Uncertainty modeling revisited: What if you don t know the probability distribution?
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1 : What if you don t know the probability distribution? Hans Schjær-Jacobsen Technical University of Denmark 15 Lautrupvang, 275 Ballerup, Denmark hschj@dtu.dk
2 Uncertain input variables Uncertain system model Uncertain output variables Uncertain system model parameters 2 DTU Diplom, Danmarks Tekniske Universitet
3 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
4 Simple test model f(x) = x(1-x) with uncertain variabel x = [; 1] 1) Monte Carlo 2) Interval Analysis, Interval Solver Result: [;,25] 4 DTU Diplom, Danmarks Tekniske Universitet
5 Rectangular input 1,2 Comparison with Uniform(;1) Rectangular input Minimum 5,4,8,6 Maximum Mean,5 Std Dev,2 Values 1,4,2 Uniform(;1) Minimum Maximum Mean,5 Std Dev,2 -,2,2,4,6,8 1,2 5 DTU Diplom, Danmarks Tekniske Universitet
6 3 f(x) = x(1-x), rectangular input f(x) = x(1-x) Minimum 4,5 Maximum,2 Mean,1 Std Dev Values ,1,15,2,25,3 6 DTU Diplom, Danmarks Tekniske Universitet
7 Triangular input 2,5 Comparison with Triang(;,2;1) 2, 1,5 Triangular input Minimum Maximum,9 Mean,4 Std Dev,2 Values 1 Triang(;,2;1) Minimum,5 Maximum Mean,4 Std Dev,2 -,2,2,4,6,8 1,2 7 DTU Diplom, Danmarks Tekniske Universitet
8 35 f(x) = x(1-x), triangular input 3 25 f(x) = x(1-x) Minimum Maximum,2 Mean,1 Std Dev Values ,1,15,2,25,3 8 DTU Diplom, Danmarks Tekniske Universitet
9 BetaGeneral input 4, Comparison with BetaGeneral(1;1;;1) 3,5 3, 2,5 2, BetaGeneral input Minimum,1 Maximum,8 Mean,5 Std Dev,1 Values 1 1,5,5 BetaGeneral(1;1;;1) Minimum Maximum Mean,5 Std Dev,1 -,2,2,4,6,8 1,2 9 DTU Diplom, Danmarks Tekniske Universitet
10 12 f(x) = x(1-x), betageneral input f(x) = x(1-x) Minimum Maximum,2 Mean,2 Std Dev Values ,1,15,2,25,3 1 DTU Diplom, Danmarks Tekniske Universitet
11 Extending to triangular uncertainty! 11 DTU Diplom, Danmarks Tekniske Universitet
12 Probability or possibility distribution? min value typical value max value uncertain input variable Triple estimate 12 DTU Diplom, Danmarks Tekniske Universitet
13 ,25 Probability μ = (a+b+c)/3 σ 2 = (a 2 +b 2 +c 2 -ab-ac-bc)/18 Possibility 1,2,2,8,15,6,1 α-cut,4 5 h = 2/(b-a),2 5 a = 7 c = 1 b = DTU Diplom, Danmarks Tekniske Universitet
14 Simple test model f(x) = x(1-x) with triangular uncertain variabel x = [;,2; 1] 1) Probability, Monte Carlo 2) Possibility, Interval Analysis, Interval Solver 14 DTU Diplom, Danmarks Tekniske Universitet
15 Uncertain input variable x = [;,2; 1] 2,4 Probability Possibility 1,2 2, 1,6,8 1,2,6,8,4,4,2 -,2,2,4,6,8 1,2 15 DTU Diplom, Danmarks Tekniske Universitet
16 Uncertain output variable f(x) = x(1-x) 35 1,2 3 Probability Possibility 25,8 2,6 15 1,4 5,2-5 5,1,15,2,25 x(1-x),3 16 DTU Diplom, Danmarks Tekniske Universitet
17 Simple test model f(x) = sin(x) with triangular uncertain variabel x 1) Probability, Monte Carlo 2) Possibility, Interval analysis, Interval Solver 17 DTU Diplom, Danmarks Tekniske Universitet
18 Uncertain input variable x = [; 2pi/16; 2pi/8] 1 π 2π DTU Diplom, Danmarks Tekniske Universitet
19 3, 2,,5 - -,5,5 sinus [; 2pi/16; 2pi/8] 19 DTU Diplom, Danmarks Tekniske Universitet
20 Uncertain input variable x = [;4pi/16;4pi/8] 1 π 2π -1 2 DTU Diplom, Danmarks Tekniske Universitet
21 1,8 1,6 1,4 1,2,8,5,6,4,2 - -,5,5 sinus [; 4pi/16; 4pi/8] 21 DTU Diplom, Danmarks Tekniske Universitet
22 Uncertain input variable x = [;6pi/16; 6pi/8] 1 π 2π DTU Diplom, Danmarks Tekniske Universitet
23 8, 6, 4,,5 2, - -,5,5 sinus [; 6pi/16; 6pi/8] 23 DTU Diplom, Danmarks Tekniske Universitet
24 Uncertain input variable x = [; 8pi/16; 8pi/8] 1 π 2π DTU Diplom, Danmarks Tekniske Universitet
25 8, 6, 4,,5 2, - -,5,5 sinus [; 8pi/16; 8pi/8] 25 DTU Diplom, Danmarks Tekniske Universitet
26 Uncertain input variable x = [; 1pi/16; 1pi/8] 1 π 2π DTU Diplom, Danmarks Tekniske Universitet
27 6, 5, 4, 3,,5 2, - -,5,5 sinus [; 1pi/16; 1pi/8] 27 DTU Diplom, Danmarks Tekniske Universitet
28 Uncertain input variable x = [; 12pi/16; 12pi/8] 1 π 2π DTU Diplom, Danmarks Tekniske Universitet
29 4, 3, 2,,5 - -,5,5 sinus [; 12pi/16; 12pi/8] 29 DTU Diplom, Danmarks Tekniske Universitet
30 Uncertain input variable x = [; 14pi/16; 14pi/8] 1 π 2π -1 3 DTU Diplom, Danmarks Tekniske Universitet
31 3, 2,,5 - -,5,5 sinus [; 14pi/16; 14pi/8] 31 DTU Diplom, Danmarks Tekniske Universitet
32 Uncertain input variable x = [; 16pi/16; 16pi/8] 1 π 2π DTU Diplom, Danmarks Tekniske Universitet
33 2,5 2, 1,5,5,5 - -,5,5 sinus [; 16pi/16; 16pi/8] 33 DTU Diplom, Danmarks Tekniske Universitet
34 CASE: Investment in a railway line Guide to Cost Benefit Analysis of Investment Projects, European Commission, 28, pp Base Case: Table 4.22, p 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: [,9; 1; 3] Calculate uncertain NPV 34 DTU Diplom, Danmarks Tekniske Universitet
35 Investment in a railway line Probability 14,8 12 Possibility 1,6 8,4 6 4, NPV 35 DTU Diplom, Danmarks Tekniske Universitet
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