Abstract
Probability and possibility theory deal with different types of uncertainty. The paper discusses the role of these formalisms in the simulation of continuous dynamical systems where parameters and/or initial conditions are uncertain.
We derive analytically the evolution law of possibility distributions in continuous dynamical system and we compare it with consolidated results in probability theory. The analysis illustrates the limits of conventional Monte Carlo techniques for the simulation of dynamical systems where the uncertainty is expressed in a non probabilistic form. We show that the interaction existing between the probability distribution and the dynamics of the system may produce, in some cases, an inaccurate outcome of the Monte Carlo simulation, making then advisable the adoption of an appropriate possibilistic technique.
Also, we propose a new algorithm for the numerical simulation of a differential system, where the uncertainty of parameters and/or initial conditions is represented by fuzzy distributions. Finally, we discuss the simulation of a dynamical system where the evolution of the possibilistic uncertainty is simulated by using both probabilistic and possibilistic techniques.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. C. Bezdek. Editorial: Fuzziness vs. probability–again (!?). IEEE Transactions on Fuzzy systems, 2 (1): 1–3, 1994.
A. Bonarini and G. Bontempi. A qualitative simulation approach for fuzzy dynamical models. ACM Transactions on Modeling and Computer Simulation (TOMACS), 4 (4), 1994.
G. Bontempi. Qua.si. iii: A software tool for simulation of fuzzy dynamical systems. In A.Javor, A. Lehmann, and I. Molnar, editors, Modeling and Simulation ESM 96 (Proceedings European Simulation Multiconference 1996), pages 615–619, Ghent, Belgium, 1996. SCS International.
D. Dubois and H. Prade. Fuzzy Sets and Systems. Academic Press, New York, 1980.
D. Dubois and H. Prade. Towards fuzzy differential calculus, part 1: integration of fuzzy mappings. Fuzzy Sets and Systems, 8: 1–17, 1982.
D. Dubois and H. Prade. Towards fuzzy differential calculus, part 2: integration on fuzzy intervals. Fuzzy Sets and Systems, 8: 105–116, 1982.
D. Dubois and H. Prade. Towards fuzzy differential calculus, part 3: differentiation. Fuzzy Sets and Systems, 8: 223–233, 1982.
P. A. Fishwick. Fuzzy simulation: Specifying and identifying qualitative models. International Journal of General Systems, 19: 295–316, 1991.
C. W. Gardiner. Handbook of stochastic methods. Springer-Verlag, Berlin, 1985.
E. Hairer, S. P. Norsett, and G. Wanner. Solving ordinary differential equations. Springer, 1987.
E. Hüllermeier. Numerical simulation methods for uncertain dynamics Technical Report tr-ri-95–170, University of Paderborn, 1995.
O. Kaleva. Fuzzy differential equations. Fuzzy Sets and Systems, 24: 301–317, 1987.
A. Kaufmann and M. Gupta. Introduction to fuzzy arithmetic: theory and applications. Van Nostran and Reinhold, New York, 1985.
G. J. Klir and T. A. Folger. Fuzzy sets, uncertainty, and information. Prentice-Hall, New York, 1988.
D. Lambert and J. D. Lambert. Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. John Wiley and Son, 1991.
R. P. Leland. Fuzzy differential systems and malliavin calculus. Fuzzy Sets and Systems, 70: 59–73, 1995.
R. E. Moore. Interval Analysis. Prentice-Hall, Englewood Cliffs, 1966.
A. Neumaier. Interval Methods for Systems of Equations. Cambridge University Press, 1990.
H. T. Nguyen. A note on the extension principle for fuzzy sets. Journal of Mathematical Analysis Applications, 64 (2): 369–380, 1978.
G. Nicolis and I. Prigogine. Self Organisation in Nonequilibrium Systems. Wiley, New York, 1977.
R. Rubinstein. Simulation and the Monte Carlo method. Wiley, 1981.
G. Shafer. A mathematical theory of evidence. Princeton University Press, Princeton, 1976.
Y.A. Shreider. Monte Carlo Methods. Pergamon Press, 1966.
P. Smets. Imperfect information: imprecision, uncertainty. Technical Report 93–3, Iridia -Université Libre de Bruxelles, Bruxelles, 1993.
K. Sobczyk. Stochastic Differential Equations with applications to physics and engineering. Kluwer Academic Publishers, Dordrecht, 1990.
J. Stoer and R. Bulirsch. Introduction to Numerical Analysis. Springer-Verlag, 2nd edition, 1993.
M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology, 1974.
M. Sugeno. Fuzzy Automata and Decision Processes, chapter Fuzzy measures and fuzzy integrals: a survey., pages 89–102. North Holland, Amsterdam, m. m. gupta and g. n. saridis and b. r. gaines edition, 1977.
Z. Wang and G. J. Klir. Fuzzy Measure Theory. Plenum Press, New York, 1992.
L. A. Zadeh. Fuzzy sets. Information and Control, 8: 338–353, 1965.
L. A. Zadeh. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1 (1): 3–28, 1978.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bontempi, G. (2004). Simulating continuous dynamical systems under conditions of uncertainty: the probability and the possibility approaches. In: Nikravesh, M., Zadeh, L.A., Korotkikh, V. (eds) Fuzzy Partial Differential Equations and Relational Equations. Studies in Fuzziness and Soft Computing, vol 142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39675-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-39675-8_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05789-2
Online ISBN: 978-3-540-39675-8
eBook Packages: Springer Book Archive