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Comparison of Fuzzy Arithmetic and Stochastic Simulation for Uncertainty Propagation in Slope Analysis Jan CAHAThis presentation is co-financed by theEuropean Social Fund and the statebudget of the Czech Republic
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Introduction uncertainty is element of data and processes associated with them propagation of uncertainty amount and character of uncertainty is substantial for decision making theories for modeling and propagation of uncertainty - probability theory, Dempster–Shafer theory, fuzzy sets theory, interval mathematics … First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Introduction Stochastic Simulation (represented by the method Monte Carlo) is often used for uncertainty propagation Monte Carlo has some undesirable properties that complicate further use of the results possible solution is utilization of Fuzzy Arithmetic fuzzy arithmetic is extension of standard arithmetic operations to fuzzy numbers First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Comparisonof Fuzzy Arithmetic and Stochastic Simulation three aspects of uncertainty that need to be considered while choosing method for modeling definitions and axiomatics semantics should define which uncertainty theory should be used there is no general agreement on the process at least two approaches – statistics, fuzzy methods different approach to the results numeric Stochastic simulation - what are the most probable outputs, it is possible that the result did not cover all the possible outcomes Fuzzy arithmetic covers all the possible outcomes including the extreme solutions First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Comparisonof Fuzzy Arithmetic and Stochastic Simulation Stochastic simulations are extremely time and computational performance demanding generation of random numbers storage of large amount of data while performing iterations Fuzzy arithmetic is less demanding smaller amount of iterations smaller demand for storage space First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Slope analysis one the basic GIS analysis of surface uncertain surface modelled by the field model Neighbourhood method 2 2 S (S E W SN S ) ( z1 2 z2 z3 ) ( z7 2 z6 z5 )SN S 2 4 d ( z3 2 z4 z5 ) ( z1 2 z8 z7 )SE W 2 4 d First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Slope analysis First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Case studies 2 case studies uncertainty of slope in one cell calculation of uncertainty for area of interest slope values are presented in percentages triangular distribution will be used for stochastic simulation Piecewiselinear Fuzzy Numbers with 10 α-cuts the presented solutions were programmed in Java First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Comparisonof Fuzzy Arithmetic and Stochastic Simulation First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Slope of the cell 3×3 cell cell size 10 meters z1–z8 have value 0 meters ±1 meter case study proves how the two methods approach uncertainty differently what is possible range of values of z9 ? First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Slope of the cell Monte Carlo Number of iterations Minimal value Maximal value Time of calculation (s-9) 100 0.18% 6.79% 9 686 759 600 0.10% 5.24% 22 836 671 1 000 0.02% 6.41% 30 969 981 100 000 000 1.18% 9.05% 134 521 347 647 Fuzzy Arithmetic – time of calculation 7 530 022 s-9 limit values - 0.0 ̶ 14.14% First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Slope of the cell First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Slope analysis of the area area of interest 4×4 km grid of size 400×400 cells cell size 10×10 meters time and storage demands of both methods First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Slope analysis of the area (m) First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Slope analysis of the area Uncertainty (m) First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Slope analysis of the area First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Slope analysis of the area - Time demands Monte Carlo Number of iterations Time of calculation (s-9) 100 4 975 466 099 600 56 907 980 483 1000 91 937 539 092 Fuzzy Arithmetic 76 523 690 406 s-9 First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Slope analysis of the area - Memory demands Monte Carlo - 1 288 512 bytes per realization 100 iterations – 128 851 200 bytes 600 iterations – 773 107 200 bytes 1000 iterations – 1 288 512 000 bytes Fuzzy Arithmetic – 28 574 944 bytes First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Conclusion methods for uncertainty propagation were compared by 3 aspects ability to provide all possible solutions time demands memory demands comparison of time demands highly depends on number of iterations and on number of alpha cuts Fuzzy arithmetic can be further optimized by different algorithms for calculation results of Fuzzy arithmetic offer much better foundation for use of the results in uncertainty analysis by containinig all possible solution results of Fuzzy Arithmetic support more appropriately decision making First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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Thank you for your attention First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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