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, 2750 Ballerup, Denmark [email protected]
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 = [0; 1] 1) Monte Carlo 2) Interval Analysis, Interval Solver Result: [0; 0,25] 4 DTU Diplom, Danmarks Tekniske Universitet
5 5 DTU Diplom, Danmarks Tekniske Universitet
6 6 DTU Diplom, Danmarks Tekniske Universitet
7 7 DTU Diplom, Danmarks Tekniske Universitet
8 8 DTU Diplom, Danmarks Tekniske Universitet
9 9 DTU Diplom, Danmarks Tekniske Universitet
10 10 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 0.25 Probability µ = (a+b+c)/3 Ã 2 = (a 2 +b 2 +c 2 -ab-ac-bc)/18 Possibility ±-cut h = 2/(b-a) a = 7 c = 10 b = DTU Diplom, Danmarks Tekniske Universitet
14 Simple test model f(x) = x(1-x) with triangular uncertain variabel x = [0; 0,2; 1] 1) Probability, Monte Carlo 2) Possibility, Interval Analysis, Interval Solver 14 DTU Diplom, Danmarks Tekniske Universitet
15 Uncertain input variable x = [0; 0,2; 1] 2.4 Probability Possibility DTU Diplom, Danmarks Tekniske Universitet
16 Uncertain output variable f(x) = x(1-x) Probability Possibility x(1-x) 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 = [0; 2pi/16; 2pi/8] π 2π DTU Diplom, Danmarks Tekniske Universitet
19 sinus [0; 2pi/16; 2pi/8] 19 DTU Diplom, Danmarks Tekniske Universitet
![2pi/8] 19 DTU Diplom,](/docs-images/45/3372609/images/page_19.jpg "Danmarks Tekniske")
20 Uncertain input variable x = [0;4pi/16;4pi/8] π 2π DTU Diplom, Danmarks Tekniske Universitet
21 sinus [0; 4pi/16; 4pi/8] 21 DTU Diplom, Danmarks Tekniske Universitet
22 Uncertain input variable x = [0;6pi/16; 6pi/8] π 2π DTU Diplom, Danmarks Tekniske Universitet
23 sinus [0; 6pi/16; 6pi/8] 23 DTU Diplom, Danmarks Tekniske Universitet
24 Uncertain input variable x = [0; 8pi/16; 8pi/8] π 2π DTU Diplom, Danmarks Tekniske Universitet
25 sinus [0; 8pi/16; 8pi/8] 25 DTU Diplom, Danmarks Tekniske Universitet
26 Uncertain input variable x = [0; 10pi/16; 10pi/8] π 2π DTU Diplom, Danmarks Tekniske Universitet
27 sinus [0; 10pi/16; 10pi/8] 27 DTU Diplom, Danmarks Tekniske Universitet
28 Uncertain input variable x = [0; 12pi/16; 12pi/8] π 2π DTU Diplom, Danmarks Tekniske Universitet
29 sinus [0; 12pi/16; 12pi/8] 29 DTU Diplom, Danmarks Tekniske Universitet
30 Uncertain input variable x = [0; 14pi/16; 14pi/8] π 2π DTU Diplom, Danmarks Tekniske Universitet
31 sinus [0; 14pi/16; 14pi/8] 31 DTU Diplom, Danmarks Tekniske Universitet
32 Uncertain input variable x = [0; 16pi/16; 16pi/8] π 2π DTU Diplom, Danmarks Tekniske Universitet
33 sinus [0; 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, 2008, 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: [0,9; 1; 3] Calculate uncertain NPV 34 DTU Diplom, Danmarks Tekniske Universitet
35 Investment in a railway line Probability Possibility NPV 35 DTU Diplom, Danmarks Tekniske Universitet
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