: What if you don t know the probability distribution? Hans Schjær-Jacobsen Technical University of Denmark 15 Lautrupvang, 2750 Ballerup, Denmark hschj@dtu.dk
Uncertain input variables Uncertain system model Uncertain output variables Uncertain system model parameters 2 DTU Diplom, Danmarks Tekniske Universitet
Calculation methods of probability output distributions Domain of input variables Infinite Finite Probability distributions of input variables Available Not available Full probability distributions of output variables by Monte Carlo simulation Not relevant Full probability distributions of output variables by Monte Carlo simulation Minimum and maximum of output variables by 1) Monte Carlo simulation or 2) Interval analysis 3 DTU Diplom, Danmarks Tekniske Universitet
Simple test model f(x) = x(1-x) with uncertain variabel x = [0; 1] 1) Monte Carlo simulation, @RISK 2) Interval Analysis, Interval Solver Result: [0; 0,25] 4 DTU Diplom, Danmarks Tekniske Universitet
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Extending to triangular uncertainty! 11 DTU Diplom, Danmarks Tekniske Universitet
Probability or possibility distribution? min value typical value max value uncertain input variable Triple estimate 12 DTU Diplom, Danmarks Tekniske Universitet
0.25 Probability µ = (a+b+c)/3 Ã 2 = (a 2 +b 2 +c 2 -ab-ac-bc)/18 Possibility 1.2 0.20 0.8 0.15 0.6 0.10 ±-cut 0.4 5 h = 2/(b-a) 0.2 0 5 a = 7 c = 10 b = 16 18 13 DTU Diplom, Danmarks Tekniske Universitet
Simple test model f(x) = x(1-x) with triangular uncertain variabel x = [0; 0,2; 1] 1) Probability, Monte Carlo simulation, @RISK 2) Possibility, Interval Analysis, Interval Solver 14 DTU Diplom, Danmarks Tekniske Universitet
Uncertain input variable x = [0; 0,2; 1] 2.4 Probability Possibility 1.2 2.0 1.6 0.8 1.2 0.6 0.8 0.4 0.4 0.2-0.2 0.2 0.4 0.6 0.8 1.2 15 DTU Diplom, Danmarks Tekniske Universitet
Uncertain output variable f(x) = x(1-x) 35 1.2 30 Probability Possibility 25 0.8 20 0.6 15 10 0.4 5 0.2 0-5 0 5 0.10 0.15 0.20 0.25 x(1-x) 0.30 16 DTU Diplom, Danmarks Tekniske Universitet
Simple test model f(x) = sin(x) with triangular uncertain variabel x 1) Probability, Monte Carlo simulation, @RISK 2) Possibility, Interval analysis, Interval Solver 17 DTU Diplom, Danmarks Tekniske Universitet
Uncertain input variable x = [0; 2pi/16; 2pi/8] 1 0 0 π 2π -1 18 DTU Diplom, Danmarks Tekniske Universitet
3.0 2.0 0.5 - -0.5 0.5 sinus [0; 2pi/16; 2pi/8] 19 DTU Diplom, Danmarks Tekniske Universitet
Uncertain input variable x = [0;4pi/16;4pi/8] 1 0 0 π 2π -1 20 DTU Diplom, Danmarks Tekniske Universitet
1.8 1.6 1.4 1.2 0.8 0.5 0.6 0.4 0.2 - -0.5 0.5 sinus [0; 4pi/16; 4pi/8] 21 DTU Diplom, Danmarks Tekniske Universitet
Uncertain input variable x = [0;6pi/16; 6pi/8] 1 0 0 π 2π -1 22 DTU Diplom, Danmarks Tekniske Universitet
8.0 6.0 4.0 0.5 2.0 - -0.5 0.5 sinus [0; 6pi/16; 6pi/8] 23 DTU Diplom, Danmarks Tekniske Universitet
Uncertain input variable x = [0; 8pi/16; 8pi/8] 1 0 0 π 2π -1 24 DTU Diplom, Danmarks Tekniske Universitet
8.0 6.0 4.0 0.5 2.0 - -0.5 0.5 sinus [0; 8pi/16; 8pi/8] 25 DTU Diplom, Danmarks Tekniske Universitet
Uncertain input variable x = [0; 10pi/16; 10pi/8] 1 0 0 π 2π -1 26 DTU Diplom, Danmarks Tekniske Universitet
6.0 5.0 4.0 3.0 0.5 2.0 - -0.5 0.5 sinus [0; 10pi/16; 10pi/8] 27 DTU Diplom, Danmarks Tekniske Universitet
Uncertain input variable x = [0; 12pi/16; 12pi/8] 1 0 0 π 2π -1 28 DTU Diplom, Danmarks Tekniske Universitet
4.0 3.0 2.0 0.5 - -0.5 0.5 sinus [0; 12pi/16; 12pi/8] 29 DTU Diplom, Danmarks Tekniske Universitet
Uncertain input variable x = [0; 14pi/16; 14pi/8] 1 0 0 π 2π -1 30 DTU Diplom, Danmarks Tekniske Universitet
3.0 2.0 0.5 - -0.5 0.5 sinus [0; 14pi/16; 14pi/8] 31 DTU Diplom, Danmarks Tekniske Universitet
Uncertain input variable x = [0; 16pi/16; 16pi/8] 1 0 0 π 2π -1 32 DTU Diplom, Danmarks Tekniske Universitet
2.5 2.0 1.5 0.5 0.5 - -0.5 0.5 sinus [0; 16pi/16; 16pi/8] 33 DTU Diplom, Danmarks Tekniske Universitet
CASE: Investment in a railway line Guide to Cost Benefit Analysis of Investment Projects, European Commission, 2008, pp. 146-157 Base Case: Table 4.22, p. 154 30 years horizon Discount rate = 5,5%, NPV = 1.953,3 12 investment cost input variables Uncertain investments: 12 independent and uncorrelated investment variables Uncertainty factor on these 12 input variables: [0,9; 1; 3] Calculate uncertain NPV 34 DTU Diplom, Danmarks Tekniske Universitet
Investment in a railway line 018 016 Probability 014 0.8 012 Possibility 010 0.6 008 0.4 006 004 0.2 002 000-1000 -500 0 500 1000 1500 2000 2500 NPV 35 DTU Diplom, Danmarks Tekniske Universitet