A Verified Realization of a Dempster-Shafer-Based Fault Tree Analysis

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1 A Verified Realization of a Dempster-Shafer-Based Fault Tree Analysis Ekaterina Auer, Wolfram Luther, Gabor Rebner Department of Computer and Cognitive Sciences (INKO) University of Duisburg-Essen Duisburg, Germany 1/26

2 Table of Contents 1 2 Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks 3 Aim Basics and Computation 4 2/26

3 Table of Contents Integrate verified in Fault tree analysis by using MATLAB and INTLAB 1 Issues 1 Use of intervals to express uncertainties in evidences 2 Computation of the verified upper bound of failure probability 3 New approach in Fault tree analysis 4 Fast evaluation of Dempster-Shafer-Functions 1 ~ rump/intlab/ 3/26

4 Table of Contents Integrate verified in Fault tree analysis by using MATLAB and INTLAB 1 Issues 1 Use of intervals to express uncertainties in evidences 2 Computation of the verified upper bound of failure probability 3 New approach in Fault tree analysis 4 Fast evaluation of Dempster-Shafer-Functions 1 ~ rump/intlab/ 3/26

5 Table of Contents Uncertainties Lack of knowledge (failure probability is not available) Variations in system tests Different evidences by experts evidence BPA1:Pl BPA1:Bel units of interest evidence BPA1:Pl BPA1:Bel units of interest 4/26

6 Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks Mathematical theory of evidence Developed by A. Dempster and G. Shafer in 1976 Features Uncertainty in evidence can be expressed as an interval Aggregation of evidence by different experts 5/26

7 Introduction to DSI Table of Contents Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks Fundamentals 2 X is the power set of all probability assignments 6/26

8 BPA Table of Contents Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks Definition Basic probability assignment (BPA) over all sets of interest (A 1,...,A n ) m :2 X [0, 1], n m(a i )=1, m( ) =0, A i 2 X i=1 In continuous case: A i = [ a i, a i ], <ai a i < + 7/26

9 Solution Space Table of Contents Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks Upper Bound Pl(X) =Plausibility ( X X ) = A i X m (A i ) Lower Bound Bel (X) =Belief ( X X ) = A i X m (A i ) 8/26

10 Solution Space Table of Contents Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks Upper Bound Pl(X) =Plausibility ( X X ) = A i X m (A i ) Lower Bound Bel (X) =Belief ( X X ) = A i X m (A i ) 8/26

11 Solution Space (Example) Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks 1 solution space of BPA A Example BPA A 1 =[10, 20] m(a 1 )=0.3 A 2 =[20, 30] m(a 2 )=0.2 A 3 =[30, 40] m(a 3 )=0.5 evidence BPA1:Pl BPA1:Bel units of interest 9/26

12 Dempster-Shafer with Intervals Toolbox Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks Implements verified Dempster-Shafer functions in MATLAB by using INTLAB Based on the IPP-toolbox for R by P. Limbourg 2 Verified Computation Sampling of cumulative distribution functions Sampling of (non)monotonous system functions Monte Carlo sampling Aggregation of BPAs / 26

13 Dempster-Shafer with Intervals Toolbox Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks Implements verified Dempster-Shafer functions in MATLAB by using INTLAB Based on the IPP-toolbox for R by P. Limbourg 2 Verified Computation Sampling of cumulative distribution functions Sampling of (non)monotonous system functions Monte Carlo sampling Aggregation of BPAs / 26

14 Dempster-Shafer with Intervals Toolbox Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks Implements verified Dempster-Shafer functions in MATLAB by using INTLAB Based on the IPP-toolbox for R by P. Limbourg 2 Verified Computation Sampling of cumulative distribution functions Sampling of (non)monotonous system functions Monte Carlo sampling Aggregation of BPAs / 26

15 Verified Normalization Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks Definition Let {+, -,, / } be interval operations Let fl Δ and fl be upward and downward directed roundings Formula m(a i ) / fl Δ n m(a j ), m(a i ) / j=1 fl n m(a j ) j=1 11 / 26

16 Verified Normalization Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks Definition Let {+, -,, / } be interval operations Let fl Δ and fl be upward and downward directed roundings Formula m(a i ) / fl Δ n m(a j ), m(a i ) / j=1 fl n m(a j ) j=1 11 / 26

17 Benchmarks Table of Contents Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks Averaged Solutions of Benchmarks in Seconds IPP (R 64Bit) DSI (Matlab64Bit) 2, , , , , ,19144 Aggregation Distribution Sampling Sampling of (non)monotonous System Functions 12 / 26

18 Comparison with IPP Introduction to the (DST) The Toolbox DSI (Dempster Shafer with Intervals) Benchmarks Comparison with IPP 1 Strict use of vector-matrix-operations (fast computation) 2 Verified computation (INTLAB) 3 Fault tree analysis 4 Fast sampling of (non)monotonous system functions 13 / 26

19 Aim of Verified Fault Tree Analysis Aim Basics and Computation Main Objective Computation of the biggest upper bound of failure probability Approach 1 Avoid limitations of floating-point numbers Round-off errors Approximation errors 2 Use of to express uncertainties 3 Use of INTLAB 14 / 26

20 Aim of Verified Fault Tree Analysis Aim Basics and Computation Main Objective Computation of the biggest upper bound of failure probability Approach 1 Avoid limitations of floating-point numbers Round-off errors Approximation errors 2 Use of to express uncertainties 3 Use of INTLAB 14 / 26

21 Aim Basics and Computation 15 / 26

22 Basic Events Table of Contents Aim Basics and Computation Evidences 1 Only one evidence is given 2 More than one evidence is given Aggregation is needed Maybe loss of information Solution Mixing based on arithmetic averaging 16 / 26

23 Basic Events Table of Contents Aim Basics and Computation Evidences 1 Only one evidence is given 2 More than one evidence is given Aggregation is needed Maybe loss of information Solution Mixing based on arithmetic averaging 16 / 26

24 Logical Gates Definitions Aim Basics and Computation Definition A 1 =[0, 2] m(a 1 )=0.5 A 2 =[2, 4] m(a 2 )=0.5 failure probability solution space of A 0.1 A:Pl A:Bel hours 17 / 26

25 Logical Gates (AND) Aim Basics and Computation AND-Gate C = A and B = A B C = A B PL(C) =PL(A) PL(B) BEL(C) =BEL(A) BEL(B) 18 / 26

26 AND-Gate (Example) Aim Basics and Computation failure probability solution space of A A:Pl A:Bel hours failure probability solutionspace of dsiand(a,a) 0.1 dsiand(a,a):pl dsiand(a,a):bel hours 19 / 26

27 OR-Gate Table of Contents Aim Basics and Computation Formula A B =1 (1 P (A)) (1 P (B)) 20 / 26

28 OR-Gate - Meaning in DST Aim Basics and Computation Evaluation Or-Gate in DST C 1 = A B C 2 = A\B C 3 = B\A PL(C 1 )=1 (1 PL(A)) (1 PL(B)) BEL(C 1 )=1 (1 BEL(A)) (1 BEL(B)) PL(C 2 )=PL(A) BEL(C 2 )=BEL(A) PL(C 3 )=PL(B) BEL(C 3 )=BEL(B) 21 / 26

29 OR-Gate - Meaning in DST Aim Basics and Computation Evaluation Or-Gate in DST C 1 = A B C 2 = A\B C 3 = B\A PL(C 1 )=1 (1 PL(A)) (1 PL(B)) BEL(C 1 )=1 (1 BEL(A)) (1 BEL(B)) PL(C 2 )=PL(A) BEL(C 2 )=BEL(A) PL(C 3 )=PL(B) BEL(C 3 )=BEL(B) 21 / 26

30 OR-Gate (Example) Table of Contents Aim Basics and Computation A B =1 (1 P (A)) (1 P (B)) failure probability solution space of A 0.1 A:Pl A:Bel hours failure probability solution space of dsior(a,a) 0.1 dsior(a,a):pl dsior(a,a):bel hours 22 / 26

31 Conclusion Table of Contents Development DSI-Toolbox: Fast and verified Dempster-Shafer computation Use of INTLAB New approach in Fault tree analysis: Verified computation of the biggest failure probability Uncertainties in failure distributions can be expressed as intervals Based on DSI (fast computation) 23 / 26

32 Further Work Further Work Combination of verified and Markov chains, with possible application to Fault tree analysis. 24 / 26

33 Literature Table of Contents Auer, K. ; Luther, W. ; Rebner, G. ; Limbourg, P.: A Verified MATLAB Toolbox for the, Limbourg,P.;Savic,R.;Petersen,J.;Kochs,H.D.: Modelling uncertainty in fault tree analyses using evidence theory. In: Journal of Risk and Reliability 222 (2008),S Rump, S.: INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages Kluwer Academic Publishers, Dordrecht, Guth, M.: A probability foundation for vagueness and imprecision in fault tree analysis. In: IEEE Transactions on Reliability (1991), Nr. 40, S Chen, Y.: Uncertainties in Fault Tree Analysis. Tonon, F.: Using random set theory to propagate epistemic uncertainty through a mechanical system. In: Reliability Engineering and System Safety 85 (2004), Nr. 1 3, S Shafer, G.: A Mathematical Theory of Evidence. Princeton, USA : Princeton University Press, 1976 Robert, C. ; Casella, G.: Monte Carlo statistical methods. 2. ed. New York, NY : Springer, 2004 (Springer texts in statistics). ISBN / 26

34 Thank you for your attention 26 / 26

Fault tree analysis in an early design stage using the Dempster-Shafer theory of evidence

Risk, Reliability and Societal Safety Aven & Vinnem (eds) 2007 Taylor & Francis Group, London, ISBN 978-0-415-44786-7 Fault tree analysis in an early design stage using the Dempster-Shafer theory of evidence