1. Comparison of Fuzzy Arithmetic
and Stochastic Simulation
for Uncertainty Propagation in Slope Analysis
Jan CAHA
This presentation is co-financed by the
European Social Fund and the state
budget of the Czech Republic

2. 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

3. 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

4. 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

5. 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

6. 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

7. Slope analysis
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc

8. 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

9. Comparison
of Fuzzy Arithmetic and Stochastic Simulation
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc

10. 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

11. 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

12. Slope of the cell
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc

13. 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

14. Slope analysis of the area
(m)
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc

15. Slope analysis of the area
Uncertainty (m)
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc

16. Slope analysis of the area
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc

17. 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

18. 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

19. 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

20. Thank you for your attention
First InDOG Doctoral Conference, 29th October - 1st November 2012, Olomouc