This presentation is co-financed by the European Social Fund and the state budget of the Czech Republic
Jan CAHA
Comparison of Fuzzy Arithmetic and Stochastic Simulation
for Uncertainty Propagation in Slope Analysis
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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
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
Comparison of 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
Comparison of 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
Slope analysis one the basic GIS analysis of surface uncertain surface modelled by the field model Neighbourhood method
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First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
Slope analysis
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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
Comparison of Fuzzy Arithmetic and Stochastic Simulation
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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
Slope of the cell Monte Carlo
Fuzzy Arithmetic – time of calculation 7 530 022 s-9
limit values - 0.0 0 14.14%
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
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
Slope of the cell
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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
Slope analysis of the area
(m)
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
Slope analysis of the area
Uncertainty (m)
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
Slope analysis of the area
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
Slope analysis of the area - Time demands Monte Carlo
Fuzzy Arithmetic 76 523 690 406 s-9
Number of iterations Time of calculation (s-9)
100 4 975 466 099
600 56 907 980 483
1000 91 937 539 092
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc
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
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
Thank you for your attention