Abstract
Fuzzy Cognitive Networks (FCN) have been introduced by the authors recently as an extension of Fuzzy Cognitive Maps (FCM). One important issue of their operation is the conditions under which they reach a certain equilibrium point after an initial perturbation. This is equivalent to studying the existence and uniqueness of solutions for their concept values. In this paper, we study the existence of solutions of FCNs equipped with continuous differentiable sigmoid functions. This is done by using an appropriately defined contraction mapping theorem. It is proved that when the weight interconnections and the chosen sigmoid function fulfill certain conditions the concept values will converge to a unique solution regardless the exact values of the initial concept values perturbations. Otherwise the existence or the uniqueness of equilibrium can not be assured. Assuming that these conditions are met, an adaptive bilinear weight estimation algorithm is proposed.
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References
Kottas, T.L., Boutalis, Y.S., Christodoulou, M.A.: Fuzzy Cognitive Networks: A General Framework. Inteligent Desicion Technologies 1(4), 183–196 (2007)
Kottas, T.L., Boutalis, Y.S., Karlis, A.D.: A New Maximum Power Point Tracker for PV Arrays Using Fuzzy Controller in Close Cooperation with Fuzzy Cognitive Networks. IEEE Transactions on Energy Conversion 21(3), 793–803 (2006)
Kosko, B.: Fuzzy Cognitive Maps. International Journal of Man-Machine Studies, 65–75 (1986)
Axelrod, R.: Structure of Decision. In: The Cognitive Maps of Political Elites. Princeton University Press, New Jersey (1976)
Kottas, T., Boutalis, Y., Devedzic, G., Mertzios, B.: A new method for reaching equilibrium points in Fuzzy Cognitive Maps. In: Proceedings of 2nd International IEEE Conference of Intelligent Systems, Varna Burgaria, pp. 53–60 (2004)
Zhang, W., Chen, S., Bezdek, J.: Pool2: A Generic System for Cognitive Map Development and Decision Analysis. IEEE Transactions on Systems, Man, and Cybernetics 19(1), 31–39 (1989)
Schneider, M., Shnaider, E., Kandel, A., Chew, G.: Constructing fuzzy cognitive maps. In: International Joint Conference of the Fourth IEEE International Conference on Fuzzy Systems and The Second International Fuzzy Engineering Symposium, vol. 4(1), pp. 2281–2288 (1995)
Boutalis, Y., Kottas, T., Christodoulou, M.: On the Existence and Uniqueness of Solutions for the Concept Values in Fuzzy Cognitive Maps. In: Proceedings of 47th IEEE Conference on Decision and Control - CDC 2008, Cancun, Mexico, December 9-11, pp. 98–104 (2008)
Rudin, W.: Principles of Mathematical Analysis, pp. 220–221. McGraw-Hill Inc., New York (1964)
Ioannou, P., Fidan, B.: Adaptive Control Tutorial. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006)
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Kottas, T., Boutalis, Y., Christodoulou, M. (2009). Bilinear Adaptive Parameter Estimation in Fuzzy Cognitive Networks. In: Alippi, C., Polycarpou, M., Panayiotou, C., Ellinas, G. (eds) Artificial Neural Networks – ICANN 2009. ICANN 2009. Lecture Notes in Computer Science, vol 5769. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04277-5_88
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DOI: https://doi.org/10.1007/978-3-642-04277-5_88
Publisher Name: Springer, Berlin, Heidelberg
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