Caha, J: Comparison of Fuzzy Arithmetic and Stochastic Simulation for Uncertainty Propagation in Slope Analysis

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

)( 22SNWE SSS

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)2()2( 567321

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)2()2( 781543

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