A (NEW) MEASURE OF FUZZY UNCERTAINTY VIA INTERVAL ANALYSIS, WHICH IS FULLY CONSISTENT WITH SHANNON THEORY

Authors

  • GUY JUMARIE Dept. of Mathematics and Computer Sci. Universitc du Quebec a Montreal, P.O. Box 8888, St A, Montreal, QUE, H3C 3P8, Canada.

DOI:

https://doi.org/10.5556/j.tkjm.22.1991.4606

Keywords:

Information, Uncertainty, Shannon entropy, Hartley entropy, Membership Functions, PossiLility

Abstract

Many authors have suggested different measures of the amount of uncertainty involved in fuzzy sets, but most of these concepts suffer from drawbacks: mainly, they are indexes of fuzziness rather than measures of uncertainty, and they are not fully consistent with Shannon theory. The question is herein once more considered by combining the information theory of deterministic functions, recently initiated by the author, with the viewpoint of interval analysis; and one so derive the new concept of "uncertainty of order c of fuzzy sets". It is shown that it satisfies the main properties which are desirable for a measure of uncertainty. Some topics are outlined, such as informational distance between fuzzy sets, and mutual infonnation between fuzzy sets for instance. One so has at hand a unified approach to Shannon information expressed in terms of probability, and to fuzzy information described by weighting coefficients commonly referred to as possibility distribution.

References

Aczel, J. and Daroczy, Z., " On measures of Information and their Characterizations", Academic Press, New York, 1975.

Arbib, M. A., Book Review, "Human Systems Managcment", Vol 4, pp 310-324, 1984.

Belis, M. and Guiasu, S.,"A qualitative-quantitative measure of information in cybernetic system", IEEE Trans. Inform. Theory, IT-14, pp 593-594, 1968.

Bouchon, B., "Entropic models", Cybernetics and Systems, Vol 18, pp 1-13, 1987.

De Luca. A. and Termini, S., "A definition of a nonprobabilistic entropy in the setting of fuzzy events", Information and Control, Vol 20, pp 301-312, 1072.

De Luca. and Termini, S., "Entropy and energy measures of a fuzzy set", in Advances in Fuzzy Set Theory and Applications, M. M. Gupta et als (Eds), pp 321-338, North-Holland, 1979.

French, S., "Fuzzy decision analysis: some criticisms", TIMS Studies in Management Sciences, Vol 20, pp 29-44, 1984.

Havrda, J. and Charvat, F., "Quantification method of classification Processes: concept of structural alpha--entropy", Cybernetica., Vol 3, pp 30-35, 1967.

Higashi, M. and Klir, G., "Measurcs of uncertainty and information based on possibility distributions", Intern. J. General Systems, Vol 9, pp 43-58, 1082.

Hirota, H., "Ambiguity based on the concept. of subjective entropy", in Fuzzy Information and Decision Processes, M.M; Gupta et als (Eds) pp 29-40, North-Holland, 1982.

Hisdal, E., "lnfinite valued logic based on two-valued logic and probability." Part 1.1. Difficulties with the present.-day fuzzy-set theory and their resortion in the TEE model, Intern. J. Man-Machine Studies, Vol 25, pp 89-111, 1986.

llisdal, E., "lnfinite valued logic based on two-valued logic and probability." Part 1.2: Different sources of fuzziness, Intern. J. Man-Machine Studies, Vol 25, pp 113-138, 1986.

Jaynes, E. T., "Information theory and statistical mechanics 1", Physical Review, Vol 106, pp 620-630,1957.

Jaynes, E. T., "Information theory and statistical mechanics 2", Physical Reviews, Vol 108, pp 171-190, 1957.

Jumarie, G., "Further advances on the general thermodynamics of open systems via information theory." Effective entropy, negative information, Intern. J. Systems Sc., Vol 6, pp 249-268, 1975.

Jumarie, G., Subjectivity, Information, Systems. Introduction to a Theory of Relativistic Cybernetics, Gordon and Breach, New York, London, 1986.

Jumarie, G., "A Minkowskian theory of observation." Application to uncertainty and fuzziness, Fuzzy Sets and Systems, Vol 24, pp 231-255, 1987.

Jumarie, G., "Further results on the information theory of deterministic functions and its applications to pat.tern recognition", Annales des Telecommunications, Vol 45, pp 66-88, 1990.

Jumarie, G., Relative Information. Theories and Applications. Springer Verlag Berlin, New York, 1990.

Jumarie, G., "Structural entropies of order s and of order (r,s) for deterministic functions", Cybernetics and Systems (to appear) 1990.

Jumarie, G., "A theory of information for vagues concepts. Outline of application to approximate reasoning", Cybernetes, (to appear), 1990.

Manes, E.G., Book Review, Bull. New Series Amer., Math. Soc., Vol 7, pp 603-612, 1982.

Moore, R. E., Interval Analysis, Prentice Hall Inc. New Jersey, 1966.

Muir, A., "Fuzzy sets and probability", Cybernetes, Vol 10,pp 197-200, 1981.

Muir, A., "Probability versus possibility: A reconciliation?" in Proceedings 2nd Napoli Metting on "The Mathematics of Fuzzy Systems", A.G.S. Vent.re (Ed) pp 89-96, 1985.

Muir , A., "Relations between probability and fuzzy theory", in The Mathematics of Fuzzy Systems, A.G.S. Vcnt1·e (Ed) pp 249-263, Verlag TUV Kheinland, 1986.

Ramer, A., "Concepts of fuzzy information measure on continuous domains", Intern. J. General Systems, Vol 17, Nos 3-4, pp 1-11, 1990.

Schumucker, K.J., Fuzzy Sets, Natural Language, Computations and Risk Analysis, Computer Science Press, Rockville, Maryland, 1984.

Toth, H., "Form fuzzy-set theory to fuzzy set-theory: some critical remarks on existing concepts", Fuzzy Sets and Systems, Vol 23, pp 219-237, 1987.

Trillas, E., Alsina , E. and Valverde, L., Do we need max, min and I - J in fuzzy set theory? in Applied systems and Cybernetics, G.E. Lasker (Ed) Vol VI, pp 2808-2813, Pergamon P1·css, 1981.

Yager, R. R., "A measurement.-informational discussion of fuzzy union and intersection", Intern. J. Man-Machine Studies, Vol 11, pp 189-200, 1919.

Zadeh, L. A., "Probability measures of fuzzy events", J. Mathematical Analysis and Applications, Vol 23, pp 421-427, 1968.

Zeleny, M., "On the (ir)relevance of fuzzy set theory", lluman Systems Management, Vol 4, pp 301-306, 1984.

Downloads

Published

1991-09-01

How to Cite

JUMARIE, G. (1991). A (NEW) MEASURE OF FUZZY UNCERTAINTY VIA INTERVAL ANALYSIS, WHICH IS FULLY CONSISTENT WITH SHANNON THEORY. Tamkang Journal of Mathematics, 22(3), 223–241. https://doi.org/10.5556/j.tkjm.22.1991.4606

Issue

Section

Papers