Data Uncertainty and Important Measures

Christophe Simon, Université de Lorraine, Centre de Recherche en Automatique de Nancy, France.Philippe Weber, Université de Lorraine, Centre de Recherche en Automatique de Nancy, France.Mohamed Sallak, PhD, Associate Professor University of Technologies of Compiègne, France.

• Contents xiForeword xiiiAcknowledgments xiiiChapter 1 Why and Where Uncertainties 11.1 Sources and forms of uncertainty 11.2 Types of uncertainty 31.3 Sources of uncertainty 31.4 Conclusion 6Chapter 2 Models and Language of Uncertainty 92.1 Introduction 92.2 Probability theory 112.2.1 Interpretations 112.2.2 Fundamental notions 132.2.3 Discussion 152.3 Belief functions theory 152.3.1 Representation of beliefs 162.3.2 Combination rules 182.3.3 Extension and marginalization 202.3.4 Pignistic transformation 202.3.5 Discussion 212.4 Fuzzy set theory 212.4.1 Basic definitions 222.4.2 Operations on fuzzy sets 222.4.3 Fuzzy relations 232.5 Fuzzy arithmetic 252.5.1 Fuzzy numbers 262.5.2 Fuzzy probabilities 282.5.3 Discussion 292.6 Possibility theory 292.6.1 Definitions 302.6.2 Possibility and necessity measures 302.6.3 Operations on possibility and necessity measures 322.7 Random set theory 322.7.1 Basic definitions 332.7.2 Expectation of random sets 342.7.3 Random intervals 352.7.4 Confidence interval 352.7.5 Discussion 362.8 Confidence structures or c-boxes 362.8.1 Basic notions 362.8.2 Confidence distributions 372.8.3 P-boxes and C-boxes 382.8.4 Discussion 402.9 Imprecise probability theory 402.9.1 Definitions 412.9.2 Basic properties 422.9.3 Discussion 442.10 Conclusion 44Chapter 3 Risk Graphs and Risk Matrices: Application of Fuzzy Sets and Belief Reasoning 473.1 SIL allocation scheme 483.1.1 Safety instrumented systems (SIS) 483.1.2 Conformity to standards ANSI/ISA S84.01-1996 and IEC 61508 493.1.3 Taxonomy of risk/SIL assessment methods 503.1.4 Risk assessment 503.1.5 SIL allocation process 523.1.6 The use of experts’ opinions 533.2 SIL allocation based on possibility theory 543.2.1 Eliciting the experts’ opinions 543.2.2 Rating scales for parameters 553.2.3 Subjective elicitation of the risk parameters 563.2.4 Calibration of experts’ opinions 593.2.5 Aggregation of the opinions 613.3 Fuzzy risk graph 653.3.1 Input fuzzy partition and fuzzification 653.3.2 Risk/SIL graph logic by fuzzy inference system 663.3.3 Output fuzzy partition and defuzzification 673.3.4 Illustration case 693.4 Risk/SIL graph: belief functions reasoning 723.4.1 Elicitation of expert opinions in the belief functions theory 723.4.2 Aggregation of expert opinions 733.5 Evidential risk graph 753.6 Numerical illustration 773.6.1 Clustering of experts’ opinions 773.6.2 Aggregation of preferences 783.6.3 Evidential risk/SIL graph 793.7 Conclusion 81Chapter 4 Dependability Assessment Considering Interval-valued Probabilities 834.1 Interval arithmetic 844.1.1 Interval-valued parameters 844.1.2 Interval-valued reliability 854.1.3 Assessing the imprecise average probability of failure on demand 864.2 Constraint arithmetic 904.3 Fuzzy arithmetic 934.3.1 Application example 954.3.2 Monte Carlo sampling approach 974.4 Discussion 994.4.1 Markov chains 1004.4.2 Multiphase Markov chains 1014.4.3 Markov chains with fuzzy numbers 1024.4.4 Fuzzy modeling of SIS characteristic parameters 1044.5 Illustration 1054.5.1 Epistemic approach 1064.5.2 Enhanced Markov analysis 1134.6 Decision-making under uncertainty 1154.7 Conclusion 117Chapter 5 Evidential Networks 1195.1 Main concepts 1195.1.1 Temporal dimension 1215.1.2 Computing believe and plausibility measures as bounds 1235.1.3 Inference 1245.1.4 Modeling imprecision and ignorance in nodes 1265.1.5 Conclusion 1285.2 Evidential Network to model and compute Fuzzy probabilities 1285.2.1 Fuzzy probability and basic probability assignment 1285.2.2 Nested interval-valued probabilities to fuzzy probability 1295.2.3 Computation mechanism 1305.3 Evidential Networks to compute p-box 1315.3.1 Connection between p-boxes and BPA 1325.3.2 P-boxes and interval-valued probabilities 1335.3.3 P-boxes and precise probabilities 1335.3.4 Time-dependent p-boxes 1345.3.5 Computation mechanism 1345.4 Modeling some reliability problems 1365.4.1 BPA for reliability problems 1365.4.2 Building Boolean CMT (AND, OR) 1375.4.3 Conditional mass table for more than two inputs (k-out-of-n:G gate) 1385.4.4 Nodes for Pls and Bel in the binary case 1405.4.5 Modeling reliability with p-boxes 1405.5 Illustration by application of Evidential Networks 1455.5.1 Reliability assessment of system 1455.5.2 Inference for failure isolation 1535.5.3 Assessing the fuzzy reliability of systems 1555.5.4 Assessing the p-box reliability by EN 1625.6 Conclusion 169Chapter 6 Reliability Uncertainty and Importance Factors 1716.1 Introduction 1716.2 Hypothesis and notation 1736.3 Probabilistic importance measures of components 1746.3.1 Birnbaum importance measure 1756.3.2 Component criticality measure 1766.3.3 Diagnostic importance measure 1766.3.4 Reliability achievement worth (RAW) 1776.3.5 Reliability reduction worth (RRW) 1776.3.6 Observations and limitations 1786.3.7 Importance measures computation 1796.4 Probabilistic importance measures of pairs and groups of components 1796.4.1 Measures on minimum cutsets/pathsets/groups 1816.4.2 Extension of RAW and RRW to pairs 1826.4.3 Joint reliability importance factor (JR) 1826.5 Uncertainty importance measures 1846.5.1 Uncertainty probabilistic importance measures 1846.5.2 Importance factors with imprecision 1866.6 Importance measures with fuzzy probabilities 1886.6.1 Fuzzy importance measures 1896.6.2 Fuzzy uncertainty measures 1906.7 Illustration 1916.7.1 Importance factors on a simple system 1926.7.2 Importance factors in a complex case 1956.7.3 Illustration of group importance measures 1976.7.4 Uncertainty importance factors 2006.7.5 Fuzzy importance measures 2036.8 Conclusion 206Conclusion 207Bibliography 211Index 225