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Uncertainty measures for interval type-2 fuzzy sets. (English) Zbl 1141.28010
Based on the representation theorem for interval type-2 fuzzy sets, 4 types of new uncertainty measures are introduced and discussed, namely cardinality, fuzziness, variance and skewness. Note that the first uncertainty measures for this type of fuzzy sets, namely the centroid, was introduced already in [{\it N. N. Karnik} and {\it J. M. Mendel}, Inf. Sci. 132, No. 1--4, 195--220 (2001; Zbl 0982.03030)]. The authors give formulae for computing the introduced uncertainty measures, discuss their properties and include several illustrative examples.

28E10Fuzzy measure theory
94D05Fuzzy sets and logic in connection with communication
Full Text: DOI
[1] Astudillo, L.; Castillo, O.; Aguilar, L. T.: Intelligent control for a perturbed autonomous wheeled mobile robot: a type-2 fuzzy logic approach. Journal of nonlinear studies 14, No. 3, 37-48 (2007) · Zbl 1117.93048
[2] Atanassov, K.; Gargov, G.: Interval valued intuitionistic fuzzy sets. Fuzzy sets and systems 31, 343-349 (1989) · Zbl 0674.03017
[3] S. Auephanwiriyakul, A. Adrian, J.M. Keller, Type-2 fuzzy set analysis in management surveys, in: Proceedings of the FUZZ-IEEE, Honolulu, HI, 2002, pp. 1321 -- 1325.
[4] Baguley, P.; Page, T.; Koliza, V.; Maropoulos, P.: Time to market prediction using type-2 fuzzy sets. Journal of manufacturing technology management 17, No. 4, 513-520 (2006)
[5] Blanchard, N.: Cardinal and ordinal theories about fuzzy sets. Fuzzy information and decision processes, 149-157 (1982) · Zbl 0508.94029
[6] P.P. Bonissone, A pattern recognition approach to the problem of linguistic approximation, in: Proceedings of the IEEE International Conference on Cybernetics and Society, Denver, CO, 1979, pp. 793 -- 798.
[7] P.P. Bonissone, A fuzzy sets based linguistic approach: theory and applications, in: Proceedings of the 12th Winter Simulation Conference, Orlando, FL, 1980, pp. 99 -- 111.
[8] A. Bouchachia, R. Mittermeir, A neural cascade architecture for document retrieval, in: Proceedings of the International Joint Conference Neural Networks, vol. 3, Portland, OR, 2003, pp. 1915 -- 1920.
[9] Burillo, P.; Bustince, H.: Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy sets and systems 78, 305-316 (1996) · Zbl 0872.94061
[10] Carlsson, C.; Fullér, R.: On possibilistic mean value and variance of fuzzy numbers. Fuzzy sets and systems 122, 315-326 (2001) · Zbl 1016.94047
[11] O. Castillo, N. Cazarez, P. Melli, Design of stable type-2 fuzzy logic controllers based on a fuzzy Lyapunov approach, in: Proceedings of the FUZZ-IEEE, Vancouver, Canada, July 2006, pp. 2331 -- 2336.
[12] O. Castillo, P. Melin, Adaptive noise cancellation using type-2 fuzzy logic and neural networks, in: Proceedings of the FUZZ-IEEE, vol. 2, Budapest, Hungary, 2004, pp. 1093 -- 1098.
[13] O. Castillo, P. Melin, Evolutionary computing for optimizing type-2 fuzzy systems in intelligent control of non-linear dynamic plants, in: Proceedings of the North American Fuzzy Information Processing Society (NAFIPS), Ann Arbor, MI, 2005, pp. 247 -- 251.
[14] Christensen, R.: Entropy minimax multivariate statistical modeling I: Theory. International journal of general systems 11, 231-277 (1985)
[15] Christensen, R.: Entropy minimax multivariate statistical modeling II: Applications. International journal of general systems 12, No. 3, 227-305 (1985)
[16] Cornelis, C.; Kerre, E.: Inclusion measures in intuitionistic fuzzy set theory. Lecture notes in computer science 2711, 345-356 (2004) · Zbl 1274.03081
[17] Cross, V. V.; Sudkamp, T. A.: Similarity and compatibility in fuzzy set theory: assessment and applications. (2002) · Zbl 0992.03066
[18] De Luca, A.; Termini, S.: A definition of nonprobabilistic entropy in the setting of fuzzy sets theory. Information and computation 20, 301-312 (1972) · Zbl 0239.94028
[19] Dubois, D.; Prade, H.: Fuzzy cardinality and the modeling of imprecise quantification. Fuzzy sets and systems 16, 199-230 (1985) · Zbl 0601.03006
[20] J. Figueroa, J. Posada, J. Soriano, M. Melgarejo, S. Rojas, A type-2 fuzzy controller for tracking mobile objects in the context of robotic soccer games, in: Proceedings of the FUZZ-IEEE, Reno, NV, 2005, pp. 359 -- 364.
[21] Geer, J. F.; Klir, G. J.: A mathematical analysis of information preserving transformations between probabilistic and possibilistic formulations of uncertainty. International journal of general systems 20, No. 2, 143-176 (1992) · Zbl 0743.94029
[22] Gottwald, S.: A note on fuzzy cardinals. Kybernetika 16, 156-158 (1980) · Zbl 0434.03038
[23] Gu, L.; Zhang, Y. Q.: Web shopping expert using new interval type-2 fuzzy reasoning. Soft computing 11, No. 8, 741-751 (2007)
[24] Hagras, H.: Type-2 flcs: a new generation of fuzzy controllers. IEEE computational intelligence magazine 2, No. 1, 30-43 (2007)
[25] D. Harmanec, Measures of uncertainty and information, http://www.sipta.org/documentation/summary_measures/main.html (1999).
[26] Higashi, M.; Klir, G.: Measures of uncertainty and information based on possibility distributions, fuzzy sets for intelligent systems. (1993) · Zbl 0497.94008
[27] Jang, L. -C.; Ralescu, D.: Cardinality concepts for type-two fuzzy sets. Fuzzy sets and systems 118, 479-487 (2001) · Zbl 0972.03054
[28] Karnik, N. N.; Mendel, J. M.: Centroid of a type-2 fuzzy set. Information sciences 132, 195-220 (2001) · Zbl 0982.03030
[29] Kaufmann, A.: Introduction to the theory of fuzzy sets. (1975) · Zbl 0332.02063
[30] Kaufmann, A.: Introduction a la theorie des sous-ensembles flous. Complement et nouvelles applications 4 (1977) · Zbl 0346.94002
[31] E.P. Klement, On the cardinality of fuzzy sets, in: Proceedings of the 6th European Meeting on Cybernetics and Systems Research, Vienna, 1982, pp. 701 -- 704. · Zbl 0527.04002
[32] Klir, G. J.: A principle of uncertainty and information invariance. International journal of general systems 17, No. 2 -- 3, 249-275 (1990) · Zbl 0703.94026
[33] Klir, G. J.: Principles of uncertainty: what are they? why do we need them?. Fuzzy sets and systems 74, 15-31 (1995)
[34] Klir, G. J.; Folger, T. A.: Fuzzy sets, uncertainty, and information. (1988) · Zbl 0675.94025
[35] Klir, G. J.; Parviz, B.: Probability -- possibility transformations: a comparison. International journal of general systems 21, No. 3, 291-310 (1992) · Zbl 0768.60003
[36] Klir, G. J.; Yuan, B.: Fuzzy sets and fuzzy logic: theory and applications. (1995) · Zbl 0915.03001
[37] Knopfmacher, J.: On measures of fuzziness. Journal of mathematics analysis and applications 49, 529-534 (1975) · Zbl 0308.02061
[38] C.-H. Lee, Y.-C. Lin, Control of nonlinear uncertain systems using type-2 fuzzy neural network and adaptive filter, in: Proceedings of the 2004 IEEE International Conference on Networking, Sensing and Control, vol. 2, Taipei, Taiwan, 2004, pp. 1177 -- 1182.
[39] Lee, E.; Li, R.: Comparison of fuzzy numbers based on the probability measure of fuzzy events. Computers & mathematics with applications 15, 887-896 (1988) · Zbl 0654.60008
[40] Q. Liang, L. Wang, Sensed signal strength forecasting for wireless sensors using interval type-2 fuzzy logic system, in: Proceedings of the FUZZ-IEEE, Reno, NV, 2005, pp. 25 -- 30.
[41] P.-Z. Lin, C.-F. Hsu, T.-T. Lee, Type-2 fuzzy logic controller design for buck DC-DC converters, in: Proceedings of the FUZZ-IEEE, Reno, NV, 2005, pp. 365 -- 370.
[42] F. Liu, J.M. Mendel, An interval approach to fuzzistics for interval type-2 fuzzy sets, IEEE Transactions on Fuzzy Systems, submitted for publication.
[43] C. Lynch, H. Hagras, V. Callaghan, Using uncertainty bounds in the design of an embedded real-time type-2 neuro-fuzzy speed controller for marine diesel engines, in: Proceedings of the FUZZ-IEEE, Vancouver, Canada, 2006, pp. 7217 -- 7224.
[44] Melin, P.; Castillo, O.: An intelligent hybrid approach for industrial quality control combining neural networks, fuzzy logic and fractal theory. Information sciences 177, No. 7, 1543-1557 (2007)
[45] Melin, P.; Urias, J.; Solano, D.; Soto, M.; Lopez, M.; Castillo, O.: Voice recognition with neural networks, type-2 fuzzy logic and genetic algorithms. Journal of engineering letters 13, No. 2, 108-116 (2006)
[46] Mendel, J. M.: Rule-based fuzzy logic systems: introduction and new directions. (2001) · Zbl 0978.03019
[47] Mendel, J. M.: Advances in type-2 fuzzy sets and systems. Information sciences 177, No. 1, 84-110 (2007) · Zbl 1117.03060
[48] J.M. Mendel, H. Hagras, R.I. John, Standard background material about interval type-2 fuzzy logic systems that can be used by all authors, http://ieee-cis.org/_files/standards.t2.win.pdf.
[49] Mendel, J. M.; John, R. I.: Type-2 fuzzy sets made simple. IEEE transactions on fuzzy systems 10, No. 2, 117-127 (2002)
[50] J.M. Mendel, H. Wu, Centroid uncertainty bounds for interval type-2 fuzzy sets: Forward and inverse problems, in: Proceedings of the FUZZ-IEEE, vol. 2, Budapest, Hungary, 2004, pp. 947 -- 952.
[51] Mendel, J. M.; Wu, H.: New results about the centroid of an interval type-2 fuzzy set, including the centroid of a fuzzy granule. Information sciences 177, 360-377 (2007) · Zbl 1111.68134
[52] Niewiadomski, A.; Bartyzel, M.: Elements of type-2 semantics in summarizing databases. Lecture notes in artificial intelligence 4029, 278-287 (2006)
[53] A. Niewiadomski, P.S. Szczepaniak, News generating based on type-2 linguistic summaries of databases, in: Proceedings of the IPMU, Paris, France, 2006, pp. 1324 -- 1331.
[54] Own, C. -M.; Tsai, H. -H.; Yu, P. -T.; Lee, Y. -J.: Adaptive type-2 fuzzy median filter design for removal of impulse noise. Imaging science 54, No. 1, 3-18 (2006)
[55] T. Ozen, J.M. Garibaldi, Effect of type-2 fuzzy membership function shape on modelling variation in human decision making, in: Proceedings of the FUZZ-IEEE, vol. 2, Budapest, Hungary, 2004, pp. 971 -- 976.
[56] Paris, J. B.: The uncertain reasoner’s companion -- A mathematical perspective. (1994) · Zbl 0838.68104
[57] Rhee, F. C. -H.: Uncertainty fuzzy clustering: insights and recommendations. IEEE computational intelligence magazine 2, No. 1, 44-56 (2007)
[58] Sepúlveda, R.; Castillo, O.; Melin, P.; Rodrı&acute, A.; Guez-Dı´ Az; Montiel, O.: Experimental study of intelligent controllers under uncertainty using type-1 and type-2 fuzzy logic. Information sciences 177, No. 10, 2023-2048 (2007)
[59] Sevastjanov, P.; Figat, P.: Aggregation of aggregating modes in MCDM: Synthesis of type 2 and level 2 fuzzy sets. Omega 35, No. 5, 505-523 (2007)
[60] H. Shu, Q. Liang, Wireless sensor network lifetime analysis using interval type-2 fuzzy logic systems, in: Proceedings of the FUZZ-IEEE, Reno, NV, 2005, pp. 19 -- 24.
[61] P. Subasic, M. Nakatsuyama, A new representational framework for fuzzy sets, in: Proceedings of the FUZZ-IEEE, Catalonia, Spain, 1997, pp. 1601 -- 1606.
[62] Szmidt, E.; Kacprzyk, J.: Entropy for intuitionistic fuzzy sets. Fuzzy sets and systems 118, 467-477 (2001) · Zbl 1045.94007
[63] Vlachos, I.; Sergiadis, G.: Submethood, entropy, and cardinality for interval-valued fuzzy sets -- an algebraic derivation. Fuzzy sets and systems 158, 1384-1396 (2007) · Zbl 1122.03055
[64] Wenstøp, F.: Quantitative analysis with linguistic values. Fuzzy sets and systems 4, 99-115 (1980)
[65] Wonneberge, S.: Generalization of an invertible mapping between probability and possibility. Fuzzy sets and systems 64, No. 2, 229-240 (1994)
[66] D. Wu, J.M. Mendel, Aggregation using the linguistic weighted average and interval type-2 fuzzy sets, IEEE Transactions on Fuzzy Systems, in press.
[67] D. Wu, J.M. Mendel, Cardinality, fuzziness, variance and skewness of interval type-2 fuzzy sets, in: Proceedings of the First IEEE Symposium on Foundations of Computational Intelligence, Honolulu, HI, 2007, pp. 375 -- 382.
[68] D. Wu, J.M. Mendel, Enhanced Karnik -- Mendel algorithms for interval type-2 fuzzy sets and systems, in: Proceedings of the NAFIPS, San Diego, CA, 2007.
[69] D. Wu, J.M. Mendel, A vector similarity measure for linguistic approximation: interval type-2 and type-1 fuzzy sets, Information Sciences, in press, doi:10.1016/j.ins.2007.04.014.
[70] Wu, D.; Tan, W. W.: Genetic learning and performance evaluation of type-2 fuzzy logic controllers. International journal of engineering applications of artificial intelligence 19, No. 8, 829-841 (2006)
[71] Wu, D.; Tan, W. W.: A simplified type-2 fuzzy controller for real-time control. ISA transactions 15, No. 4, 503-516 (2006)
[72] H. Wu, J.M. Mendel, Antecedent connector word models for interval type-2 fuzzy logic systems, in: Proceedings of the FUZZ-IEEE, vol. 2, Budapest, Hungary, 2004, pp. 1099 -- 1104.
[73] Wygralak, M.: A new approach to the fuzzy cardinality of finite fuzzy sets. Busefal 15, 72-75 (1983) · Zbl 0531.04001
[74] Wygralak, M.: Cardinalities of fuzzy sets. (2003) · Zbl 1028.03043
[75] Yager, R. R.: A measurement-informational discussion of fuzzy union and fuzzy intersection. International journal of man -- machine studies 11, 189-200 (1979) · Zbl 0403.03044
[76] Yager, R. R.: Fuzzy subsets of type-2 in decisions. Journal of cybernetics 10, 137-159 (1980)
[77] R.R. Yager, Quantified propositions in a linguistic logic, in: Proceedings of the 2nd International Seminar on Fuzzy Set Theory, Linz, Austria, 1980, pp. 69 -- 124.
[78] Zadeh, L.: Fuzzy sets and information granularity. Advances in fuzzy set theory and applications, 3-18 (1979)
[79] Zadeh, L. A.: The concept of a linguistic variable and its application to approximate reasoning -- 1. Information sciences 8, 199-249 (1975) · Zbl 0397.68071
[80] Zadeh, L. A.: PRUF, a meaning representation language for natural languages. International journal of man -- machine studies 10, 395-460 (1978) · Zbl 0406.68063
[81] Zadeh, L. A.: A theory of approximate reasoning. Machine intelligence, 149-194 (1979)
[82] Zadeh, L. A.: Possibility theory and soft data analysis. Mathematical frontiers of the social and policy sciences, 69-129 (1981)
[83] Zadeh, L. A.: Test-score semantics for natural languages and meaning representation via PRUF. Empirical semantics, 281-349 (1981)
[84] L.A. Zadeh, Fuzzy probabilities and their role in decision analysis, in: Proceedings of the IFAC Symposium on Theory and Application of Digital Control, New Delhi, 1982, pp. 15 -- 23. · Zbl 0532.90003
[85] Zadeh, L. A.: A computational approach to fuzzy quantifiers in natural languages. Computers and mathematics with applications 9, 149-184 (1983) · Zbl 0517.94028
[86] Zadeh, L. A.: Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy sets and systems 19, 111-127 (1997) · Zbl 0988.03040
[87] Zadeh, L. A.: Toward a generalized theory of uncertainty (GTU) -- an outline. Information sciences 172, 1-40 (2005) · Zbl 1074.94021
[88] Zeng, J.; Liu, Z. -Q.: Type-2 fuzzy hidden Markov models and their applications to speech recognition. IEEE transactions on fuzzy systems 14, No. 3, 454-467 (2006)
[89] Zeng, W.; Li, H.: Relationship between similarity measure and entropy of interval valued fuzzy sets. Fuzzy sets and systems 157, 1477-1484 (2006) · Zbl 1093.94038