Computational Procedures of the Evidence Theory for Interval and Fuzzy Assignments of the Basic Probability Masses for Focal Elements
DOI:
https://doi.org/10.7250/itms-2019-0007Keywords:
Belief function, data incompleteness, evidence theory, frame of discernment, focal elements, fuzzy value, inaccuracy, interval probability, interval value, membership function, plausibility function, probability mass, uncertaintyAbstract
The evidence theory is ascribed to a specific kind of uncertainty. In this theory, uncertainty refers to the fact that the element of our interest (the true world) may be included in subsets of other similar elements (possible worlds). In the original evidence theory, the estimates of the basic probability masses for the focal elements are given in an unambiguous form. In practice, to obtain such estimates is often difficult or even impossible. In such a situation, the relevant estimates are given in the interval or fuzzy form. The goal of the paper is to present and analyse the calculation procedures for determination of the belief functions and plausibility functions in the evidence theory for cases when the initial estimates are given in the interval or fuzzy form.
References
N. Burrus, D. Lesage, “Theory of Evidence”, Laboratorie de Reserche et Développment de lʹEpita, France, Tech. Rep., 2011.
G. Shafer, A mathematical theory of evidence. Princeton University Press, 297 p., 1976.
P. Dempster, “Upper and lower probabilities induced by a multivalued mapping”, Annals of Mathematical Statistics, vol. 38, no. 2, pp. 325–339, 1967. https://doi.org/10.1214/aoms/1177698950
O. Uzhga-Rebrov, Uncertainties management. Part 3. Modern nonprobabilistic methods. Rezekne, RA Izdevniecība, 560 p., 2010. (In Russian).
L. Zadeh, “Fuzzy sets and information granularity”, in Advances in Fuzzy Sets Theory and Applications, R. K. Ragade, M. M. Gupta, and R. R. Yager (editors). 1979, pp. 3–18.
J. Yen, “Generalizing the Dempster-Shafer Theory to Fuzzy Sets”, IEEE Transactions on Systems, Man and Cybernetics, vol. 20, no. 3, pp. 559–570, 1990. https://doi.org/10.1109/21.57269
M.-Sh. Yang, T.-Ch. Chen, K.-L. Wu, “Generalized Belief Function, Plausibility Function, and Dempster’s Combinational Rule to Fuzzy Sets”, International Journal of Intelligent Systems, vol. 18, no. 8, pp. 925–937, 2003. https://doi.org/10.1002/int.10126
P. Walley, Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, etc., 706 p., 1991.
T. Denoeux, “Reasoning with imprecise belief structures”, International Journal of Approximate Reasoning, vol. 20, no. 1, pp. 79–111, 1999. https://doi.org/10.1016/S0888-613X(98)10023-3
K. Weichselberger, “The Theory of Interval-Probability as a Unifying Concept for Uncertainty”, International Journal of Approximate Reasoning, vol. 24, no. 2–3, pp. 149–170, May 2000. https://doi.org/10.1016/S0888-613X(00)00032-3
K. Weichselberger, Th. Augustin, “On the Symbiosis of Two Concepts of Conditional Interval Probability”, in ISIPTA’03, J. Bernard, T. Seidenfeld, and M. Zaffalon (Editors), Waterloo, Carleton Scientific, 2003, pp. 608–630.
L. Zadeh, “Fuzzy Sets”, Information and Control, vol. 8, no. 3, pp. 338–353, 1965. https://doi.org/10.1016/S0019-9958(65)90241-X