . . . . 101
4.4.3.Markovchainswithfuzzynumbers . . . . . . . . . . . . . . 102
4.4.4.FuzzymodelingofSIScharacteristic
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.5.Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.5.1.Epistemicapproach . . . . . . . . . . . . . . . . . . . . . . . 106
4.5.2.EnhancedMarkovanalysis . . . . . . . . . . . . . . . . . . . 113
4.6.Decision-makingunderuncertainty . . . . . . . . . . . . . . . . 115
4.7.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Chapter 5. Evidential Networks . . . . . . . . . . . . . . . . . . . . 119
5.1.Mainconcepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.1.1.Temporaldimension . . . . . . . . . . . . . . . . . . . . . . . 121
5.1.2.Computingbelieveandplausibilitymeasures
asbounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.1.3.Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.1.4.Modelingimprecisionandignoranceinnodes . . . . . . . . 126
5.1.5.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.2.EvidentialNetworktomodelandcompute
Fuzzyprobabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.2.1.Fuzzyprobabilityandbasicprobability
assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.2.2.Nestedinterval-valuedprobabilities
tofuzzyprobability . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2.3.Computationmechanism . . . . . . . . . . . . . . . . . . . . 130
5.3.EvidentialNetworkstocomputep-box . . . . . . . . . . . . . . 131
5.3.1.Connectionbetweenp-boxesandBPA . . . . . . . . . . . . 132
5.3.2.P-boxesandinterval-valuedprobabilities . . . . . . . . . . . 133
5.3.3.P-boxesandpreciseprobabilities . . . . . . . . . . . . . . . 133
5.3.4.Time-dependentp-boxes . . . . . . . . . . . . . . . . . . . . 134
5.3.5.Computationmechanism . . . . . . . . . . . . . . . . . . . . 134
5.4.Modelingsomereliabilityproblems . . . . . . . . . . . . . . . . 136
5.4.1.BPAforreliabilityproblems . . . . . . . . . . . . . . . . . . 136
5.4.2.BuildingBooleanCMT(AND,OR) . . . . . . . . . . . . . . 137
5.4.3.Conditionalmasstableformorethantwoinputs
(k-out-of-n:Ggate) . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4.4.NodesforPlsandBelinthebinarycase . . . . . . . . . . . 140
5.4.5.Modelingreliabilitywithp-boxes . . . . . . . . . . . . . . . 140
5.5.IllustrationbyapplicationofEvidentialNetworks . . . . . . . . 145
5.5.1.Reliabilityassessmentofsystem . . . . . . . . . . . . . . . . 145
5.5.2.Inferenceforfailureisolation . . . . . . . . . . . . . . . . . . 153
5.5.3.Assessingthefuzzyreliabilityofsystems . . . . . . . . . . . 155
5.5.4.Assessingthep-boxreliabilitybyEN . . . . . . . . . . . . . 162
5.6.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Chapter 6. Reliability Uncertainty and Importance
Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.2.Hypothesisandnotation. . . . . . . . . . . . . . . . . . . . . . . 173
6.3.Probabilisticimportancemeasuresofcomponents . . . . . . . . 174
6.3.1.Birnbaumimportancemeasure . . . . . . . . . . . . . . . . . 175
6.3.2.Componentcriticalitymeasure . . . . . . . . . . . . . . . . . 176
6.3.3.Diagnosticimportancemeasure . . . . . . . . . . . . . . . . 176
6.3.4.Reliabilityachievementworth(RAW). . . . . . . . . . . . . 177
6.3.5.Reliabilityreductionworth(RRW) . . . . . . . . . . . . . . 177
6.3.6.Observationsandlimitations . . . . . . . . . . . . . . . . . . 178
6.3.7.Importancemeasurescomputation . . . . . . . . . . . . . . . 179
6.4.Probabilisticimportancemeasuresofpairsandgroups
ofcomponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.4.1.Measuresonminimumcutsets/pathsets/groups . . . . . . . . 181
6.4.2.ExtensionofRAWandRRWtopairs . . . . . . . . . . . . . 182
6.4.3.Jointreliabilityimportancefactor(JR) . . . . . . . . . . . . 182
6.5.Uncertaintyimportancemeasures . . . . . . . . . . . . . . . . . 184
6.5.1.Uncertaintyprobabilisticimportancemeasures . . . . . . . . 184
6.5.2.Importancefactorswithimprecision . . . . . . . . . . . . . . 186
6.6.Importancemeasureswithfuzzyprobabilities . . . . . . . . . . 188
6.6.1.Fuzzyimportancemeasures . . . . . . . . . . . . . . . . . . 189
6.6.2.Fuzzyuncertaintymeasures . . . . . . . . . . . . . . . . . . 190
6.7.Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.7.1.Importancefactorsonasimplesystem . . . . . . . . . . . . 192
6.7.2.Importancefactorsinacomplexcase . . . . . . . . . . . . . 195
6.7.3.Illustrationofgroupimportancemeasures . . . . . . . . . . 197
6.7.4.Uncertaintyimportancefactors . . . . . . . . . . . . . . . . . 200
6.7.5.Fuzzyimportancemeasures . . . . . . . . . . . . . . . . . . 203
6.8.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Foreword
The probabilistic quantitative assessment of a system is a problem born
at the same time as the first computers with the aim of a reduction of their
failure probability. The proposed models and methods were inspired by the
developmentofdigitalelectronics.Thelimitationsofthesemodelsweremore
or less consciously admitted and today other approaches are available as the
firstthreebooksoftheseriesSystemsReliabilityAssessmenthaveshown.The
presentbookisthefourthoneofthisseries.
For almost two decades, the pioneers of reliability concentrated their
effortsontheprobabilisticcalculusofsystemdependabilitywithoutworrying
about the calculus sensitivity to its different influence factors. In the sixties,
the importance measures appear in the reliability literature and remain
associatedwiththenamesofBirnbaum,LambertandVesely.
ThequestioningofthesemodelsbasedonBooleanstructurefunctionsand
their translation in the probability space took place at the end of the 20th
Century when the probabilistic models of some failure event became
incredible. Could in fact the uncertain knowledge of some events be
legitimately modeled by a probability distribution? We then saw the
development of attempts to represent human reasoning by fuzzy sets and
approximatereasoning.
Interested in the problem of how to design and assess dependable control
systems during the eighties, I was then confronted with the question of how
qualify an instrumented system dedicated to safety application. Until then,
based on qualitative requirements, regulations in the subject evolved in the
sense of requesting a more quantitative assessment of the risk level. Some
studies quickly showed that the assessment of the ability of such a system to
reduce the risk level of the process under supervision was very sensitive to a
variation in the estimation of one or more input parameters. The
representation of this variation by interval-valued representation in the risk
matrix method was much too abrupt and not very relevant to express the
expertadvicewhichisoftenassortedofshades.
Therefore, the idea of confronting the probability assessment of the fuzzy
modelingcamenaturallytome asameansto abettercontrolof theinfluence
parameters. In 2004, I proposed to Christophe Simon, who had a good
expertiseinfuzzysettheory,tobetheco-supervisorofaPhDonthissubject.
ThisresultedthreeyearslaterinthethesisdefenseofMohamedSallak.Since
then,bothhavecontinuedtoworkinthisfieldandpresentedmanysignificant
publications.
Similar to the approaches of reliability assessment by graph theory,
stochastic automata, Petri nets or Bayesian models, this contribution to the
uncertaintymodelingisoneoftherepresentativeaspectsoftheDependability
Nancy School of thought! Who better than Christophe Simon and Mohamed
Sallak, reinforced by Philippe Weber for probabilistic graphical models for
theaspectsrelativetothebelieffunctionstheory,towritesuchabook?
More than a collection of research work results, this book is also a
preciouseducationaldocumentwherethefoundationsofthevariousconcepts
are clearly presented. It contains as well a set of practical implementations of
the proposed approaches, especially in the relevant field of safety integrated
systems. No doubt that students, safety and dependability engineers and even
teachersinthefieldwouldfindalotofinterestingandstrongresourcesinthis
book.
Jean-FrançoisAUBRY
ProfessorEmeritus
UniversityofLorraine,France