Abstract
Probabilistic fuzzy systems combine a linguistic description of the system behaviour with statistical properties of data. It was originally derived based on Zadeh’s concept of probability of a fuzzy event. Two possible and equivalent additive reasoning schemes were proposed, that lead to the estimation of the output’s conditional probability density. In this work we take a complementary approach and derive a probabilistic fuzzy system from an additive fuzzy system. We show that some fuzzy systems with universal approximation capabilities can compute the same expected output value as probabilistic fuzzy systems and discuss some similarities and differences between them. A practical relevance of this functional equivalence result is that learning algorithms, optimization techniques and design issues can, under certain circumstances, be transferred across different paradigms.
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References
Meghdadi, A.H., Akbarzadeh-T., M.R.: Probabilistic fuzzy logic and probabilistic fuzzy systems. In: Proceedings of the Tenth IEEE International Conference on Fuzzy Systems, Melbourne, Australia, pp. 1127–1130 (2001)
Liu, Z., Li, H.X.: A probabilistic fuzzy logic system for modeling and control. IEEE Transactions on Fuzzy Systems 13, 848–859 (2005)
Zhang, G., Li, H.X.: A probabilistic fuzzy learning system for pattern classification. In: 2010 IEEE International Conference on Systems Man and Cybernetics (SMC), pp. 2336–2341 (2010)
Zhang, G., Li, H.X.: An efficient configuration for probabilistic fuzzy logic system. IEEE Transactions on Fuzzy Systems 20, 898–909 (2012)
Zadeh, L.A.: Probability measures of fuzzy events. Journal of Mathematical Analysis and Applications 23, 421–427 (1968)
Zadeh, L.A.: Discussion: Probability theory and fuzzy logic are complementary rather than competitive. Technometrics 37, 271–276 (1995)
Dubois, D., Prade, H.: The three semantics of fuzzy sets. Fuzzy Sets and Systems 90, 141–150 (1997)
Bertoluzza, C., Gil, M.A., Ralescu, D.A. (eds.): Statistical Modeling, Analysis and Management of Fuzzy Data. STUDFUZZ. Physica Verlag, Heidelberg (2002)
Mamdani, E.H., Gaines, B.R. (eds.): Fuzzy Reasoning and its Applications. Academic Press, London (1981)
Takagi, T., Sugeno, M.: Fuzzy identification of systems and its applications to modelling and control. IEEE Transactions on Systems, Man and Cybernetics 15, 116–132 (1985)
Pedrycz, W.: Applications of fuzzy relational equations for methods of reasoning in presence of fuzzy data. Fuzzy Sets and Systems 16, 163–175 (1985)
van den Berg, J., Kaymak, U., Almeida, R.J.: Conditional density estimation using probabilistic fuzzy systems. Accepted for Publication in IEEE Transactions on Fuzzy Systems (2013)
Dubois, D., Prade, H.: Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets. Fuzzy Sets and Systems 192, 3–24 (2012)
van den Berg, J., van den Bergh, W.M., Kaymak, U.: Probabilistic and statistical fuzzy set foundations of competitive exception learning. In: Proceedings of the Tenth IEEE International Conference on Fuzzy Systems, Melbourne, Australia, vol. 2, pp. 1035–1038 (2001)
Waltman, L., Kaymak, U., van den Berg, J.: Fuzzy histograms: A statistical analysis. In: EUSFLAT-LFA 2005 Joint 4th EUSFLAT 11th LFA Conference, pp. 605–610 (2005)
Kaymak, U., van den Bergh, W.M., van den Berg, J.: A fuzzy additive reasoning scheme for probabilistic Mamdani fuzzy systems. In: Proceedings of the 2003 IEEE International Conference on Fuzzy Systems, St. Louis, USA, vol. 1, pp. 331–336 (2003)
Xu, D., Kaymak, U.: Value-at-risk estimation by using probabilistic fuzzy systems. In: Proceedings of the 2008 World Congress on Computational Intelligence, Hong-Kong, pp. 2109–2116 (2008)
Almeida, R., Kaymak, U.: Probabilistic fuzzy systems in Value-at-Risk estimation. Intelligent Systems in Accounting, Finance & Management 16, 49–70 (2009)
van den Berg, J., Kaymak, U., van den Bergh, W.M.: Financial markets analysis by using a probabilistic fuzzy modelling approach. International Journal of Approximate Reasoning 35, 291–305 (2004)
Kosko, B.: Fuzzy Engineering. Prentice Hall, Upper Saddle River (1997)
Kosko, B.: Global stability of generalized additive fuzzy systems. IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications and Reviews 28, 441–452 (1998)
de Finetti, B.: Sull’impostazione assiomatica del calcolo delle probabilità. Annali Univ. Trieste 19, 3–55 (1949); (English translation: Probability, Induction, Statistics. Wiley, London, 1972 (Chapter 5)
Popper, K.: The Logic of Scientific Discovery. Routledge, London (1959)
Coletti, G., Scozzafava, R.: Conditional probability and fuzzy information. Computational Statistics & Data Analysis 51, 115–132 (2006)
Baldwin, J., Lawry, J., Martin, T.: A mass assignment theory of the probability of fuzzy events. Fuzzy Sets and Systems 83, 353–367 (1996)
Singpurwalla, N.D., Booker, J.M.: Membership functions and probability measures of fuzzy sets. Journal of the American Statistical Association 99, 867–877 (2004)
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Almeida, R.J., Verbeek, N., Kaymak, U., Da Costa Sousa, J.M. (2014). Probabilistic Fuzzy Systems as Additive Fuzzy Systems. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2014. Communications in Computer and Information Science, vol 442. Springer, Cham. https://doi.org/10.1007/978-3-319-08795-5_58
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DOI: https://doi.org/10.1007/978-3-319-08795-5_58
Publisher Name: Springer, Cham
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